AD-A271 647

Water Resources Support Center

Institute for Water Resources

COMPENDIUM ON WATERWAY

TRANSPORTATION RELIABILITY:

LOCK CONGESTION AND LOCK QUEUES

DTIC

ELECTF

OCT2 91993

A

This document has been approved for public release and saol; its

distribution is un.ii.fited.

93-26025

APRIL 1993 IWR REPORT 93-R-9

93 10 26 061
REPORT DOCUMENTATION PAGE Ioa f., oV08

IUh(IM .algortonq bardan fot the,s IclonlK• Of fnfonnetiOe, Iit matm4Ed to a'vaelt I hO~Jejr atraipOte~ta including th, tiern fot rcieewlrg instruclc, ,.aichingaerng da~lta ,ovtcOaa.

qath~mcn efld maintalnlrl the data nwadad. and cOrnfai4tlnq and ratv~awlnq tha collectione of ,nfO~rnatetO# Sand cOmmants raearalen thtb• 1 a astkm al t al|l an o t hay aic Of t/ed

(olne(oE Of intoematlon. tudltlg eejo]attons fot ucduetheis bu rden, ftO Wik•'Nto Steat. Diractoraa of o4aalqm.anate tion Operauion• = Nat t 1o is5 J1ffa'io"

oo)viq ti, taP 1204. AthInton, V 2 22202-4302. =nd t the Oicte of Managaeamnt and ludgat. Pope work Reducio Prolat (0704-018S), Waohntton. DC 20503.

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4. TITLE AND SUBTITLE S. FUNDING NUMBERS

COMPENDIUM ON WATERWAY TRANSPORTATION RELIABILITY: LOCK

CONGESTION AND LOCK QUEUES

6. AUTHOR(S)

VARIOUS

7. PERFORMING ORGANIZATION NAME(S) AND ADORESS(ES) 8. PERFORMING ORGANIZATION

U.S. Army Corps of Engineers REPORT NUMBER

Institute for Water Resources

Navigation Division IWR Report 93-R-9

Casey Building

Ft. Belvoir, VA 22060-5586

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Available from Technical Information Service, 5285 Port Royal Road, Springfield,

Virginia 22161 - (703) 487-4650

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Unlimited/Unclassified

13. ABSTRACT (Maximum 200 words)

The papers presented in this Compendium report research results on problems and

proposed solutions on lock congestion and lock queues. The research results present

suggestions or alternative solutions for improving the reliability of the inland

waterways transportation system. We believe the research provides a stimulus for

useful discussion at the Division and District levels. Any comments on information

presented in the articles should be addressed to the Navigation Division of the

Institute for Water Resources.

14. SUBJECT TERMS Simulation of waterway transportatiL., model of IS. NUMBER OF PAGES

lockage stalls, lock interdependence, tow delays, lock congestion 286

approximating delays, prioritizing & scheduling, numerical model, 16. PRICE CODE

investments. metamodels. queues_

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT

OF REPORT OF THIS PAGE OF ABSTRACT

Unclassified Unclassified Unclassified Unlimited

NSN 7540-01-280-5500 Standard Form 298 (Rev 2-89) Precr,wed by ANSI Sid 139-18
COMPENDIUM ON WATERWAY

TRANSPORTATION RELIABILITY:

LOCK CONGESTION AND LOCK QUEUES

Accesion For

NTIS C RA,.I

by .. .. ...

byy by , •~~B.._........ 2: .........

NAVIGATION DIVISION D! .t ib-e.;I

U.S. Army Corps of Engineers

Water Resources Support Center Aaiti ,-,x6C

Institute for Water Resources Avail ±'d °'

Casey Building Dist pica

Fort Belvoir, VA 22060-5586

lDTIC QUALITly .

April 1993 IWR Report 93-R-9
1!
FOREWORD

The papers presented in this Compendium report research results on problems and

proposed solutions on lock congestion and lock queues. This research was done under separate

Corps of Engineers contracts with Dr. Paul Schonfeld, Department of Civil Engineering, and Dr.

Harry Kelejian, Department of Economics, University of Maryland, College Park, MD.

The Editorial Board of the Transportation Research Record gave special permission for

several of the articles to be reproduced in this Compendium. We are appreciative of Ms. Nancy

Ackerman, Director of Reports and Editorial Services for the National Research Council,

Transportation Research Board (TRB), Washington, D.C. for her support.

The research results present suggestions or alternative solutions for improving the

reliability of the inland waterways transportation system. We believe the research provides a

stimulus for useful discussion at the Division and District levels. Any comments on information

presented in the articles should be addressed to the Navigation Division of the Institute for Water

Resources.

QYD G. ANTLE

Chief, Navigation Division

Institute for Water Resources

ii1
iv
TABLE OF CONTENTS

PAGE

FO REW O RD ........................................................ iii

SIMULATION OF WATERWAY TRANSPORTATION RELIABILITY

Melody D. M. Dai and Paul Schonfeld ..................................... I

PROBABILITY MODEL OF LOCKAGE STALLS AND INTERFERENCES

Harry H. Kelejian . ................................................... 11

EFFECTS OF LOCK INTERDEPENDENCE ON TOW DELAYS

Steven I. J. Chien and Paul M. Schonfeld .................................. 21

EFFECTS OF LOCK CONGESTION AND RELIABILITY ON OPTIMAL

WATERWAY TRAVEL TIMES

Melody D. M. Dai, Paul Schonfeld

and George Antle . ................................................. 51

PRIORITIZING AND SCHEDULING INTERDEPENDENT LOCK

IMPROVEMENT PROJECTS

David Martinelli and Paul Schonfeld . ..................................... 79

APPROXIMATING DELAYS AT INTERDEPENDENT LOCKS

David Martinelli and Paul Schonfeld ...................................... 111

APPROXIMATE DELAYS DUE TO LOCK SERVICE INTERRUPTIONS

Venkatesh Ramanathan and Paul Schonfeld ................................ 155

COMPUTATIONAL CHARACTERISTICS OF A NUMERICAL MODEL

FOR SERIES OF WATERWAY QUEUES

Chung-Hung Wei, Melody D. M. Dai, and

Paul Schonfeld . ................................................... 173

A METHODOLOGY FOR PLANNING EFFICIENT INVESTMENTS ON

INLAND WATERWAYS

David Martinelli, Melody D. M. Dai, Paul Schonfeld

and George Antle . ................................................. 211

METAMODELS FOR ESTIMATING WATERWAY DELAYS THROUGH A

SERIES OF QUEUES

Melody D. M. Dai and Paul Schonfeld ................................... 249

v
vi
SIMULATION OF WATERWAY TRANSPORTATION RELIABILITY

by

Melody D. M. Dai and Paul Schonfeld

Published in Transportation Research Record

Volume No. 1313

S . i.... " . i . i " - : I - " i l • i ii l i ili1
2
rRANSPOR TA TION RESEARCH RECORD 1313

Simulation of Waterway Transportation

Reliability

MELODY D. M. DAI AND PAUL SCHONFELD

A microscopic model for simulating barge traffic through a series Baumol's model. Nason and Kullman (8) developed a total

of locks has been developed and tested with data for a section logistics cost model to predict inland diversions from waterof the Ohio River. The model was designed primarily to analyze ways.

the economic effects of waterway congestion and service relia- Two models based on queueing theory have oeen found for

bilitv. The results indicate that the model is capable of simulating

the system performance sufficiently well for analytic purposes. estimating lock delays. DeSalvo (9) models lock operation as

The results also indicate to what extent coaL stockouts would a simple singLe-server queueing process with Poisson distnbincrease at a power plant, or alternatively, how safety stocks uted arrivals and exponentially distributed service times (i.e..

would have to be increased, as traffic volumes approach capacity. MIM/I queues). Wilson's model (10) extends DeSalvo's by

treating Lhe service processes as general distributions (M/G/I

The reliability of service times on inland waterways signifi- queues). Both models are designed for analyzing single lock

delays. However. the assumption of exponentially distributed

cantlv influences barge fleet requirements. operating costs. service times is not consistent with empirical data (11) and

inventory costs. and stock out costs for customers. Therefore. the Poisson arrivals assumption is also unreliable. Carroll and

the service reliability influences the competitive position and Desai (12.13) studied the arrival processes at 40 locks on the

market share of inland waterway transportation. Illinois. Mississippi. and Ohio river systems, and found that

To analyze the effects of congestion and service time vari- 13 of the 40 locks had non-Poisson arrivals at the five percent

ability, a simulation model has been developed. In its earliest

applications for which results are presented. the model is used significance level. toestimate the relations among capacity and service time ThreutfoM/1qeesiDeavsmdl()ae to ederived on the basis of first-in-first-out (FIFO) service disvariance at successive locks, stock-out probabilities and du- cipline although the actual discipline is primarily one-up-onerations, and inventory safety stocks for an electric power plant down. This assumption can still generate reasonable results

supplied with coal through the Ohio S~since River. This model will delays mainly depend on volume to capacity ratios. Wilsoon be usable for estimating the benefits and costs of alter- son (10) modeled the service processes more realistically with

native plans for maintaining and improving the waterway sys-

"tem. a general rather than an exponential distribution. However.

arrivals are still assumed to be Poisson distributed at all locks

and no exact queueing results are available for locks with two

LITERATURE REVIEW chambers in parallel. Since analytic queueing models must be

kept simple to be suivable. the above two models also neglect

The research most relevant here regards the economic costs the interdependence among serial locks and the stalls (i.e..

service interruptions at locks). Both of these factors signifi- of lock delays, lock delay models, and waterway simuationreliabilit.

models. The estimation of economic benefits is essential for c The system simulation models developed to anilyze lock

selecting and scheduling lock improvement projects. The U.S. delays ard two travel times originated mainly fromzHowe's

Army Corps of Engineers. which is the agency responsible for U.S. waterways, usually estimates the economic benefits microscopic model (14). In that model service times are derived from empirically determined frequency distributions. To

ofavoid some troublesome problems and errors associated with

between barges and the next cheapest mode (1-3). Such eval- the rume toublane len rfows d issowe smdl

uatinmortnt oitssom ogisicscoss (~g. fo lager the requirement to balance iong-run flows in Howe's model.

uation omits some important logistics costs (e.g.. for larger Carroll and Bronzini developed another waterway system siminventories and barge fleet sizes) used to hedge against un- ulation model (15). It provides detailed outputs on such vardeliveries. nreliable iables as two traffic volumes. delays, processing times, transit

In systems with unreliable deliveries stockouts may occur.and transit

There are situations in which the on-site stocks are not sufficient to satisfy the demand (4). Stock-out costs include du- times. queue lengths, and lock utili ation ratios. Both of these

plicate ordering costs from another source or mode and fore- models simulate waterway operations in detail but require

gone profits (5.6). Baumol and Vinod (7) indicate that delays considerable amounts of data and computer time. which limit

can increase the shippers' inventorv costs which include on their applicability for problems with large networks with nu- n- merous combinations of improvement alternatives. Both models

site carrying costs and • stock-out penalties. On the basis of assume mer o c a Poisson ina distribution tion of imprveietgal for two trip generations. erative B s. hmoe which

Department of Civil Engineenng. University of Maryland. College is not always realistic. More important for reliability analyses.

Park. Md. 20742. neither of these models explicitly accounts for stalls, which

3
)aw and )chon,'eld 99

re different in frequency and duration from other events and for all coal d stinations every unit time. This provides detailed

tave significant effects on overall transit time reliabilttv, information on inventory levels for i..1 coal destinations.

Hence a waterway simulatinro model that explicitly accounts A fifth type of event is a lock stall. Whenever a stall occurs.

or stalls and estimates the ettects of service unreliability of the affected chamber becomes unavailable until the end of

riventorv costs is desirable for evaluating and scheduling lock the stall.

mprovemenl projects. The size of problem that the model can handle is limited

by the computer capacity and the storage capacity of the

Fortran compiler or linker. There are no restrictions on the

IMULATION MODEL number of locks, chambers. cuts. waterway links. tows. utility

plantc. origin-destination (O-D) pairs. and simulation time

"purpose priods. This model can simulate two way operation on a

mainline waterway.

\ waterway simulation model was developed to analyze the This model is programmed in Fortran-77. which allows the

ions between tow trips, travel times, delays, lock oper- simulation of relatively complex operations. The following is

itions. coal consumption. and coal inventoris while taking a more detailed description of how tow trip generation. tow

Lccount of stochastic effects and seasonal variations. This sim- -ravel times, and coal inventory levels ire computed. The

ilation mndel enables the estimation of inventory levels and overall structure of the simulation model is displayed in

-xpected stock-out amounts for coal. tow travel times along Figure 1.

he waterway, and tow delays under a variety of assumptions

ibout tow trip generation, tow motion, lock service, lock

)peration discipline, coal inventory level, and coal consum- Tow Frip Generation

ion. These estimates are useful for estimating economic ben4fits of lock improvements. Tow trips are generated randomly. but the mean of their

generating distribution is constant for each O-D pair over

each simulation time period. The distribution for tow trip

Features generation is represented by a table. It is a.sumed that the

distribution of trip generation times is similar to the distriburhis simulation model is focused on how variations in lock tion of trip arrival times to locks. (for which data are avail-

;ervice times affect tow delays and how variations in tow able).

ielays affect coal inventories. The output of this mode! can This model assumes that each tow will maintain its size

:rovide the necessary information to estimate inventory costs, through its trip. As in trip generation. tow siz.es (numbers of

;tock-out costs, and expected benefits resulting frot lock barges per tow) are also generated randomly. The distribution

-ehabilitation or lock construction. of tow sizes is represented by a table and is assumed to be

This simulation model is microscopic. It traces the motion the same for each O-D pair. The tow size table is determined

md records the characteristics of each tow. The characteristics from input data and can represent tow size distribution.

)f tows include their number of barges, commodity types. Tow traffic is divided into coal and non-coal traffic. There-

,peed. crigin and detination. direction of motion, and arrival "nre. for the same O-D pairs. there may be different trip rates

ime at various points. In addition. the model determines

:umulative deliveries, cumulative consumption, and actual INPUT:

Link A Lock Characteristics nventories at various plants. Traffic Demand

Probability Distributions This is an event-scanning simulation model-the status of inventoriet & Consumption

vhich is updated by events. There are five types of events.

)ne is the generation of tow trips, which are generated sto-

:hasticallv on the basis of actually observed traffic distribu- PROCESS:

ions. The model uses a table to represent the trip generation Origin Nodes: generating tow trips

,attern and is. therefore. not limited to standard mathemat- Oestinatlon Nodes: updating: cumulative deliveries

cal probability distributions. Locks: assigning chamber inventory levels

A second type of event is the tow entrance in a lock. which determining number of cuts "determining lock service times

s determined bv tow arrival time at that lock. the times when calculating queueing times Sdetermining Links: detrmining travealing arrival times tinesto vnet locks or hainbers become available, and the chamber assignment dis- deot iningatriv iononx lcso

ipline. If a tow arrives before the lock is available, it needs destinations

o wait in the queue storage area. Otherwise, it is served

,ccording to the chamber assignment discipline, discussed later.

n general. tb: lock service is presently -first come first serve." OUTPUT:

Cumulative deliveries, cumulative consumption. inventory levels

ubject to the chamber assignment procedure. Queueing time per tow at each lock fcr different 0-0 pairs "tv iAverage speed A third type of event is a coal tow's arrival at its destination, Total numer of tow trips for different 0-0 pairs Total gueueing times for different locks vhich increases the cumulative deliveries by the amount of Total lock service tines for different ;ocks and chambers

oal that tow is carrying. The cumulative consumption and Total tow travel times I distances

Total dclei times

iventory at the destination are also updated then.

A fourth type of event is the update in the status of cu- FIGURE I Structure and elements of the

iulative consumptions. inventories, and consumption rates simulation model.

4
100 TRANSPORTATION RESEARCH RECORD 1313

for coal and non-coal traffic. When coal tows arrive at their and are defined in terms of durations and frequencies. which

destinations, the model updates inventory levels. It is assumed depend on weather conditions and lock conditions at each

that only a specified fraction of the barges on a coal tow are chamber. The model assumes that stalls occur stochasticallv

carrying coal. with an exponential distribution.

Tow Travel Times Inventory Levels

Tow travel times are estimated separately for each waterway Inventory levels are represented by the difference between

section, queue storage area. and lock. Section travel times cumulative deliveries and cumulative consumption. Whenbetween locks and/or piers are determined by speeds and ever inventory levels drop to negative values, this model corndistances to be covered. Tow speeds are specified as an input putes stock-out amounts and durations for the analysis of total

to the model in the form of a probability distribution. The costs. This modc. updates cumulative deliveries and cumudistribution of speeds is assumed to be normal. The model lative consumption whenever coal tows arrive at destinations.

assumes that tows maintain constant speeds between origins Cumulative deliveries are determined from initial inventory

and destinations and that backhaul speeds are a constant ratio levels, inter-delivery times, and delivery amounts. The initial

of linehaul speeds. inventory level is exogenously specified for each destination

To avoid generating extreme speed values, a speed range (utility plant). The interdelivery time is generated by the simis specified. If speeds are lower than the 2.5 percentile speed ulation model. The delivered amount is determined from the

or zero, or higher than the 97.5 percentile speed. the speeds barge payload and the number of arriving coal barges. The

are regenerated. barge payload is currently assumed to be constant for each

Queueing times at locks are a major focus of this simulation tow. The number of coal barges is currently assumed to be a

model. Such queueing delays may occur well before traffic constant fraction of tow size. The coal barge fractions vary

levels approach lock capacity since tow arrivals and lock ser- for different O-D pairs. Although coal barge fractions are

vice times are not uniform. These delay times are computed constant throughout the simulated period. the amount delivfrom the difference between the tow arrival times at the queue ered by each tow is not constant since tow sizes are randomly

storage area and their departure from the queue to enter the generated.

lock. The storage area has unlimited capacity and is adjacent Cumulative consumption is a function of consumption rate

to the lock. and time. The mean consumption rate is constant for each

Lock service times are generated from a specified distribu- utility plant during each simulation period. although it fluction table. The distribution table can directly reflect actually tuates randomly around its mean. However. a constant rate

observed service times. Therefore, the model can be applied is assumed within each period. The consumption rate is upto any type of locks. Lock service times will be affected by dated every time unit and is. therefore, a step-wise linear

lock improvements, which are represented by smaller average distribution over time. whose slopes are consumption rates.

lock service times or reduced service time distributions. The

average lock service times vary for differentl locks, chambers.

and numbers of cuts. Input Requirements

The number of cuts is determined by chamber and tow sizes.

The maximum cut size (barges handled simultaneously) is Generally, the model requires four types of inputs related to

exogenously specified for each chamber. A tow may be di- (a) link and lock characteristics. (b) traffic demand between

vided into different numbers of cuts at different lock chain- origins and destinations. (c) probability distributions. and (d)

bers. inventories and consumption.

If a lock has more than one chamber in parallel. (main and

auxiliary chambers are usually provided), it is currently assumed that the main chamber will be preferred. unless the Link and Lock Characteristics

additional wait time it requires (compared to the auxiliary

chamber) exceeds a specified level. This lock selection bias The following kinds of information are needed for each link:

factor reflects the additional work and delays required to (a) end nodes. (b) link length. (c) distances between the end

break tows into more (and smaller) cuts. move them sepa- nodes and the lock. (d) number of chambers. (e) average

rately through the auxiliary chamber and then reassemble frequencies and durations of stalls. (f) maximum cut sizes of

them. This bias factor has been estimated separately for var- chambers. (g) average service times of chambers for cuts of

ious locks from empirical data. various sizes. (h) maximum number of barges for each cut

The lock service discipline is currently -'first come first serve.'" size at each chamber. (i) bias time for each auxiliary chamber.

It is expected that the -N up-N down" service discipline will and (j) random number seeds.

be simulated later.

Traffic Demand

Stalls

Traffic demand in tows per day is expressed in the form of

Stalls are failure conditions in which chambers are not avail- O-D matrices by time periods. The lengths of time periods

able to serve tows. Stall characteristics ditfer among chambers may be different and need to be specified. Additional infor5
Dai ana Wchonteld 101

mation needed includes Ia) dwell time at origins and desti- chambers are the smallest. A new Gallipolis lock chamber is

nations (both average and standard deviationM: (b) average under construction. The physical characteristics of these six

number of barges per tow for each O-D pair: (c) fractions of locks are given in Table 1.

coal barges in a tow for each O-D pair: (d) payload in short- In general. a new lock will provide better service quality

tons: tel speed (both average and standard deviation): and by reducing service time and improving reliability. The prior

Ij) ratio ot backhaul speed to linehaul speed (empty'full or expectation is that electric utility plants served by a waterway

upstreamdownstream). may be able to reduce the required inventory levels and the

expected stock-out costs if the service reliability on the waterway is improved.

Probabilirv Distributions The objective of this case study is to compare the inventory

levels and expecied stock-out amounts of a utility plant downProbability distributions are specified in this model for (a) stream of Gallipolis for cases with and without a new Gallilock service timers. (b) trip generation. (c0 tow composition polis lock.

(barges per tow). and (d) coal consumption at power plants. The Stuart utility plant. which belongs to Dayton Power

The probability distribution tables represent cumulative and Light Co.. was chosen for this case study. It is located

distribution curves, wherein the abscissas are cumulative fre- between the Greenup and Meldahl locks. It is 63.5 mi downquency. and the ordinates represent the ratio of the tabulated stream from Greenup and 31.7 mi upstream from Meldahl.

variable to its mean. To reduce the input complexity and

specify only ordinates, a specified number of equal intervals

is currently used for any cumulative frequency distribution. Model Application

This case study focuses on the Ohio river between the BelleInventories and Consumption ville and Greenup locks. Although the model can simulate

multiple plants. only one utility plant was analyzed. It inInitial inventorv levels in short-tons for different nodes (utility cluded O-D pairs. The simulation period is 1 year.

plants) must be specified. In addition, consumption rates in

short-tons per day are expressed in the form of node matrices

by time period. The information on cumulative deliveries. Link and Lock Characteristics

cumulative consumption. and inventory levels, is provided for

intervals whose duration in days must be specified. To simulate the operation between Belleville and Greenup.

five nodes and four links are used. The link characteristics

are shown in Table 2. The lock characteristics are shown in

Model Output Table 3.

It is noted that except for Node 5. which represents the

This model prints out the following results: (a) total tow travel Stuart utility plant. all nodes are null nodes that are used as

time (not including the queueing time. lockage time. and dwell the origins and destinations of non-coal traffic to generate

time) in days: (b) total tow travel distances in 1.000 mi: (c) equivalent volumes and congestion levels.

total dwell times at origins and destinations in days: (d) total For existing locks, the average lock service times are dequeueing times in days for different locks and chambers: (e) termined according to the 1984 lock data. Because the new

total lock service times in days for different locks, chambers Gallipolis Lock is still under construction, its service times

and cuts: !f) total number of tow trips for different O-D pairs: were not available and had to be estimated. The estimated

(g) average speed in mi per day: (h) queueing time (both values are slightly smaller than those of the four older locks.

average and standard deviation) in days per tow at each lock which have similar chamber sizes. because the newer lock is

for different O-D pairs: (i) monthly cumulative deliveries, assumed to improve service.

cumulative consumption. and inventory levels tables in 1.000

short-tons for different utility plants: (j) cumulative deliveries.

cumulative consumption. and inventory levels tables for spec- TABLE 1 PHYSICAL CHARACTERISTICS OF LOCKS

itied intervals in 1.00M short-tons for different utilitv companies: (k) graphs of cumulative deliveries and cumulative Chambers

consumption by specified time intervals for different utility Year Width Length Lift

plants: and (1) graphs of inventory level by specified time Lock Name Opened (ft) (ft) (ft)

interval for different utility plants. Belleville 1968 110 1200 22

1968 110 600 22

Racine 1971 110 1200 22

CASESTUDY 1971 110 600 22

Gallipolis 1937 110 600 23

1937 110 360 23

A five-lock section of the Ohio River. centered on the Gal- Gallipolis 1991 110 1200 23

lipolis Lock was selected for a case study because that lock (new) 1991 110 600 23

Greenup 1959 110 1200 30

constitutes a relative bottleneck in the water capacity. Coin- 1959 110 600 30

pared with the four locks nearest to it. (Belleville. Racine. Meldahi 1962 110 1200 30

Greenup. and Meldahl). Gallipolis is the oldest and its two 1962 110 600 30

6
102 TRANSPORTATION RESEARCH RECORD 1313

TABLE 2 LINK CHARACTERISTICS

Distance

Between In

Lock N Length Node & lock

Link Name In Out (mi) (e1)

I Belleville 1 2 37.9 21.1

2 Racine 2 3 37.6 16.8

3 Gallipolis 3 4 51.8 20.9

4 Greenup 4 5 94.4 30.9

TABLE 3 LOCK CHARACTERISTICS power plant were determined from 1984 coal consumption

Average Upper Limit data and are shown in Table 6.

Service time of

(in days per cut) Cut size

Lock Name 1 cut 2 cuts (in barges per cut) Other Parameters

Belleville .03512 .09823 18"

.02389 .07682 8"

Racine .03425 .09579 18" The mean and standard deviation of downstream tow speeds

.02427 .07805 8" are 9.02 and 2.82 mph (216.48 and 67.68 mi/day). respectively.

Gall ipol is .03563 .07840 6'

.02088 .06173 3" The ratio of upstream speeds to downstream speeds is 0.83.

Gallipolis .03000 .09000 is'" These values were developed on the basis of 1983 statistical

(new) .01600 .07000 9dn

Greenup .03267 .09213 18" data of vessel performance on inland waterways. The barge

.02027 .08108 8' payload was assumed to be 1.400 long tons or 1568 short tons.

based on P1S data (One long ton= 2.240 lbs whereas a short ton= 2.000 lbs.)

**: based on chamber dimensions

Model Validation

Traffic Demand and Consumption The ability of the model to realistically simulate actual opThere were five O-D pairs in this case study. O-D Pair I erating conditions may be assessed by comparing predictions

represents coal traffic for the Stuart plant. The other five with actual data. Tables 7 through 9 show such comparisons

O-D pairs are non-coal traffic or coal traffic for other utility between results of simulation runs of one year and actual data

plants. from 1984. Table 7 shows that traffic volumes are predicted

The baseline values for average trip rates and tow sizes are quite accurately by the model, with an average deviation of

determined from 1984 data, and are shown in Tables 4 and 1.53 percent. Table 8 shows that the waiting time in queues

5. The average consumption rates over 12 mo for the Stuart

TABLE 6 AVERAGE

TABLE ,1 AVERAGE TRIP RATES CONSUMPTION RATES AT THE

STUART POWER PLANT

Trip Rate (tows/day) Consumption Rate

0-0 pair Month (1000 short-tons/day)

Month 1-5 1-2 2-3 3-4 4-5

Jan. 17.23

Jan. 1.98 3.06 3.23 3.36 4.88 Feb. 18.03

Feb. 1.97 3.48 3.46 3.76 5.65 Mar. 15.26

Mar. 1.36 4.43 4.77 4.43 6.06 Apr. 14.90

Apr. 1.57 4.68 4.63 3.74 6.31 May 18.35

May 2.27 4.31 4.39 4.33 5.52 June 16.70

June 1.91 6.31 5.76 5.91 10.44 July 17.32

July 2.20 5.55 5.20 5.15 9.10 Aug. 18.52

Aug. 1.98 5.63 5.84 5.54 8.36 Sep. 18.80

Sep. 2.08 5.07 5.61 5.42 8.71 Oct. 16.29

Oct. 2.40 2.44 3.24 3.37 6.55 Nov. 15.33

Nov. 2.13 2.43 2.92 2.95 5.82 Dec. 16.48

Dec. 1.22 2.77 3.14 3.96 6.30

TABLE 7 TRAFFIC VOLUME COMPARISON

TABLE 5 AVERAGE TOW SIZES

Volume (tows/year)

0-0 Pair Tow Size (barges/tow) Lock Deviation

Data Model (%)

1-5 6.8

1-2 9.1 Belleville 4466 4292 3.90

2-3 9.4 Racine 4591 4580 0.24

3-4 8.4 GallIpolis 4575 4622 1.03

4-5 6.7 Greenup 6511 6450 0.94

7
Da, anda Schonteld 103

TABLE 8 WAITING TIME COMPARISON in 1991. which will match the capacity of Gallipolis to that of

Wait Time(min/tow) the other locks in the senes. we present simulation results for

Lock Deviation both the old and new locks.

Data Model (%) Table 10 shows the effects of traffic volumes and safety

Belleville 21.45 21.81 1.68 stocks on expected stock-out amounts. It is evident that as

Racine 17.26 15.22 11.82 volumes (both coal traffic and non-coal traffic) increase from

Gallipolis 200.53 137.33 31.52 baseline levels (1.0) to levels 50 and 100 percent higher (i.e.. Greenup 14.46 13.31 7.95 volume ratios of 1.5 and 2.0. respectively), the stock-out

amounts increase more than proportionately. As safety stock

levels are increased from 0 to 150.000 and 300.000 tons, the

TABLE 9 RELATIVE UTILIZATION OF LOCK stock-out amounts consistently decrease. The rate of decrease

CHAMBERS (VOLUMES ARE GIVEN IN TOWS/YEAR) tapers off (to zero, eventually) as safety stocks are increased.

Data Model The effect on stock outs of the new higher capacity GalliLock Main Total Main Total Deviation polis Lock is nearly negligible at current volumes (volume

Chamber Lock %Nain Chamber Lock %Main i%) ratio = 1.0). However as volumes double, its effect becomes

Belleville 3332 4466 74.61 3134 4292 73.02 2.13 quite significant. since the old lock would reach a utilization

Racine 3848 4591 83.82 3851 4580 84.08 0.31 rate of 82.85 percent (i.e.. 83 percent of capacity). In this

3al I ipol is 3656 4575 79.91 3488 4622 75.47 5.56 case, the decrease in stock-outs ranges from 60,850 tons/day

Greenup 4500 6511 69.11 4891 6450 75.83 9.72( = 363, 010 - 302.160) or 16.76 percent at zero safety stock

to 54.790 tons/day or 40.73 percent at a safety stock of 300.000

tons.

is predicted reasonably well by the model. although the model Table 11 shows the effects on stock outs of stalls (failures)

significantly underestimates the delays at the Gallipolis Lock. at locks. The stalls column indicates stall frequency. Thus 1

That lock has unusual operating characteristics because it re- indicates baseline conditions (i.e.. frequency based on 1980-

quires disassembly of tows into exceptionally small and oddly 1987 data), whereas 2 and 3 indicate that frequency is doubled

composed cuts. A more detailed analysis of operations at and tripled. respectively. The predicted stock-out amounts

Gallipolis may be required to more accurately model its pecu- are & -- for both the old and new Gallipolis Locks in the

liarities. Table 9 shows that the model can satisfactorily es- format in,-. -w. The results show that stall duration and fretimate the relative utilization of the two chambers at each quencies have relatively slight effects on stock outs when vollock. with an average deviation of 4.43 percent. It should be umes are low. that is. when comparing Case 2 or Case 4 with

noted that the model predictions are not only close to actual the baseline Case 1. However at high volumes (Cases 9-12),

observation, but are also not systematically biased in any when the system operates closer to its capacity, the effects of

particular direction. stalls become significant and the advantage of the higher capacity of the new Gallipolis Lock is quite substantial.

System Congestion and Reliability

In waterways, as in other transportation systems, delays in- Total System Costs

crease much faster than volumes as the capacity is approached

and tend toward infinite values. Moreover. the relative var- The results of this work show how expected stock-out levels

iance of service times (e.g.. the coefficient of variation = increase disproportionately with congestion levels (i.e., volstandard deviation divided by the mean) is expected to in- ume to capacity ratios) and decrease (with diminishing recrease faster than the average service times. with unfavorable turns) as safety stocks are increased. Figure 2 shows how the

effects on system reliability. In a linear network such as that total system costs depend on holding costs and stock-out costs.

in our case study, the capacity of the entire system is limited Holding costs, which include storage costs and interest charges

by the capacity of the most constrictive element in the series, on the safety stock are indicated by the linear function H in

namely the Gallipolis Lock. Because a new lock will be opened the Figure 2. The holding cost is assumed to be $0.10/tonTABLE 10 EXPECTED STOCK-OUT AMOUNTS FOR VARIOUS

SAFETY STOCK LEVELS AND VOLUMES

Utilization Expected Stock-Out Amount

Gallipolis Volume of (1000 short-tons/daY)

Lock Ratio Gallipolis

lock Safety Stock(1000 short-tons)

% 0 150 300

Old 1.0 38.19 220.88 91.41 7.28

1.5 59.19 258.58 125.33 29.23

2.0 82.85 363.01 236.17 134.52

New 1.0 18.73 219.74 90.50 6.97

1.5 27.42 254.04 121.26 26.73

2.0 35.92 302.16 176.38 79.73

8
104 IrRANSPORTATIO%" RESEARCH RECORD 1313

TABLE I1 EXPECTED STOCK-OUT AMOUNTS iN I.tA) TONS,

DAY)

Multiplier Starting Inventory (1000 tons) Case

Stall

Volume Stalls Duration 0 150 300

1 1 1 1 220.1/219.7 Q1.41/90.50 7.28/6.97

2 1 1 2 221.1/220.0 91.56/90.71 7.29/7.01

3 1 1 3 222.4/220.5 97.69/91.29 7.54/7.10 4 1 2 1 221.2/220.0 91.66/90.73 7.37/7.00

5 1 3 1 221.5/220.2 91.86/90.89 7.49/7.13

6 1.5 1 1 257.2/254.0 124.1/121.3 28.30/26.73 7 1.5 2 1 259.1/254.6 125.9/121.8 29.14/26.89

8 1.5 3 1 261.6/255.8 128.3/122.9 30.89/27.64

9 2 1 1 363.0/302.2 236.2/176.4 134.5/79.73

10 2 2 1 416.4/302.8 282.0/177.0 174.9/83.10

11 2 3 1 579.6/306:2 435.8/180.0 311.7/82.54

12 2 3 2 823.1/320.8 678.5/193.7 553.0/92.80

Key: Expected stock-out amounts given with OLD/NEW Gallipolis Lock. Multipliers are ratios of ASSUMED/BASELINE values.

Case I represents baseline values.

200- 600.000 tons and that total system costs would more than

double from approximately $33.000/dav to $69.000/day. If.

H: Holdng Cost however, the new Gallipolis Lock was operational. the opS2: Stock Out Cost (l1/11/1LD) •S2: Stock Out Cost (2/211/01.13) timal safety stock level would only be approximately 450.000

150s3: Stock Out Cost (2/2i1/NEW) tons and the system cost would be approximately $51.000/

TI: Total Cost 11/1/l/0D0) day. despite doubled volumes and stall frequencies. The curves S T2: Totat Cost (212/1/OLD) in Figure 2 show quite clearly the tradeoffs between increased

T3: Total Cost (2/2/l/NEW) safety locks and increased stock out costs.

0too Figures 3 through 5 repeat the analysis of Figure 2 with "%3 various assumptions about the cost of holding safety stock

; T and the cost of stocking out. They show that as stock out

o costs increase relative to holding costs. the optimal amounts

a so of safety stocks should increase.

It should be noted that the only sources of delivery unreliability modeled so far are lock operations and lock failures.

Safety stock policies of utilities might also be affected by other

factors such as probabilistic expectations of coal mine strikes.

H 200 400 600 8oo 1000 frozen waterways and coal price changes. It is possible that

such factors may dominate the effects of lock performance

Safety Stock (1000 tons) analyzed to date.

FIGURE 2 Effect of holding costs and stock-out costs on total system costs. 300 o

Hokling Cost /Stock Out Cost

S/ton-day / S/ton

day. If that holding cost were doubled, the slope of the func- -5.--- 0.1 0.4

tion H would double. 0.1/0.e

Figure 2 shows the stock out costs for three combinations --- o 0.11/12 ,a200 - of parameters, using the key VOLUME/STALL FRE- a 0.2/0.4

QUENCY/STALL DURATION/GALLIPOLIS LOCK. Thus. 0.2/0.8

according to this key, 2/2/1/OLD means that volumes and stall g= 0.2/1.2

frequencies are twice the baseline values, stall durations are

equal to baseline values, and the old Gallipolis lock is being

simulated. It should be remembered that our baseline volumes 100

represent 1984 data. A cost of $0.40/ton is assumed in computing the stock out cost curves of Figure 2. 1.-

The total system cost is obtained by adding the holding cost So

to the stock out costs. Because the holding cost is the same

for all cases in Figure 2. we obtain, one total system cost

function for each of the three stock out cost functions. The 0 total cost curves show that as volumes and stall frequencies 0 200 40 600 60'0 4100

double (from I/I/1/OLD to 2/2/1/OLD) the optimal safety Safety Stoek (1000 tons)

stock levels should approximately double from 300.000 to FIGURE 3 Total system costs (I/i/i/OLD).

9
0)ai and .Shijnreld 105

bkX Hoodng costi, Slak out cost purposes. alrhough its computer requirements are quite mod- o c s o cest for a microscopic simulation model. The model's accuracy

S/ton-day / S/ton might be improved by improvements in traffic generation.

400-- 01/04 tow composition. lock selection, and failure generation func-

-"-- 1/08 

tions. These improvements might be developed on the basis

350 - 0 11 2 of a more extensive analysis of empirical data and, possibly.

- --- 02/0.4 on lock maintenance and failure research. The model may 300 02/0's also be extended to translate physical performance measures

250, - - 02 1.2 such as fleet requirements. delays, safety stocks, and stock

outs into monetary costs And benefits. Finally. more macro200 scopic versions of the model are being developed to efficiently

+5C !analyze alte-native investment and maintenance strategies for

the national waterway system.

00

50

ACKNOWLEDGMENTS

0 200 400 600 8oo 1000 The authors wish to thank L. George Antle and Charles Yoe

of the Institute of Water Resources. U.S. Army Corps of

Safety Stock (1000 tone) Engineers. for their very helpful advice and support in this

FIGURE 4 Total system costs (2/2/1/OLD). work.

400

Holding Cost / Stock Out Cost

$/ton-day / S/ton REFERENCES

0 0.1/0.4

3 0- 0.1/ 0.8 1. Gallipolis Lock and Dam Replacement. Ohio River. U.S. Army -- --- 01 / 1.2 Corps of Engineers. Nov. 1980.

250 -4-- 0.210.4 2. C. Yoe. Lock Rehabilitation A Public Infrastructure Problem: -6-- 02/0.8 The Value of Increased Productivity in Mean Lockage Perfor200 - 02/1.2 mance. U.S. Army Corps of Engineers. Jan. 1987.

3. J. G. Crew and K. H. Horn. Logistics Total Cost Prescriptions: S 150 Do the Theoretical Results Necessarily Conform with Reality. 0 Transportation Research Forum. 1980. pp. 319-327.

S too. 4. D. J. Bowersox. Logistical Management. 2nd Edition. Macmillan o Publishing Co.. Inc.. 1978.

50' 5. M. L. Fair and E. W. Williams. Economics of Transportation and Logistics. Business Publications. Inc.. 1975.

0 26. 1. L. Heskett. N. A. Glaskowskv and R. M. Ivie. Business Lo. St Sto0k (00 600 tgistics. 8001000 2nd Edition. The Ronald Press Company. 1973. Safety Stock (1000 tons) 7. W. J. Baumol and H. D. Vinod. An Inventory Theoretic Model

FIGURE 5 Total system costs (2/2/I/NEW). of Freight Transportation Demand. Management Science. 1970.

pp. 413-421.

8. S. D. Nason and B. C. Kullman. An Analysis of Inland Waterway

CONCLUSIONS Diversions Using a Total Logistics Cost Model. Transportation Research Forum. 1979. pp. 138-144.

Waterway Congestion and Reliability 9. J. S. DeSalvo and L. B. Lave. An Analysis of Towboat Delays. W. Transpn Economic Policy 2. 1968. pp. 232-241,

10. H. G. Wilson. On the Applicability of Queueing Theory to Lock

The results of this work show how expected stock-out levels Capacity Analysis. Transportation Research. Vol. 12. 1978. pp.

increase disproportionately with congestion levels (i.e.. vol- 175-180.

ume to capacity ratios) and decrese (with diminishing returns) It. J. L. Carroll and H. G. Wilson. A Zero-sum User Charge System

as safety stocks are increased. Such results provide the basis for Rationing the Use of Inland Waterway Locks. Proc.. Interfor tradeoffs between inventory holding costs and stock-out .national Conference on Transportation Research. 1973.

12. J. L. Carroll. M. S. Bronzini and J. C. Hayward. Analysis of costs. The optimized safety stocks resulting from such trade- Waterway Systems: Summary Report. Pennsylvania Transportaoffs. and hence their holding costs, would increase as conges- lion and Traffic Safety Center. Sept. 1972.

tion increases and transit time reliability decreases in the svs- 13. K. J. Desai. Analysis of Waterway Systems.: A Study of Tow

tem. Such effects are relatively slight when volume to capacity Arrival Processes at Inland Waterway Locks. Pennsylvania Transtern vlportation and Traffic Safety Center. Sept. 1972.

ratios are small. If and when volumes increase substantially 14. C. W. Howe et al. Inland "Vaterway Trinsportation Studies in

above present levels, reliability benefits can justify capacity Public and Private Management and Investment Decisions. Reimprovements such as the new Gallipolis Lock. sourres for the Future. Inc.. 1969.

15. J. . , -,r-' ind M. S. Bronzini. Waterway Transportation Simultion Models: Development and Application. Waterway

Model Capability Resources Research. Vol. 9. No. 1. 1973. pp. 51-63.

The simulation model provides estimates of system perfor- Publication of this paper sponsored by Committee on Inland Water

mance that are sufficiently detailed and accurate for analytic Transportation.

10
PROBABILITY MODEL OF LOCKAGE STALLS

AND INTERFERENCES

by

Harry H. Kelejian

Published in Transportation Research Record

Volume No. 1313

11
12
ER.4ANPI'ORA .TION RENF.AR(H RH ORO I 111 91

Probability Model of Lockage Stalls and

Interferences

HARRY H. KELEJIAN

A model of lock failures as manifested by %talls or interferences lock operation. An interference is an occurrence which slows

and specified in terms of a logit formulation is presented in this lock operation during a lockage. For more detail see the Corps*

paper. Stalls or interferences that correspond to commercial tow User's Manual for Data Analhsis (2).

and recreational vessel lockages and that result from lock hardware problems or to testing or maintaining the lock or its equipment are first considered. The expected frequency of such lock stalls) that correspond to commercial tow and recreational

failures relative to the number of commercial tow and recreational vessel lockages and that result from lock hardware problems

vessel lockages is then explained. These expected frequencies can or to testing or maintaining the lock or its equipment. It then

be viewed as measures of reliability or interpreted as the prob- explains the expected frequency of such lock failures relative

ability that such failure% will occur on any given commercial tow to the number of commercial tow and recreational vessel lock- or recreational vessel lockage. The qualitative results correspond- ages. This can be viewed as a measure of reliability. It can

ing to the underlying variables are consistent with expectations. -

The usefulness and flexibility of the model in evaluating changes also be interpreted as the probability that such a failure will

in the values of these variables is demonstrated. Among other occur on any given commercial tow or recreational vessel

things, this demonstration suggests that many major maintenance lockage.

projects relating to lock chambers can be evaluated by their con- The model is specified in terms of a logit formulation. The

sequent effect on lock failure probabilities. It is demonstrated explanatory variables relate to characteristics of the lock

that the extent of the renewal of a chamber in response to major chambers, to the extent of major maintenance (if any), and

maintenance can be calculated. to variables which identify the Corps of Engineer district the

lock chamber is associated with. Among other things. the

The following scenario was suggested in a recent study (Charles usefulness of the model a, a tool of prediction and as an

Yoe, unpublished data). The Army Corps of Engineers op- instrument for allocating major maintenance funds is demerates and maintains 260 lock chambers and 536 dams at 5% onstrated.

sites. These structures are in various states of repair. performance, and obsolescence. Many of them are older than their

original 50-year design life. Maintenance. repair. major main- DATA ISSUES

tenance, and replacement of these facilities are becoming increasingly necessary and increasingly costly. Furthermore. the The data underlying this study were taken from the U.S.

recent inland navigation investment program. as reflected by Army Corps of Engineers' Lock Performance Monitoring Systotal appropriations for general construction and operations tern (PMS) data tapes. details of which are reported elsewhere

and maintenance has declined from $689 million in fiscal yeat (2). The data taken from these tapes relate to lockages at 125

1980 to $655 million in fiscal year 1987. After adjusting for lock chambers for 1981 through 1986. These lock chambers

price level differences, this 5 percent nominal decline becomes correspond to 14 Corps of Engineer districts. The 125 lock

a 35 percent real decline. Continued and even increasing strain chambers were chosen from the entire list of lock chambers

on fiscal resources is expected for the foreseeable future. described in the PMS tapes because the corresponding data

Further details are given elsewhere (1). were of a higher quality in the sense that fewer errors were

As a result of increasing needs and decreasing fiscal re- present and more complete in terms of having fewer missing

sources to meet those needs the Corps' decision problem is observations. Data relating to an individual lockage at these

how best to allocate scarce resources to operation. mainte- 125 chambers were not used unless observations on all of the

nance. repair. major maintenance, and replacement of struc- relevant variables were available.

tures on the inland waterway. In evaluating the economic A description of the 125 lock chambers is contained in

impacts of many of these investment decisions, it is necessary Kelejian (3). The 14 districts corresponding to these 125 lock

to quantify the costs of increasingly unreliable or insufficient chambers are listed in Table 1. It became convenient to deservice at locks and/or the benefits of improving reliability or scribe each district by a number (i.e.. District 1. District 2,

increasing capacity. This analysis of reliability generally re- etc.). These district numbers are also listed in Table 1.

quires an effort to quantify the probabilities of impaired lock The data file used to estimate the model contained two

services with and without proposed projects. types of PMS data. The first relates to individual lockages.

This study presents a model of lock failures as manifested The second relates to calendar year sums (e.g., total stall

by stalls or interferences. A stall is an occurrence which stops time). The individual lockage data represents a one-out-ofDepartment of Economics. University of Maryland College Park. twelve sample from the original PMS data tape. The annual

Md. 20742. sums are based on a 100 percent sample.

13
92 TRANSPOR TA lION RLSEARC 'H RECORD 1313

TABLE I DISTRICTS AND THEIR ASSOCIATED P,,. Therefore. variables that are components of 1,, that inNUMBERS crease P,, should have positive weights; negative weights corAScOeatecl Associated respond to variables which decrease P,.

District Number District Number

Pittsbmgh 1 Hutigton 8 Details of the index

Mobile 2 St. LoUIS 9

Nashville 3 St. Paul 10

Walla Waet 4 Little Rock 11 In this study, the index relating to the ith chamber at time t.

Wilrington 5 Tulsa 12 namely I,,. is

LOW" 6 Vicksxurg 13

Rock W" 7 Seattle 14 b,, bo + b, Age,, + b, MPT,,-, + b3 ICE,,.

+ b4 AIT,, + b, Maint,, + b. ST,,

The data file also contained information pertinent to lock

chambers described in (1). Among other things, this information relates to the age of lock chambers and the cost of + a, DD7, + a, DD8, + a, DDIO, (3)

completed major maintenance projects. The cost of the maintenance projects were given in current dollars. These data where b ..... b,. a, . a, are parameters to be estimated

were converted into constant 1982 dollars by deflating by the and all of the remaining terms on the right hand side of expresConstruction Cost Index. Data on this index were supplied sion 3 are explanatory variables whose definitions are given

by Corps personnel. A more complete discussion of the data in Table 2.

and their original source is given in Kelejian (3). In expression 3 Maint,, represents the extent of a major

maintenance. if any. It was formulated as

LOGIT MODEL OF STALL PROBABILITY Maint,, = 1 - EXP(-cost,,) (4)

Basic Formulation: An Overview where cost,, = 0 if. up through time t. lock chamber i did not

have major maintenance; if such maintenance did take place.

There are typically many lockages that take place during a cost,, is its 1982 dollar cost. The specification in expression 4

year at a given lock chamber. Corresponding to each of these implies that Maint,, = 0 if cost,, = 0. This is the case in which

lockages there is a probability that a stall will occur. a major maintenance did not take place. If it did take place.

Let P,, be the probability that a commercial tow or recre- cost,, > 0 and so Maint,, > 0. and the more extensive it was

ational vessel lockage taking place during the tth year at lock (the higher is cost,,), the higher is Maint,,. In this sense, the

chamber i results in a stall. Note that P,, is indexed to vary variable Maint,, is a positive measure of the extent of a major

from lock chamber to lock chamber (over i) and from year maintenance.

to year, but not from one lockage to another within a year A number of other variables were also considered but found

at a given chamber. not to be statistically significant. Results relating to these

The assumption that the probability of a stall is the same other variables can be found elsewhere (3).

for all lockages taking place within a year at a given chamber Since age. other things being equal. is associated with lock

is clearly an approximation. For example, a lock chamber deterioration, one would expect b, > 0. Similarly. higher

ages continuously and, therefore, from lockage to lockage. values of mean processing time may be indicative of equipHowever. one might view the effective aging of a chamber as ment which is not in top operating condition and so one exbeing very gradual and therefore reasonably well approximated by the age of the chamber as measured in years. If so.

and if the other relevant factors change gradually from lockage TABLE 2 DEFINITIONS OF EXPLANATORY

to lockage. the assumption of a constant stall probability within VARIABLES

a year at a given chamber is reasonable.

Let X,, be a vector of variables corresponding to the ith Van*i oDeio

lock at time t, which might be taken to explain P,,. Let B be Aqe,, , The sp of b i at t

a corresponding vector of parameters such that Mt ,... W p Wm ,mm - d ica dwi• i at &. 1..

I,, = X,,B (I) Ice,,. Ther Mer01tce dwv at &dito idberlifyeart -.

AfT,, Averag do ile 4 lWM dr dwin" raiig i ym" L

can be taken to be an index determining P,,. Then, in the logit u,10,, A vama ke frt mr" . drear vaJe al a nmior

formulation P,, is related to 1,, as l ,f Ntc ) ol ak doimb ,.

ST,., Tow re Wie due Io to ew or mlnasrm 01 10a ock dof 

',Or el eqiQflWW~mln Ig you t-1.

P,, = EXP(1,)/[1 + EXP(I,,)j (2) o ft equorncy urmg ar t-1.

SF_1t 71* I 9W bquory me 1= o• toml i at year t- 1.

oe sfreqmny is on raw 01 "f 1o locgao.

It is not difficult to show that P,, lies between zero and unity cou A &mly w aMiu a unity SIg Mi 10a dw 6b 4 n

for all possible values of the index I,,. In addition (dP,/dlJ) uario J. wid er a

> 0 for all 1I, so that the larger is the index !i, the higher is

14
Kelepaw 93

pects b, > 0. One would also expect b, > 0 and b, > 0 because expression in expression 3 yields

ice formation accelerates decay and a previous stall frequency

is indicated of general conditions, which are not radically log`(SF,,/(I - SF,)) - bo + b, Age,, + b, MPT,,

different from one year to the next.

One would expect b, < 0. The reason for this is that idle + b3 ICE,, + b, AIT,,

time could be used to perform minor maintenance and repair. + b, Maint,, + b, ST,, _,

and so on. Thus. higher values of AIT,, should lower the index,

/,,. and hence lower the probability of a stall. Similarly. for + b, SF, -- + a, DD1,

very evident reasons one expects b, < 0. On a somewhat

more moderated scale, one would also expect b, < 0. That + a. DD3, + a3 DD7,

is. the more testing and maintenance, and corresponding minor repairs, of the lock chamber and its equipment in one + a4 DD8 + a, DDI0,

year. the better the condition (other things equal) of that + uit (9)

chamber in the following year. For the readers' convenience.

the sign expectations relating to the coefficients of expression It can be shown that if the number of lockages during year

3 are summarized in expression 5 as follows: t at chamber i is "*large". the term u,, has a mean and variance

which are approximated by the following expression

b, > 0, b. > 0. b, > 0. b, > 0i.

b,<0.b,<0,b< 0(5) Eu,, = 0. Var(u,) = IL,, P,, (1 - P,,)j" (10)

The implication of expression 10 is that expression 9 can be cateswheth oeffinot e of dumyprobabliny e sa coprespiond3ngd viewed as a regression model having a heteroskedastic error

cates whether or not the stall probability corresponding to tr.Bcuetevrac fuivle ,.wihi o

that district is higher (if positive) or lower (if negative) than known, the appropriate estimation procedure is a feasible

in the districts not represented in expression 3 after the effects form of generalized least squares that is based on an estimated

of the other variables in the index have been accounted for. value of the variance of ua,, this estimated value would be

Conceptual arguments do not suggest the signs of these coef- based on an estimate of P,,.

ficients. In implementing this procedure for the PMS data. two complications arose. The first is that for certain years at certain

lock chambers SF,, is zero. In these cases, the dependent

The Issue of Estimation variable in expression 9 is not defined. The second complication is that in certain years the number of lockages at certain

Assume that P,, is neither zero nor unity. Then from expres- lock chambers. L,,. is small. In these cases the large sample

sion 2 it can be shown that P,,I(1 - P,) = EXP(I,,) so that approximations in expression 10 are not appropriate and so.

therefore, neither is the model in expression 9.

log (P,,/( I - P,,)) -- 1,, (c) The discussion in Kelejian (3) suggests that if the number

of Iockages for each chamber in each time period is large. the

The result in expression 6 is useful in that it leads to a first of these problems can be overcome by replacing the

relatively simple procedure for estimating the parameters de- depen e vrible in expression 9 by

termining the index I,, as given in expression 3. For example.

let SF,, be the number of stalls at lock chamber i during year Y,, = lo&I(SF,, + (2L,,) ')/I[l - SF,, + (2L,,)- 111 (1)

t. Then SF,, may be expressed as SF,, = S,,IL,, where S,, is

the number of stalls of the type being considered at lock The reason for this is that Y,, is defined for all values of SF,

chamber i during year t. and L,, is the corresponding number in the interval 0 :- SF,, s 1; furthermore, under reasonable

of lockages. Because the probability of a stall on any given conditions. Y,, and log[SF,,/(l - SF,,)] converge in probability

lockage is assumed to be the same for all lockages during the as L,, increases beyond limit.

year at a given chamber. SF,, can be taken as an estimate of The procedure that was followed in this study is on the basis

Pi,. The reason for this is that SF,, can be viewed as the ratio of a variant of expression 11 and is defcribed in steps detailed

of the number of successes (stalls) to the number of trials in the following. Note that the estimators so obtained are

(lockages). asymptotically efficient because they are equivalent to the

For ease of presentation, suppose that SF,, is neither zero corresponding maximum likelihood estimators.

nor unity. Then let

u, = logjSF,,/(1 - SFJ, - log,(P,,/(l - P,,)) (7) Details of the Procedure

so that

Step I

log&(SF,,I(l - SF,,)) = Io&lP,,I(i - P,,)J + u,, (8)

Some restriction on the original PMS sample was necessary

The first term on the right hand side of expression 8 is equal because the number of lockages in certain years at certain

to the index I,, via expression 6. Replacing this index by its chambers (henceforth. cells) was very small (e.g., as low as

15
94 TRANSPORTATION RESEARCH RECORD 1313

4). The restriction that was imposed was that Jnly data re- estimates are based on a consistent and efficient procedure.

lating to cells for which L,, > Q were considered in the es- the final estimate of the stall probability for the ith lock chamtimation procedure. Q was taken as the largest multiple of ber at time I was taken as

50 such that at least 2/3 of the original cells remain in the

revised sample. It turned out that Q = 150. and the smallest P, - EXP(!,,)/[1 + EXP(i,,)] (15)

value of L,,, say MIN, satisfying L,, > 150 was MIN = 151.

The cut off value of 150 is reasonably large, but not overly where

restrictive in terms of the scope of the revised sample. For

example, the condition L,, > 150 only eliminates reference I,, = bo + 6, Age,, + 6, MPI,_

to 32 lock chambers, thus leaving 93 such chambers in the

sample. The number of cells in the revised sample is 499 which + b3 ICE,,- + b, AIT,, + 6, Maint,,

is roughly 67 percent of the original number of cells. + 66 ST.,,, + 6,SF,,_I + d, DD1,

Step 2 + d, DD3, + d3 DD7,

+ d. DD8, + d. DDiO, (16)

Because the number of lockages in each cell of the sample

constructed in Step I varied from 151 to 1.355. a modified

form of Y,, in expression 11 was considered; namely EMPIRICAL RESULTS

"V, = log.{SF,, + (2 x MIN)-'/[I - SF,, Results Relating to the Probability Model

+(2 MIN)-]) (12) The empirical r-sults obtained by the procedure described in

Y,, was considered for two reasons. First, unlike Y,,. 'Y,, does Steps I through 5 are given in expression 17. The figures in

not induce an artificial variation in the dependent variable, parentheses beneath the parameter estimates are the absolute

which is due solely wcorrelationecsoffiyienttbetween to the wide range of values of the number values of the corresponding t-ratios. thegobservedustall i2 is the square frequency. of the

of lockages. Second. there is no penalty in terms of asymptotic correlation coefficient between the observed stall frequency.

efficiency in the use of "Y,, as compared with Y,, because 'Y,, SF,,. and its model predicted value P,, (see expression 15).

and Y,, converge as L,, increases beyond limit. I,, = - 5.956 + .0102 Age,, + .0071 MPT,, -,

Step 3 (41.31) (5.130) (2.997)

+ .0066 ICE,, - .0899 AIT,

Taking Y,, as the dependent variable, expression 9 was first

estimated by least squares. This provided a consistent estimate (4.458) (1.599)

of the index 1,,. say IE,,, for each of the 499 cells of the sample. - 1.197 Maint,, - .0048 ST,, + 23.09 SF,,

Step 4 (I.81R (3.997) (8881)

- .2422 DDI, + .8219 DD3, - .2121 DD7,

The estimated index, IE,,. was then used to obtain an initial

but consistent estimate. PE,,. of the stall probability P,, for (2.651) (7.292) (2.716)

each of the 499 cells + .3416 DD8, - .3167 DDIO, : = .366

PEt = EXP(IE.J/I1 + EXP(IE,,)J] (13) (3.888) (3.093) (17)

Correspondingly. the variance of u,, as given in expression i0 The units of measurement underlying expression 17 are: Age

was then estimated as is in years: MPTis in minutes per lockage; ICE is in days per

year; AIT is in hundreds of minutes; Maint = 1 - EXP( - C)

*'ar(u,,) = [L,, PE,,(1 - PE,,)J - (14) where C is in hundreds of millions of 1982 dollars: SF is the

observed stall frequency; ST is in thousands of minutes.

The value of ;2 = .366 suggests that, overall, the model

Step 5 offers a reasonable explanation of stall probabilities associated with individual lockages. In interpreting this figure one

Finally. with "Y,, in expression 12 taken as the dependent should note that stall probabilities, as measured by stall frevariable, expression 9 was reestimated by least squares after quencies, vary widely across lock chambers and time, and

deflating each variable by the square root of ý'ar(u,,) in expres- therefore are not easily explained. For example, the R- stasion 14. This is the feasible generalized least squares proce- tistic (over the sample underlying expression 17) between the

dure. annual stall frequency at a lock chamber, and its age is only

Let the estimates of the parameters of expression 9 obtained .015. More extensive results along these lines are given in

in Step 5 be bo. b7, a, ... , as. Then, because these Kelejian (3). Nevertheless, • = .366 does imply that 63.4

16
Kelepan 95

percent of the variation in stall probabilities is unexplained. TABLE 3 DISTRICT EFFECTS AT SAMPLE

and so further studit along these lines could be of value. MEAN VALUES

On a qualitative level, note that the sign of each estimate Probability Probablty

given in expression 17 is consistent with plior expectations as Ditict Changes Change (%)

described in expression 5. Also note that each of these estimates if considered alone, is statistically significant at the Nashvillb .084 -7

one-tail .05 level with the sole exception of the coefficient of Huntin3ton .0041 50 Rock Island -.0013 -21

the average idle time variable. The sign of this coefficient is Pittsburgh -.0012 -24

negative, as anticipated, but its one-tail significance level is St. Paul -.0043 -31

.0548. Because strong prior Bayesian beliefs suggest that average idle time is important, and because the one-tail significance level is quite close to .05, the idle time variabe was

not dropped from the model. TABLE 4 STALL PROBABILITIES AND ELASTICITIES

There are no prior sign expectations for the coefficients of

the district dummy variables and therefore a test of signifi- Stl Probabiliy Elasticites

cance would be determined by a two-tail procedure: clearly Lowest Highest Mean AGE MPT ICE AIT MAINT ST

the results in expression 17 imply that if these vaiiables are

considered individually, each and every one of them would 002 03'4 005 .86 287 106 -079 -009 .027

be statistically significant at the two-tail .05 level. The joint

significance of the district dummy variables is confirmed by chamber will have a stall on a given lockage. Conversely. th.

t,, corresponding F test. namely F = 15.18 > F(.95/5.486) probability that a stall will not take place at such a typical

= 2.23. chamber is .9944. This probability is so high that even if a

Districts that are represented in the sample but for which reasonably large number of lockages take place over a given

there are no dummy, variables in expression 17 are Mobile. period of time, the probability that a stall will not occur during

Walla Walla. Louisville, St. Louis, Little Rock. and Seattle. that time could remain non-negligible.

Therefore. if a coefficient corresponding to a dummy variable The table also gives the lowest and highest values of the

in expression 17 is posities the stall probability in the cor- stall probability based on the values of the index over the

responding district is higher than in the excluded 5 districts chambers and years in the sample. These figures. namely

for given and equal values of the other variables in expression .0023 and .0374, correspond. respectively, to Chamber 1 of

17. Districts 3 and 8 (Nashville and Huntington) fall into this the Old River Lock on the Mississippi River (ORLMR) for

category. Similarly. if such a coefficient in expression 17 is 1982. and Chamber I at the Gallipolis Locks and Dam on the

negativ,. the stall probability in the corresponding district is Ohio River (GLDOR) in 1986. These figures differ by more

lower than iii the excluded five districts for given and equal than a factor of ten. As an indication of time variation, the

values of the other variables in expression 17. Districts 1, 7, stall probability at the ORLMR for 1986 is .0025: the stall

and 10 (Pittsburgh. Rock Island, and St. Paul) fall into this probability i, .0096 for 1982 at the GLDOR. Among other

category. things, these results suggest that stall potentials, as measured

One measure of the magnitude of these district effects is by stall probabilities, vary considerably from chamber to

the consequent change in the stall probability. For example. chamber, as well as over time. Given the results in expression

in District 1, (Pittsburgh). the sample mean of the index in 17, and the model in expression 15. the stall probability can

expression 17 is I, = - 5.447: the corresponding stall prob- be calculated for any chamber, for any year, as long as the

ability is P, = .00429. If District I were typical. as say de- values of the independent variables are known. Clearly, the

scribed by the five excluded districts, the coefficient of its calculation of such stall probabilities should be helpful in aldummy variable would be zero. In this case. the sample mean locating scarce major maintenance funds.

of its index would be IE = - 5.2048, and the corresponding Table 4 also gives estimates of the elasticities of the stall

stall probability would be PE, = .00546. Therefore, whatever probability with respect to six of the index variables, again at

the special effects associated with District 1, they lead to a sample mean values. These elasticities were calculated as

reduction of .00117 in the stall probability. Since these probabilities are small, this small change represents a large per- dlog(P,)/dlog,(Zh,)

centage change. Specifically. taking (P, + PE,)/2 as the base.

the district effec* (at the sample mean) associated with District "Z' b,/(I + EXP('I)). j = I. 6 (18)

1 leads to a 24 percent reduction in the stall probability.

Corresponding figures for Districts 3, 7, 8, and 10 are given where Zl,, is the jth explanatory variable (excluding the inin Table 3. Consistent with the results for District 1, a glance tercept) in expression 16; h, is its corresponding estimated

at the table suggests that these districts also have effects that coefficient given in expression 17, and 71 and "! are the

are important in percentage terms concerning stall probabil- sample averages of Z), and !,,.

ities. The elasticities in Table 4 indicate the relative sensitivity

Further results relating to the empirical model are given in of the stall probability with respect to a given percentage

Table 4. Specifically, the table gives the stall probability cor- change in the value of the corresponding explanatory variable

responding to sample mean values of the variables determin- at sample mean values. For example, the elasticity with reing the index in expression 17. This figure, namely .0056. can spect to the age variable is .386. This figure suggests that. at

be interpreted as the probability that the average or typical sample mean values, a I percent (a 10 percent) increase in

17
96 TR.4 V'-tPOR TA TION RESEARCH RECORD 1313

the age of a chamber would (other thih~gs equal) lead to a 90

.386 percent (a 3.86 percent) increase in the stall probability 80

Among other things. the figures in Table 4 suggest that stall

probabilities for a typical lock chamber are more sensitive to B 700L

small percentage changes in the age of the chamber, than to M

small percentage changes in the other variables of the index. 1 600

Figures 1 through 4 give further insights concerning the B t00

probability model. Figure 1 describes the relationship between the stall probability and the age of the chamber at 400

sample mean values ot the other variables involved in the

index. Again, since these sample mean values could be viewed .300

as typical, Figure I essentially describes a time profile of a 200l -_ _ _

stall probability for a typical chamber. As the chamber ages, 10 40 70 100 130

the probability increases. Calculations based on the diagram Mean Processing Ti,•* pe Lockage (Miules)

suggest that this probability is roughly 20 percent higher when FIGURE 3 Effect of mean processing time

the chamber is 60 as compared with 40 years old. on the predicted probability of a stall.

Figure 2 describes the relationship between the stall probability and the extent of major maintenance, as measured by

its 1982 dollar cost, again at sample mean values. As expected, 900

the more extensive the maintenance, the lower the probability. Calculations performed on the basis o1 the diagram sug- 8.00 wt uao,

gest that, for a typical chamber, a 30 million 1982 dollar major

maintenance reduces the stall probability by, roughly, 35 per-, 70

•5 600

10.00 - 5.00

9600 - 400 m

&OD - 300

1 21 41 61 81

600. Age (Years)

t SFIGURE 4 Effect of major maintenance on

the effective age of a lock.

S4.00 a.

300

21 41 61 8 cent. Similarly, a 20 million 1982 dollar major maintenance

Age(Years) reduces this probability by. roughly. 20 percent. Figure 3

describes the rehtionship between the stall probability and probability of a stall o mean processing time. at sample mean values. The figure

suggests that. for the :ypical chamber, stall probabilities are

roughly 45 percent more likely when the mean processing

700 time is I(X) minutes per lockage than when it is 40 minutes

per lockage.

600 Finally, Figure 4 describes how major maintenance reduces

0 the effective age of a chamber. The upper curve in that figure

outlines the relationship between the stall probability and the

5w 500 age of the chamber if there is no major maintenance and the

other relevant variables are equal to their sample means. The

4W -lower curve descrit s the change in the probabilities outlined

by the upper curve if a 20 million 1982 dollar major maintenance were undertaken when the chamber is 50 years old.

S3,00 A 20 million 1982 dollar major maintenance was considered

in this illustration because it is. roughly. the average cost of

such maintenance completed during or before 1987.

200 - If the lower curve, at any age exceeding 50 years, is hori0 20 40 60 80

Coot of Maio Maintenance zontally extended to the left, it will intersect the upper curve (Millions of 6 Dollars) corresponding to an age which is. roughly, between 20 and

FIGURE 2 Effect of major maintenance on 25 years earlier. The suggestion is that, for th,; typical chainthe predicted probability of a stall. ber. a 20 million 1982 dollar major maintenance, undertaken

18
Kreljan 97

when the chamber is 50 years old, reduces the effective age again be evaluated. Comparisons between the two cases should

of that chamber by. roughly. 20 to 25 years. be of interest.

The redu-tion in the effective age of a particular chamber

corresponding to a proposed major maintenance of a certain SUMMARY AND CONCLUSIONS

dollar magnitude can be done in a similar, but more exact

way. Specifically. let 1,1 be the value of the index in expression A probability model of lock failures has been presented. The

17 for lock Chamber i at time I before the major maintenance, qualitative results corresponding to the underlying variables

Let I1 be the value of that index after the major maintenance, are consistent with expectations. The usefulness and flexibility

Because the index is reduced if major maintenance is under- of the model in evaluating changes in the values of these taken. jA < iI and so the "after" stall probability would be variables has been demonstrated. ,mong other things. this

less than the "before" stall probability. The effective age of demonstration suggests that many major maintenance projthe chamber after the maintenance is the value of the age ects relating to lock chambers can be evaluated in terms of

variable that equates the before index. i|B. to the after index, their consequent effect on lock failure probabilities. It was

iA. That is. let IB, be the net sum of the right hand side of also demonstrated that the extent of the renewal of a chamber

expression 17. before the major maintenance, with the ex- in response to major maintenance can be calculated.

ception of the age variable: !,B = . + .0102 * Age,,. Then

the effective age of chamber i at time t is Age. where ACKNOWLEDGMENTS

AgeE = (I;A - P,)/.0102 (19) I would like to thank Craig Hiemstra. Michael Wagner. and

Charles Yoe for useful discussions and comments on an earlier

The reduction in the effective age is therefore Age,, - version of this paper. I would also like to thank Craig HiemAge,. stra and Michael Wagner for help at various computer-related

stages of this study. This study would not have been possible

without the support of George Antle of the Army Corps and

Suggestions Concerning Further Calculations I anm grateful for that support. I would also like to thank

Yvonne Sisler who cheerfully typed, and retyped, the various

Calculations concerning chambers in districts which are not drafts of this paper. Finally, I would like to thank the Comin the sample require an assumption concerning the district puter Science Center at the Univetsity of Maryland for its

effect as described in the index (see expression 17). One pos- generous support of computer time.

sibility is that Corps personnel could use expert opinion to

determine the district effect: given this. stall probabilities could

be evaluated for any chamber of interest, for any year. as REFERENCES

long as the values of the variables determining the index in 1. Institute for Water Resources. The 1988 Inland Waterway Review.

expression 17 are known. The magnitude of the district effects Report 88-R-7. U.S. Army Corps of Engineers. 1988.

of Districts 1. 3. 7. 8. and 10 should offer guidance if this 2. Institute for Water Resources. Lock Performance Monitoring Sysroute is taken. tern: User's Manual for Data Analysis. User Manual 85-UM-2.

Another possibility is to assume that the district effect cor- U.S. Army Corps of Engineers. 1985.

3. H. H. Kelejian. A Study of Lockage Stall Probabilities. Report to responding to a chamber of interest, which is not in the sam- Institute for Water Resources, Fort Belvoir. U.S. Army Corps of

pie. is equal to the average of the effects of those for Districts Engineers. 1989.

1. 3. 7. 8. and 10. Still another possibility is to consider worst 4. G. S. Maddala. Limited Dependent and Qualitative Variables in

and best case scenarios. For example. the district effect could Econometrics. Cambridge University Press. New York. 1983

be taken to be equal to that of District 3. which would be a

worst case scenario. Given this. it policy could " be evaluated ~are The not views necessarilyv expressed those jn this of tire studv Arrm)' are solely Corps.those of tie author, and

in terms of its effect on the stall probability. The district effect

could then be taken to be equal to that of District 10. which Publication of this paper sponsored by Committee on Inland Water

would be a best case scenario. Given this, the policy could Transportation.

19
20
EFFECTS OF LOCK INTERDEPENDENCE ON TOW DELAYS

by

Steven I. J. Chien and Paul M. Schonfeld

21
Effects of Lock Interdependence on Tow Delays

Steven I. J. Chien and Paul M. Schonfeld

September 21, 1992

Transportation Studies Center

University of Maryland

College Park MD 20742

22
Introduction

This study examines the effect on tow delays of interactions among successive locks.

Relatively simple queueing models may be used for analyzing single-lock delays. However,

considerable interdependence may exist among locks in a series, especially if the locks are

relatively closely spaced and if congestion levels are high. The tow departure distributions at a

lock differ from the arrival distributions at that lock since the service-time distributions change the

tow headways. The departures from one lock usually affect the arrivals at the next lock. Thus, it

is risky to assume that the locks are independent. The interdependence among locks increases the

difficulty in estimating delays for a system of locks, since it is necessary at each lock to identify the

interarrival-time distributions of flows from adjacent locks.

Analysis Method

A simulation model which reflects the interaction among locks is used here to compare

results obtained by assuming (a) isolated locks or (b) interdependent locks. This model (Dai 1991,

Dai and Schonfeld 1991) is microscopic, event scanning, and programmed in FORTRAN. Its

most detailed documentation is provided in Dai (1991). By running the model with actual

distances among locks, interdependence effects are automatically considered. As distances among

locks are artificially increased, the locks approach an isolated condition. Isolation may also be

modeled by generating tow arrivals at each lock according to a Poisson distribution, which is

equivalent to generating exponential interarrival times. As the distances between successive locks

increase, and as the effect of the departure distributions is gradually randomized, the arrivals

should asymptotically approach a Poisson distribution.

Node I Lock 22 Node 2 Lock 24 Node 3 Lock 25 Node 4

I-u11.85 bI. 13.90,.,1 13.90 16.00,.1 16.00 19.25

(miles)

Figure 1

The Geometric Configuration for the Segment

on the Mississippi River

23
1987 PMS data from Mississippi Locks 22, 24, and 25 were used in the experiments

reported here. The geometric configuration of this system is shown in Figure 1. Four nodes and

three locks are included in this system. The average trip rate differed for each lock and is shown in

Table 1. Results have been compared for two different traffic volumes representing (1) 1987 levels

(from PMS data), which correspond to a 0.65 V/C (volume/capacity) ratio at the critical Lock 22,

and (2) a more congested hypothetical 0.95 V/C ratio at Lock 22. For the 0.95 V/C ratio case, all

volumes are increased by the same multiplier (0.95/0.65) as shown in Figures 2 and 3. Traffic

volumes are equal in the two directiors.

Lock No. Lock 22 Lock 24 Lock 25

Trip Rate (65%) 5.25 5.5 5.87

Trip Rate (95%) 7.67 8.03 8.57

Table 1. Trip Rates for Each Lock (tps/day)

The trip rates among the various O/D pairs which actually exchange traffic are shown in

Table 2.

O/D pair No. 1-4 1-2 2-3 3-4

Trip Rate (65%) 5.08 0.17 0.42 0.79

Trip Rate (95%) 7.42 0.25 0.61 1.15

(trips/day)

Table 2. Trip Rates for Each O/D Pair

** 1-4 means from node I to node 4

The actual node distances are 25.75, 29.90, and 35.25 miles form node 1 to node 2, from

node 2 to node 3, and from node 3 to node 4, respectively. The average numbers of stall events

are 30, 40, and 49 per year at Locks 22, 24, and 25, respectively. The average tow speed is

203.76 miles per day with a standard deviation 81.36 miles per day. Time periods of 2100 days

are simulated, after a steady state condition is reached. At each lock, the data collected include the

directional interdeparture-time distributions, the overall interarrival-time distributions, and the

24
average wait time. The corresponding trip rate, V/C ratio, lock service time distribution, distance

and the tow speed distribution were also recorded. To obtain the required information, the

simulation program was run for 30 independent replications in each relevant case, i.e. 30x2100

days. To insure that the results were for a steady state, each simulation run discarded the first

20,000 observations and collected the next 40,000 values for evaluating the results.

5.08 trips/day

0.17 trips/day 0.42 trips/day 0.79 trips/day•

Nodel Lock 22 Node 2 Lock 24 Node 3 Lock 25 Node 4

Figure 2

The Average Trip Rate at 65% Service Capacity of

the Bottleneck Lock Both Directions

7.42 trips/day

0.25 trips/day 0.61 trips/day 1.15 trips/day•

Node I Lock 22 Node 2 Lock 24 Node 3 Lock 25 Node 4

Figure 3

The Average Trip Rate at 95% Service Capacity of the

Bottleneck Lock Both Directions

Results

The overall results obtained from running the simulation model are shown in Appendices 1

and 2. The average delays, standard deviations, and standard errors of the mean for different trip

rates and lock distance are summarized in Tables 3 and 4.

25
Distance Multiplier Lock 22 lock 24 Lock 25 Total System

101.92* 76.95 86.70 265.67

0.1 (4.26)** (2.87) (3.72) (7.97)

[0.78J*** [0.521 [0.681 [1.46]

106.80 83.29 90.70 280.79

1 (4.53) (2.84) (3.75) (8.00)

[0.831 [0.521 [0.68] (1.46]

113.87 95.58 96.74 306.19

10 (3.50) (3.58) (3.70) (7.64)

[0.641 [0.65] [0.68] [1.40]

110.69 96.71 95.82 303.22

100 (4.19) (3.10) (2.77) (7.28)

[0.761 [0.57] [0.51] [1.33]

* mean value

** standard deviation

***: standard error of the mean

Table 3 Effects of Lock Separation on Delays (V/C = 0.65 at Lock 22)

Distance Multiplier Lock 22 lock 24 Lock 25 Total System

716.94 454.40 406.84 1578.18

0.1 (74.87) (50.20) (41.82) (123.98)

[13.67] [9.16] [7.63] [22.64]

737.39 475.86 430.07 1643.32

1 (72.63) (52.65) (42.55) (122.10)

[13.26] [9.61] [7.77] [22.29]

805.62 543.84 489.51 1838.97

10 (85.66) (55.99) (44.64) (134.66)

[15.64] [10.22] [8.15] [24.59]

851.98 647.16 534.71 2043.85

100 (82.19) (64.52) (49.78) (140.93)

[15.01] [11.78] [9.09] [25.73]

847.04 659.31 526.84 2043.19

200 (100.56) (74.89) (52.26) (158.11)

[18.36] [13.67] [9.54] [28.87]

Table 4 Effects of Lock Separation on Delays (V/C = 0.95 at Lock 22)

26
The distance multipliers are used to vary all inter-lock distances proportionally. 1

represents actual distances. Distance multipliers of 100 or 200 are, for all practical purposes,

equivalent to infinite distances, i.e. complete isolation of individual locks.

The results in Tables 1 and 2 confirm, as expected, that delays asymptotically approach the

value for isolated locks as distances increase and interdependence becomes negligible. However,

at actual distances, considerable errors in delay estimation would be made if the interdependence is

neglected. To test whether the locks are essentially isolated (i.e. whether the arrivals are Poisson

distributed), the tow interarrival times at each lock have been extracted and analyzed.

When the distance multiplier reaches 100 at a 65% volume/capacity ratio at Lock 22, the

average tow delays converge to 111, 97, and 96 minutes per tow at Locks 22, 24, and 25,

respectively. Similarly, when the distance multiplier reaches 200 at the 95% V/C ratio at Lock 22,

the average tow delays converge to 850, 650, and 530 minutes per tow at Locks 22, 24, 25,

respectively. Furthermore, the average system delay per tow converges to 303 and 2040 minutes

per tow at 65% and 95% V/C ratios, respectively. We found that at large distance multipliers (i.e.

100 or 200) the tow interarrival times at each lock are indeed exponentially distributed, and the

numbers of tow arrival events in every small interval is Poisson distributed, as would be

theoretically expected if the locks were isolated and independent.

The results in Table 5 show that for individual locks in this system, the average delays are

overestimated from 4% (Lock V/C=0.65) to 39% (Lock 24, V/C=0.95). For the entire 3-lock

series the overestimation error is 8% at V/C = 0.65 and 24% at V/C = 0.95. As expected, the

results confirm that delay overestimation errors increase significantly as the system approaches

capacity. Therefore it is very important to explicitly consider the effects of interdependence among

locks, especially as congestion increases and/or when locks are relatively closely spaced. Based

on the results in Tables 3 and 4, the average tow delays at each lock have been plotted in Figures 4,

5, and 6. The effect of lock separations on combined delays in the 3-lock system is shown Figure

7. It clearly shows that the interdependence effects, and the errors resulting from neglecting them

are much greater as congestion increases in the system. The standard deviation o f combined

27
delays for various V/C ratios (5%, 35%, 65%, and 95%) at Lock 22, 24, and 25 is shown in

Figure 8. It points out that when the congestion becomen serious at the bottleneck lock, the

standard deviation of the tow delays increase drastically.

Lock V/C = 0.65 at Lock 22 V/C = 0.95 at Lock 22

Interdependence* Isolation*" Interdependence Isolation ***

Lock 22 106.80 110.69 737.39 847.04

(100%) (104%) (100%) (115%)

Lock 24 83.29 96.71 475.86 659.31

(100%) (116%) (100%) (139%)

Lock 2S 90.70 95.82 430.07 526.84

(100%) (106%) (100%) (122%)

Total 280.79 303.22 1643.32 2043.19

System (100%) (108%) (100%) (124%)

* Interdependence: actual distances among locks

** Isolation at V/C = 0.65: 100 x (actual distances among locks)

* Isolation at V/C = 0.95 : 200 x (actual distances among locks)

Table 5 Summry Result for Delay Estimation

Conclusion

The results of this analysis are simply summarized in Table 5 and in Figure 7. They show

that lock interactions may be quite significant and that considerable errors may be introduced when

such interdependence is neglected for the sake of simplified models. The interdependence effects,

and the associated errors of neglecting them increase (1) as locks are located closer together and (2)

as congestion levels increase in the system.

28
1(000

goo-' V/C 0.95 at Lock 22

6 00-

418

o 400-

ta 200,

V/C 0.65 at Lock 22

0

0 100 200

Distance Multiplier

Figure 4

Average Tow Delays for Different Trip Rates

at Lock 22 on Mississippi River

29
700

V/C 0.95 Loa 22

5 00- S 300-

•, V/C =0.65 at Lock 22

"0- -,

0 100 200

Distance Multiplier

Figure 5

Average Tow Delays for Different Trip Rates

at Lock 24 on Mississippi River

30
500

*- 400

300

• 200

.. V/C =.65 at Lock 22

> 100

0

0 100 200

Distance Multiplier

Figure 6

Average Tow Delays for Different Trip Rates

at Lock 25 on Mississippi River

31
f

2000 -

'I

a 1000

V/C = 0.65 at Lock 22

0'

0.1 tO 100 200

Distance Multiplier

Actual Distance

Figure 7

Effect of Lock Separations on Combined

Delays at Locks 22, 24, and 25

32
Figure 8

Standard Deviations of Combined Delays

for Various V/C Ratios at Lock 22, 24, and 25

150'

o 100,

4...

O - A.D.

0

50O

V

0

.0.5 .35 .65 .95

Volume/Capacity (V/C) Ratios

33
References

[1] Dai, M.D.M., "Delay Estimation on Congested Waterways", Ph.D. Dissertation,

University of Maryland, College Park, 1991.

[21 Dai, M. D. M. and Schonfeld, P. M., "Simulation of Waterway Transportation Reliability",

Transportaon Research Record, No. 1313, pp. 98-105, 1991

34

1.• r.. .dda
Appendix 1

Simulated average tow delays at 5% of capacity (30 replications)

(1) Trip Rate : 5% of capacity at Lock 22

Distance : 0.1 x actual distance

lock 22 lock 24 lock 25 Total System

---------------- ------------ ------------ ------------

6.22 3.97 6.97 17.16

6.28 4.61 7.63 18.52

:.85 5.43 6.52 17.80

5.31 3.78 7.40 16.49

6.03 4.98 7.54 18.55

5.45 3.03 6.57 15.05

6.25 3.77 7.26 17.28

4.99 4.22 5.67 14.88

4.34 3.63 7.21 15.18

4.82 3.93 8.62 17.37

5.09 4.75 6.86 16.70

9.66 3.67 7.07 20.40

4.89 4.36 7.01 16.26

3.85 3.88 8.67 16.40

6.58 3.21 7.76 17.55

4.66 3.56 6.64 14.86

4.65 4.12 7.89 16.66

6.67 3.46 11.92 22.05

5.29 4.23 6.18 15.70

5.51 5.15 8.02 18.68

4.84 3.37 5.31 13.52

5.83 3.09 5.83 14.75

6.67 3.44 5.96 16.07

5.82 4.18 6.50 16.50

6.20 6.38 6.08 18.66

4.74 4.46 7.58 16.78

5.26 3.80 7.76 16.82

4.56 3.05 7.25 14.86

6.04 3.77 8.31 18.12

5.50 3.29 6.91 15.70

------------------------------------------------------

mean mean mean mean

5.59 4.02 7.23 16.84

------------------------------------------------------

std std std std

1.05 .75 1.20 1.76

------------------------------------------------------

std error std error std error std erro

.19 .14 .22 .32

"* mean : average value

"* std : standard deviation

"* std error : standard error of the mean

35
(2) Trip Rate : 5% of capacity at Lock 22

Distance : 1 x actual distance

lock 22 lock 24 lock 25 Total System ---------------- ------------ ------------ ------------

5.14 3.40 6.43 14.97

5.90 4.09 6.52 16.51

6.26 3.58 6.23 16.07

5.07 3.36 7.67 16.10

6.41 4.35 6.31 17.07

4.23 3.91 4.95 13.09

5.34 3.48 8.25 17.07

6.51 4.51 5.59 16.61

5.25 4.07 9.06 18.38

4.73 3.50 7.09 15.32

5.43 4.24 6.22 15.89

7.87 3.48 6.82 18.17

5.89 4.03 6.95 16.87

3.57 3.43 6.46 13.46

5.71 3.70 7.84 17.25

5.28 3.94 6.00 15.22

4.85 3.42 7.89 16.16

5.40 4.68 9.18 19.26

5.38 4.23 6.15 15.76

4.53 3.45 7.27 15.25

4.52 3.18 6.39 14.09

5.45 3.10 5.30 13.85

6.87 4.02 6.56 17.45

5.54 4.55 7.92 18.01

6.20 4.81 7.52 18.53

5.50 4.99 5.47 15.96

5.97 3.39 5.40 14.76

4.00 4.15 8.24 16.39

5.86 3.52 8.17 17.55

5.09 4.60 6.68 16.37

---------------- ------------ ------------ ------------

mean mean mean mean

5.46 3.91 6.88 16.25

---------------- ------------ ------------ ------------

std std std std

.86 .51 1.08 1.50

std error std error std error std error

.16 .09 .20 .27

* mean : average value

* std : standard deviation

* std error : standard error of the mean

36
(3) Trip Rate : 5% of capacity at Lock 22

Distance : 10 x actual distance

lock 22 lock 24 lock 25 Total System

---------------- ------------ ------------ ------------

5.56 3.49 7.99 17.04

7.19 3.99 8.77 19.95

5.50 3.51 7.97 16.98

4.16 3.30 6.65 14.11

6.69 5.15 7.35 19.19

6.03 3.92 5.88 15.83

4.07 3.92 9.32 17.31

6.62 4.24 6.77 17.63

6.09 3.59 7.73 17.41

7.04 4.26 5.76 17.06

5.48 3.30 6.83 15.61

5.70 4.37 8.79 18.86

5.28 4.63 6.44 16.35

5.19 4.06 7.35 16.60

5.71 3.94 6.81 16.46

8.09 4.52 5.98 18.59

4.69 3.90 7.82 16.41

6.76 3.00 6.41 16.17

5.38 5.01 6.34 16.73

6.71 3.53 7.59 17.83

5.93 4.61 7.05 17.59

5.40 5.83 7.24 18.47

6.14 4.05 8.13 18.32

4.93 4.95 5.69 15.57

5.87 4.55 6.75 17.17

5.36 3.68 4.08 13.12

6.54 4.23 6.18 16.95

4.95 3.85 7.30 16.10

3.42 3.33 11.68 18.43

5.11 5.08 8.59 18.78

---------------- ------------ ------------ ------------

mean mean mean mean

5.72 4.13 7.24 17.09

---------------- ------------ ------------ ------------

std std std std

.98 .64 1.36 1.43

---------------- ------------ ------------ ------------

std error std error std error std error

.18 .12 .25 .26

"* mean : average value

"* std : standard deviation

"* std error : standard error of the mean

37
(4) Trip Rate : 5% of capacity at Lock 22

Distance : 100 x actual distance

lock 22 lock 24 lock 25 Total System

4.13 3.84 7.77 15.74

6.75 4.64 5.82 17.21

6.10 4.49 9.87 20.46

5.72 3.77 8.20 17.69

4.84 4.51 5.93 15.28

5.65 3.30 7.71 16.66

4.42 4.16 8.00 16.58

3.93 3.80 5.45 13.18

5.75 5.00 5.58 16.33

6.76 3.28 6.02 16.06

6.39 3.39 7.09 16.87

8.39 3.46 4.99 16.84

4.66 4.73 4.88 14.27

4.84 3.37 7.49 15.70

3.61 3.15 5.68 12.44

4.29 5.29 5.08 14.66

6.05 3.46 7.49 17.00

6.44 4.52 9.55 20.51

5.13 4.06 5.79 14.98

5.64 3.79 8.22 17.65

6.79 3.34 5.38 15.51

6.76 4.63 5.95 17.34

5.91 4.63 7.43 17.97

5.96 3.55 6.80 16.31

5.55 3.69 6.20 15.44

5.38 3.40 5.57 14.35

5.71 4.94 7.09 17.74

4.28 2.90 9.57 16.75

6.35 3.74 7.80 17.89

3.90 4.76 9.72 18.38

----------------- ------------ ------------ ------------

mean mean mean mean

5.54 3.99 6.94 16.46

---------------- ------------ ------------ -------------

std std std std

1.08 .64 1.47 1.76

----------------- ------------ ------------ ------------

std error std error std error std error

.20 .12 .27 .32

"* mean : average value

"* std : standard deviation

"* std error : standard error of the mean

38
Appendix 2

Simulated average tow delays at 5% of capacity (30 replications)

(1) Trip Rate : 35% of capacity at Lock 22

Distance : 0.1 x actual distance

lock 22 lock 24 lock 25 Total System

------------------------------------------------------

32.30 26.42 32.35 91.07

36.25 26.26 30.49 93.00

33.59 27.11 32.83 93.53

36.73 26.99 30.98 94.70

34.10 28.01 30.50 92.61

33.43 27.29 31.52 92.24

35.54 27.07 34.30 96.91

31.96 26.10 29.10 87.16

37.85 27.05 31.38 96.28

33.87 27.90 30.07 91.84

34.85 26.63 31.17 92.65

35.22 25.79 32.59 93.60

34.83 28.31 32.18 95.32

33.61 26.28 31.72 91.61

34.49 27.25 35.09 96.83

33.43 26.70 31.02 91.15

35.53 25.73 31.88 93.14

37.89 27.84 32.30 98.03

32.53 25.54 29.85 87.92

33.28 27.12 35.31 95.71

31.63 24.C5 30.27 86.75

36.41 26.52 31.91 94.84

34.65 27.05 31.23 92.93

34.08 27.58 30.68 92.34

35.57 28.57 31.77 95.91

34.68 27.20 33.53 95.41

37.96 27.23 30.36 95.55

35.81 27.64 34.54 97.99

34.37 27.35 32.83 94.55

34.89 27.13 32.25 94.27

------------------------------------------------------

mean mean mean mean

34.71 26.95 31.87 93.53

------------------------------------------------------

std std std std

1.64 .82 1.52 2.81

---------------- ------------ ------------ ------------

std error std error std error std error

.30 .15 .28 .51

* mean: average value

* std : standard deviation

* std error: standard error of the mean

39
(2) Trip Rate : 35% of capacity at Lock 22

Distance : 1 x actual distance

lock 22 lock 24 lock 25 Total System

---------------- ------------ ------------ ------------

34.66 28.37 33.38 96.41

35.91 29.09 30.67 95.67

32.89 28.11 33.88 94.88

36.07 28.43 31.80 96.30

35.93 29.45 30.74 96.12

33.53 27.22 31.78 92.53

34.37 28.83 34.65 97.85

32.59 27.63 31.04 91.26

36.19 27.66 31.88 95.73

36.36 27.80 30.68 94.84

35.13 28.40 30.32 93.85

34.71 28.40 34.05 97.16

35.19 30.20 32.98 98.37

33.57 27.49 33.54 94.60

36.36 27.88 37.03 101.27

34.05 29.56 31.25 94.86

35.05 27.72 33.40 96.17

37.63 29.95 34.02 101.60

33.97 27.46 30.18 91.61

35.17 29.26 35.14 99.57

33.46 27.06 31.49 92.01

36.85 28.32 31.67 96.84

35.38 27.51 31.28 94.17

33.08 28.81 30.94 92.83

35.30 29.73 32.95 97.98

35.39 29.07 33.11 97.57

36.99 29.59 31.46 98.04

34.50 28.74 33.78 97.02

33.44 28.14 34.95 96.53

36.03 28.22 33.07 97.32

---------------- ------------ ------------ ------------

mean mean mean mean

34.99 28.47 32.57 96.03

---------------- ------------ ------------ ------------

std std std std

1.29 .84 1.65 2.52

------------------------------------------------------

std error std error std error std error

.23 .15 .30 .46

"* mean: average value

"* std: standard deviation

"* std error : standard error of the mean

40
(3) Trip Rate : 35% of capacity at Lock 22

Distance : 10 x actual distance

lock 22 lock 24 lock 25 Total System

---------------- ------------ ------------ ------------

33.96 30.18 33.93 98.07

35.32 31.41 30.43 97.16

35.01 30.31 35.61 100.93

36.86 29.36 32.02 98.24

37.32 31.06 30.00 98.38

34.29 29.68 31.32 95.29

34.38 29.98 32.78 97.14

35.53 30.89 32.50 98.92

38.12 29.52 33.09 100.73

35.30 28.27 31.74 95.31

34.35 30.75 31.30 96.40

35.17 27.65 34.71 97.53

33.22 32.11 33.63 98.96

33.95 30.38 34.05 98.38

35.88 29.83 35.49 101.20

34.49 29.04 30.63 94.16

36.28 33.03 31.12 100.43

36.79 30.28 33.27 100.34

34.95 30.39 30.42 95.76

34.81 30.35 35.43 100.59

33.29 28.85 32.14 94.28

37.68 28.90 34.56 101.14

36.29 28.71 33.90 98.90

35.12 28.34 33.03 96.49

35.16 29.77 32.49 97.42

35.35 30.33 30.68 96.36

39.34 30.44 32.41 102.19

36.78 29.40 33.66 99.84

35.81 29.71 35.41 100.93

35.91 30.01 33.38 99.30

---------------- ------------ ------------ ------------

mean mean mean mean

35.56 29.96 32.84 98.36

---------------- ------------ ------------ ------------

std std std std

1.39 1.11 1.63 2.18

---------------- ------------ ------------ ------------

std error std error std error std error

.25 .20 .30 .40

"* mean : average value

"* std : standard deviation

"* std error : standard error of the mean

41
(4) Trip Rate : 35% of capacity at Lock 22

Distance : 100 x actual distance

lock 22 lock 24 lock 25 Total System

36.20 29.69 33.61 99.50

37.88 30.66 31.68 100.22

34.45 28.78 33.33 96.56

34.40 30.58 33.27 98.25

37.92 33.25 33.01 104.18

33.27 28.67 32.15 94.09

35.50 28.80 35.40 99.70

35.11 28.87 31.01 94.99

33.86 29.62 33.94 97.42

34.86 28.76 33.32 96.94

34.58 30.52 31.66 96.76

35.80 30.39 34.08 100.27

35.60 30.81 34.44 100.85

32.92 27.09 33.69 93.70

36.56 30.69 34.55 101.80

32.75 28.52 32.18 93.45

38.03 30.92 31.05 100.00

36.93 31.84 33.25 102.02

35.96 30.67 30.70 97.33

34.88 31.08 34.42 100.38

32.86 27.80 31.90 92.56

36.11 29.63 33.43 99.17

36.24 30.15 32.13 98.52

35.28 29.98 32.07 97.33

35.67 28.67 32.17 96.51

34.21 30.06 31.23 95.50

38.75 28.86 31.01 98.62

35.45 30.80 32.59 98.84

34.12 29.99 35.60 99.71

33.89 30.87 30.45 95.21

mean mean mean mean

35.33 29.90 32.78 98.01

std std std std

1.55 1.24 1.37 2.73

std error std error std error std error

.28 .23 .25 .50

* mean : average value

* std : standard deviation

* std error : standard error of the mean

42
Appendix 3

Simulated average tow delays at 65% of capacity (30 replications)

(1) Trip Rate : 65% of capacity at Lock 22

Distance : 0.1 x actual distance

lock 22 lock 24 lock 25 Total System

101.48 79.78 90.77 272.03

102.41 78.23 84.54 265.18

98.43 81.33 94.60 274.36

107.31 79.13 89.69 276.13

105.57 78.55 81.85 265.97

92.93 73.20 83.53 249.66

95.15 74.13 84.11 253.39

96.00 73.98 81.82 251.80

112.69 80.55 88.20 281.44

105.37 78.33 85.12 268.82

100.84 75.46 87.27 263.57

99.64 73.63 88.15 261.42

104.26 78.68 83.59 266.53

99.33 77.55 91.48 268.36

102.95 81.04 92.33 276.32

99.61 76.76 82.32 258.69

104.65 81.66 87.85 274.16

108.54 78.88 85.05 272.47

101.85 73.91 82.33 258.09

102.03 80.70 92.63 275.36

95.57 73.66 85.54 254.77

107.83 74.06 83.75 265.64

102.47 75.70 89.45 267.62

98.77 75.38 82.00 256.15

99.50 77.73 85.17 262.40

103.55 75.73 87.23 266.51

103.78 76.16 81.66 261.60

99.44 74.80 92.85 267.09

99.96 70.25 87.90 258.11

105.80 79.48 88.20 273.48

mean mean mean mean

101.92 76.95 86.70 265.57

std std std std

4.26 2.87 3.72 7.79

std error std error std error std error

.78 .52 .68m 1.46

"* mean : average value

"* std : standard deviation

"* std error: standard error of the mean

43
(2) Trip Rate : 65% of capacity at Lock 22

Distance : actual distance

lock 22 lock 24 lock 25 Total System

----------------- ------------ ------------ ------------

107.66 86.40 95.05 289.11

106.99 87.82 88.41 283.22

100.27 86.84 95.58 282.69

109.89 83.77 95.87 289.53

109.09 84.56 86.43 280.08

96.74 80.11 86.60 263.45

99.69 79.39 86.38 265.46

104.77 81.69 85.42 271.88

117.64 85.67 90.63 293.94

114.31 82.70 90.12 287.13

108.33 84.24 93.06 285.63

100.64 78.20 94.70 273.54

108.35 86.76 88.04 283.15

104.71 82.13 97.18 284.02

110.36 87.77 97,02 295.15

102.45 82.86 86.64 271.95

108.76 86.45 91.21 286.42

111.72 84.80 89.80 286.32

110.77 79.31 86.97 277.05

107.77 86.94 97.35 292.06

102.31 80.22 88.74 271.27

110.16 82.67 88.02 280.85

108.29 79.57 92.28 280.14

104.50 80.07 84.73 269.30

102.47 85.36 91.43 279.26

108.02 81.39 89.92 279.33

105.30 83.56 86.78 275.64

103.63 82.54 94.22 280.39

106.04 79.48 90.69 276.21

112.45 85.41 91.86 289.72

----------------- ------------ ------------ ------------

mean mean mean mean

106.80 83.29 90.70 280.80

----------------- ------------ ------------ ------------

std std std std

4.53 2.84 3.75 8.00

----------------- ------------ ------------ ------------

std error std error std error std error

.83 .52 .68 1.46

* mean: average value

* std : standard deviation

* std error : standard error of the mean

44
(3) Trip Rate : 65% of capacity at Lock 22

Distance : 10 x actual distance

lock 22 lock 24 lock 25 Total System

113.22 98.70 101.51 313.43

115.20 101.56 98.61 315.37

112.81 100.90 101.65 315.36

115.57 99.01 96.56 311.14

117.15 93.41 97.55 308.11

109.33 87.53 89.43 286.29

106.72 91.65 96.01 294.38

114.10 94.01 99.14 307.25

119.09 101.84 100.53 321.46

115.97 94.57 90.79 301.33

111.26 101.46 98.64 311.36

112.23 94.97 96.89 304.09

112.96 96.76 96.60 306.32

113.06 98.41 100.08 311.55

114.49 97.13 100.98 312.60

113.45 93.29 92.42 299.16

114.10 97.33 97.55 308.98

117.95 94.17 95.00 307.12

116.48 90.83 93.20 300.51

114.06 98.57 104.96 317.59

107.50 93.95 92.78 294.23

122.74 93.51 94A47 310.72

114.64 92.44 92.57 299.65

116.43 92.99 93.86 303.28

114.01 94.98 98.47 307.46

111.48 94.16 93.98 299.62

119.39 96.38 91.61 307.38

108.15 91.59 96.31 296.05

110.07 91.23 102.24 303.54

112.51 100.16 97.79 310.46

mean mean mean mean

113.87 95.58 96.74 306.19

std std std std

3.50 3.58 3.70 7.64

std error std error std error std error

.64 .65 .68 1.40

Smean : average value

* std : standard deviation

* std error : standard error of the mean

45
(4) Trip Rate : 65% of capacity at Lock 22

Distance : 100 x actual distance

lock 22 lock 24 lock 25 Total System

------------------------------------------------

111.32 100.56 97.05 30d.93

121.00 99.27 95.53 315.80

107.25 98.85 101.28 307.38

110.73 98.31 98.30 307.34

110.01 97.04 94.43 301.48

105.38 90.20 94.44 290.02

106.47 95.20 94.51 296.18

113.19 102.20 90.98 306.37

114.25 103.27 100.03 317.55

111.36 93.67 96.79 301.82

113.17 95.36 93.36 301.89

107.96 91.69 99.23 298.88

117.39 100.53 95.54 313.46

106.42 94.45 96.33 297.20

116.45 96.30 94.65 307.40

107.05 96.93 90.08 294.06

110.69 101.84 96.91 309.44

114.41 96.43 96.88 307.72

108.43 94.73 91.71 294.87

110.08 98.66 100.29 309.03

101.67 92.78 95.26 289.71

112.57 97.72 97.39 307.68

116.05 96.77 95.05 307.87

106.62 91.92 90.99 289.53

105.40 97.46 94.21 297.07

110.10 95.40 94.48 299.98

114.68 94.09 94.00 302.77

114.00 95.36 99.10 308.46

107.36 96.64 97.11 301.11

109.12 97.81 98.54 305.47

----------------- ------------ ------------ ------------

mean mean mean mean

110.69 96.71 95.82 303.22

----------------- ------------ ------------ ------------

std std std std

4.19 3.10 2.77 7.28

----------------- ------------ ------------ ------------

std error std error std error std error

.76 .57 .51 1.33

"* mean : average value

"* std : standard deviation

"* std error: standard error of the mean

46
Appendix 4

Simulated average tow delays at 95% of capacity (30 replications)

(1) Trip Rate : 95% of capacity at Lock 22

Distance : 0.1 x actual distance

lock 22 lock 24 lock 25 Total System

---------------- ------------ ------------ ------------

624.45 477.81 390.64 1492.90

798.69 504.71 454.41 1757.81

642.50 466.00 475.44 1583.94

742.37 457.81 430.48 1630.66

697.38 493.31 377.38 1568.07

680.39 364.21 350.03 1394.63

574.04 395.44 355.05 1324.53

707.62 434.43 391.80 1533.85

800.64 514.25 476.69 1791.58

746.49 393.78 380.79 1521.06

720.63 462.17 396.95 1579.75

683.77 473.61 406.68 1564.06

726.91 529.83 387.80 1644.54

691.44 451.61 403.11 1546.16

882.73 569.67 395.74 1848.14

676.43 356.05 345.31 1377.79

712.74 458.29 409.66 1580.69

722.54 532.11 449.49 1704.14

778.27 391.68 364.57 1534.52

704.11 505.22 414.47 1623.80

702.10 412.68 521.97 1636.75

689.67 450.74 377.98 1518.39

752.52 416.58 406.23 1575.33

578.22 393.34 369.57 1341.13

716.64 470.73 389.50 1576.87

665.10 441.95 369.59 1476.64

751.57 426.31 387.25 1565.13

665.63 450.43 484.87 1600.93

948.51 442.39 417.41 1808.31

723.96 494.98 424.33 1643.27

---------------- ------------ ------------ ------------

mean mean mean mean

716.94 454.40 406.84 1578.18 --------- --------- -------- ------------

std std std std

74.87 50.20 41.82 123.98

---------------- ------------ ------------ ------------

std error std error std error std error

13.67 9.16 7.63 22.64

"* mean: average value

"* std : standard deviation

"* std error : standard error of the mean

47
(2) Trip Rate : 95% of capacity at Lock 22

Distance : actual distance

lock 22 lock 24 lock 25 Total System

652.45 497.68 414.84 1564.97

828.91 556.82 478.58 1864.31

662.96 502.78 510.04 1675.78

775.40 485.59 454.98 1715.97

713.71 540.17 397.43 1651.31

712.20 384.52 369.34 1466.06

600.94 426.32 391.44 1418.70

727.58 456.33 416.76 1600.67

826.92 544.43 504.30 1875.65

762.34 423.25 401.13 1586.72

760.19 485.30 416.70 1662.19

682.26 495.62 434.17 1612.05

749.33 539.84 410.70 1699.87

724.02 459.42 424.32 1607.76

888.01 584.55 417.93 1890.49

698.46 369.99 364.21 1432.66

729.31 478.99 430.29 1638.59

729.36 551.66 462.82 1743.84

805.84 412.82 383.62 1602.28

710.15 524.30 429.76 1664.21

708.16 426.27 552.00 1686.43

722.71 454.19 411.89 1588.79

788.51 430.27 423.94 1642.72

607.63 410.51 395.01 1413.15

733.09 489.25 414.32 1636.66

669.03 463.39 391.96 1524.38

764.75 444.20 408.79 1617.74

688.22 464.61 496.37 1649.20

949.37 458.05 445.02 1852.44

749.86 514.82 449.56 1714.24

mean mean mean mean

737.39 475.86 430.07 1643.33

std std std std

72.63 52.65 42.55 122.10

std error std error std error std error

13.26 9.61 7.77 22.29

* mean: average value

* std : standard deviation

* std errc : standard error of the mean

48
(3) Trip Rate : 95% of capacity at Lock 22

Distance : 10 x actual distance

lock 22 lock 24 lock 25 Total System

---------------- ------------ ------------ ------------

689.74 549.73 455.38 1694.85

881.01 605.02 538.54 2024.57

714.45 578.39 544.25 1837.09

888.27 534.86 529.60 1952.73

740.05 613.49 483.10 1836.64

745.49 445.53 436.75 1627.77

692.93 501.90 442.96 1637.79

773.00 509.13 464.96 1747.09

894.26 612.08 554.47 2060.81

873.23 509.32 479.61 1862.16

802.84 560.04 465.73 1828.61

708.46 593.98 467.39 1769.83

858.83 609.89 489.76 1958.48

776.64 524.55 470.04 1771.25

916.39 692.51 502.8P 2111.70

769.38 474.91 397.! 1642.07

802.14 522.19 504.96 1829.29

854.03 626.72 517.85 1998.60

912.33 477.49 436.15 1825.97

785.39 597.00 494.14 1876.53

738.26 474.66 630.86 1843.78

759.70 557.64 482.14 1799.48

841.29 496.15 461.81 1799.25

687.25 471.40 444.47 1603.12

835.45 517.09 478.99 1831.53

676.20 519.75 466.21 1662.16

869.71 519.01 488.54 1877.26

790.72 508.61 551.46 1850.79

1063.55 536.07 510.04 2109.66

827.72 576.00 494.62 1898.34

---------------- ------------ ------------ ------------

mean mean mean mean

805.62 543.84 489.51 1838.97

---------------- ------------ ------------ ------------

std std std std

85.66 55.99 44.64 134.66

---------------- ------------ ------------ ------------

std error std error std error std error

15.64 10.22 8.15 24.59

* mean : average value

* std : standard deviation

* std error : standard error of the mean

49
(4) Trip Rate : 95% of capacity at Lock 22

Distance : 100 x actual distance

lock 22 lock 24 lock 25 Total System

977.50 676.46 496.40 2150.36

780.71 686.89 540.88 2008.48

824.33 600.16 668.05 2092.54

955.95 689.59 554.55 2200.09

934.33 665.27 488.65 2088.25

832.52 503.39 489.42 1825.33

764.69 651.04 487.71 1903.44

920.18 684.88 488.79 2093.85

857.02 796.42 604.53 2257.97

821.57 585.55 476.31 1883.43

887.42 606.37 596.00 2089.79

840.67 627.98 501.74 1970.39

851.37 661.30 539.57 2052.24

743.33 640.47 564.93 1948.73

1000.82 710.17 544.80 2255.79

693.99 558.20 455.70 1707.89

962.09 665.20 619.67 2246.96

895.74 786.02 542.88 2224.64

919.06 681.54 538.55 2139.15

862.81 723.42 618.15 2204.38

815.33 607.06 605.63 2028.02

837.13 586.65 498.82 1922.60

892.06 611.93 487.18 1991.17

804.13 522.79 534.20 1861.12

805.50 623.12 527.63 1956.25

795.79 667.42 499.07 1962.28

876.36 618.43 505.03 1999.82

671.39 618.12 520.79 1810.30

982.65 644.73 511.50 2138.88

752.81 714.29 534.24 2001.34

mean mean mean mean

851.98 647.16 534.71 2033.85

std std std std

82.19 64.52 49.78 140.93

std error std error std error std error

15.01 11.78 9.09 25.73

* mean: average value

* std : standard deviation

* std error : standard error of the mean

50
EFFECTS OF LOCK CONGESTION AND RELIABILITY

ON OPTIMAL WATERWAY TRAVEL TIMES

by

Melody D. M. Dai, Paul Schonfeld and

George Antle

Published in Transportation Research Record

Paper No. 930596

51
52
EFFECIS OF LOCK CONGESTION AND RELIABILITY ON

OPTIMAL WATERWAY TRAVEL TIMES

by Melody D.M. Dai, Paul Schonfeld and George Antle

Transportation Studies Center Working Paper 92-25

University of Maryland

College Park

Submitted for Presentation at the

72nd Annual Meeting of the Transportation Research Board

July 1992

53
EFFECTS OF LOCK CONGESTION AND RELIABILITY ON

OPTIMAL WATERWAY TRAVEL TIMES

by Melody D.M. Dai', Paul Schonfeld' and George Ant.'

Abstract

The congestion and variability of service times at locks significantly affect the cost and

reliability of waterway transportation. This paper considers the effects of lock congestion levels

and reliability on the operating cost of tows, assuming that tow operators have the opportumty

to optimize speed in response to the delays they have already experienced and the delays they

expect to encounter. The analysis method in this paper is useful for evaluating long-t-rm

consequences of lock improvements, as well as for optimizing speed from the viewpoint of

operators.

This analysis method optimizes tow operations in two stages. The first stage finds the

optimal speeds for each individual tow, re-optimizing the speed after every lock. The second

stage determines the optimal allowed delivery times and associated optimal speeds based on the

lock transit time distributions. The optimization is guided by a total cost objecti-'; function

which includes penalties for late deliveries.

A four-lock section on the Ohio River is used for a case study in which various

congestion levels and speed limits are tested. The resulting total cost functions are U-shaped

with respect to the allowed delivery times. At given congestion levels, the optimal allowed

delivery times and costs decrease as speed limits increase. The results also show how the

optimal allowed delive.y tmes ?nd costs increase as congestion becomes severe.

Associate Professor, Department of Transportation and Management Science, National

Cheng-Kung University, Tainan, Taiwan, Tel. 011-886-6-2757575-53200.

2 Associate Professor, Department of Civil Engineering, University of Maryland, College

Park, MD 20742, Tel. (301) 405-1954.

Chief, Navigation Branch, Institute for Water Resources, U.S. Army Corps of Engineers,

Casey Building, 22060-5586, Fort Belvoir, Virginia, Tel. (703)355-3083.

54
1. Introduct4m

The cost and reliability of waterway transportation is influenced considerably by

congestion and by the variability of service times at locks. These locks enable or at least greatly

facilitate navigation on waterways which may not be navigable otherwise. However, they may

also constitute severe bottlenecks on those waterways as volumes increase.

The effects of congestion on mean travel times have been explored in various previous

studies, including earlier parts of our work (e.g., Dai [2], Dai and Schonfeld [3]). The effects

of variability in lock service times (and in consumption rates by end users) on inventory

requirements and logistic costs was explored by Dai and Schonfeld [41. This paper considers the

effects of congestion levels and lock service time variability on the costs of operating barge tows,

assuming that operators have the opportunity to optimize speed in response to expected delays.

The analysis method presented here is intended for use in evaluating the long-term consequences

of lock improvements, as well as for optimizing speeds in the short-term from the viewpoint of

carriers. For the latter purpose, a more precise operating cost function than that provided so far

would be required.

Although speed and reliability of travel times are less important for waterway

transportation than for some other transportation modes, these variables significantly affect the

costs of carriers and shippers. As speeds increase, power requirements and fuel costs increase

far more than linearly but other costs such as labor and equipment depreciation per output unit

(e.g. per ton mile) decrease. Higher speeds may also reduce the probability of late deliveries.

Late deliveries impose significant costs on customers by increasing stock-outs as well as required

inventories [4]. The costs associated with late deliveries may, at least sometimes, be passed

55
along to tow operators in the form of penalties. As speeds increase, it is also possible to reduce

the time allowed for deliveries, which presumably includes a scheduled travel time based on

expected sped and delays and a safety factor based on the variances of total travel time,

including delays. Alternatively, for a given allowed delivery time, increased speed reduces the

probability of late deliveries and penalties. Therefore, the optimal speeds and optimal allowed

delivery times should be compromises among the costs of fuel, fleet and penalty costs. A total

cost function formulated below is used to optimize tow speeds and allowed delivery times under

changing traffic conditions.

The complete delivery time includes tow travel times and lock transit times (including

lock service times and waiting times). In this study, the lock transit times are treated as

probability distributions derived from a waterway simulation model [2] which has been validated

against PMS (lock Performance Monitoring System) [6] data. Thus, each tow experiences a

different transit time even at same lock. Such an assumption, which increases somewhat the

complexity of this study, is more realistic than using only the average lock transit times. In

addition, it allows us to study the relations among penalty costs, transit times, speeds and allowed

delivery times which affect system reliability.

A case study of a four lock section on the Ohio River is presented below. Various

volume/capacity (V/C) ratios and speed limits are tested. The total cost functions are U-shaped

functions of the allowed delivery times. The results show how various volume to capacity (V/C)

ratios and speed limits influence the optimal allowed delivery times and operating speeds.

56
2. Literature Review

Several analytic models and simulation models are available for estimating lock transit

times. The characteristics of these models are discussed in this section.

2.1 Lock Delay Models

Two models based on the application of queuing theory have been found for estimating

lock delays. DeSalvo and Lave modeled lock operation as an M/M/1 (Poisson arrivals/

exponential service times/i server) queuing station [5]. Wilson modified this model as an M/G/i

(Poisson arrivals/General service times/i server) queue [9]. Thus, the difference between these

two models is the service time distribution. DeSalvo and Lave assumed that service times were

exponentially distributed while Wilson relaxed this assumption by using generally distributed

service times, which are more realistic. However, both models are limited to Poisson distributed

arrivals, which may not be realistic even if locks are isolated and are quite unreasonable for

closely spaced locks. In addition, the delays in both models were only analyzed for independent

single-chamber locks without stalls (failures). In waterways, many locks may have two dissimilar

chambers in parallel and the delays at adjacent locks may be highly related. Moreover, the stall

occurrences interrupt lock operations and thus increase delays. Therefore, it is desirable to use

a model which can analyze lock delays for entire interrelated series of locks and predict the

effects of stalls and dissimilar parallel chambers.

57
2.2 Waterway Simulation Models

Howe [7] developed a system simulation model for analyzing lock delays and tow travel

times. The service times in this model were based on empirically-determined frequency

distributions. To avoid some troublesome problems and errors associated with the requirement

to balance long-run flows in Howe's model, Carroll and Bronzini [I] developed another waterway

system simulation model. Both models simulate waterway operations in detail but require

considerable amounts of data and computer time, which limit their applicability for problems with

large networks and numerous combinations of improvement alternatives. Both models assumed

Poisson distributions for tow trip generation, which is not always realistic. More importantly for

reliability analyses, neither of these models explicitly accounts for stalls, which are very different

in frequency and duration from other events and affect overall transit time reliability.

Hence a waterway simulation model that explicitly accounts for stalls is desirable for

evaluating the impacts of waterway reliability.

3. Simulation Model

A simulation model developed for related waterway studies (Dai [2], Dai and Schonfeld

[3]) is used in this work. It may be used to determine the relations among delays, tow trips,

distributions of generated tow trips, lock operations, lock service time distributions, travel times,

as well as coal consumption and inventories at power plants supplied by tows. This simulation

model, which is developed on the basis of PMS (lock Performance Monitoring System) data [6],

can take into account stochastic effects such as stalls, randomly distributed arrivals and service

times as well as seasonal variations.

58
It is a microscopic, event-scanning simulation model. It traces the movement of each

individual tow and records its characteristics. It can handle any distributions for trip generation,

travel speeds, lock service times and tow sizes. These distributions can be specified for each

interval in tables or by standard statistical distributions. Currently, travel speeds are assumed to

be normally distributed, while general distributions based on empirical observations are used for

other input variables. Tows are allowed to overtake other tows. A FIFO (First-In-First-Out)

service discipline is currently employed. The model simulates two-way traffic through common

servers and accounts for stalls.

This simulation model is programmed in Fortran-77, which allows us to simulate

relatively complex operations. The size of waterway systems that can be modeled is only limited

by the computer capacity and the storage capacity of the Fortran compiler or linker. The

simulation model has been developed with "dynamic dimensioning" to the degree allowed by the

computer system available. Parameter statements are used so that the dimensions, and hence

capacities, of the model components may readily be modified. This allows the maximum

flexibility of waterway system representation and the most efficient computer utilization. Thus,

the dynamic dimensioning programming technique allows flexibility in the number of locks,

chambers, cuts, waterway links, tows, utility plants, origin-destination (O-D) pairs and simulation

time periods.

Detailed descriptions and vaidation results for this simulation model are provided in Dai

(2].

59
4. Methodology

4.1 Background

For this analysis it may be assumed that operators dispatch tows to meet certain delivery

deadlines. If tows fail to meet their deadlines, penalties must be paid for the excess time.

However, if operators try to avoid penalties by increasing their speeds, they incur higher fuel

costs, which increase disproportionately with speed. Also, when speeds increase, the total

delivery times decrease, thereby reducing fleet costs (i.e., equipment depreciation and labor).

Thus, the operating speeds must be optimized through trade-offs among fuel costs, penalty costs,

and fleet costs.

In this study, it is assumed that tow boats always try to operate at the most economic

speeds to minimize the total costs. However, since lock transit times are somewhat uncertain,

the speed for the remainder of a trip should be re-optimized after passing each lock, i.e., after

that part of the uncertainty has been resolved. The new speeds would be based on the scheduled

delivery times, expected remaining total lock transit times (including waiting times in queues and

service times at locks), the remaining travel distance, unit fuel costs, unit penalty costs, and unit

fleet costs. Sometimes, however, the most economic speeds may not be attainable due to

physical or regulatory constraints.

This study assumed that tows with the same origin and destination have the same

maximum allowed delivery times. The tow operator may optimize these by deciding how far in

advance of the delivery deadline to dispatch a tow, i.e., how much travel time should be prescheduled. It is noted that the fleet size and the required delivery frequency affect the maximum

allowed delivery times. When the delivery frequency remains constant, a larger fleet can allow

60
longer delivery times. When the fleet size remains constant, reduced delivery frequencies allow

longer delivery times. As allowed delivery times increase, the probability of missed delivery

deadlines decreases and optimal tow operating speeds may also decrease. Therefore, long

allowed delivery times require large fleet costs, but small fuel costs and penalty costs. This study

aims to help carriers determine the optimal allowed delivery times and the optimal speeds

between locks. The optimization is conducted in two stages. The first stage finds the optimal

speeds for each individual tow, re-optimizing the speed after every lock. The second stage

determines the optimal allowed delivery times for all tows serving a given origin destination pair.

The optimal speeds and optimal allowed delivery times are determined based on the total cost

function.

4.2 Optimal Speeds

The optimal speeds are determined for each individual tow boat after passing each lock

since different tow boats may experience different lock transit times. The total cost function for

optimizing speeds of the remaining delivery includes three components: (1) fuel cost, (2) penalty

cost, and (3) fleet cost.

=CCf. + P + C(1)

where

Cq: total cost for the remaining delivery distance

C : fuel cost for the remaining delivery distance

Cp penalty cost for the remaining delivery distance

Cg: fleet ccst for the remaining delivery distance

61
The fuel cost increases as the speed increases. In this study, it is assumed that the unit

fuel cost is proportional to the square of the speed. The fuel cost for the remaining delivery

distance can be represented as:

CN = c,(V) DR (2)

where

ch(V) : unit fuel cc st when a tow is operating at speed V, $/tow-mi

V : speed, mi/hr

DR : remaining travel distance, mi

The unit fuel cost is assumed to be a constant k multiplied by the square of the speed:

ch(V) = k V2 (3)

The constant k is estimated from cost values in the Reebie [8] study of barge operating

costs. An improved cost fur•:tion may easily be substituted in this model when it becomes

available.

The penalty cost is charged when the delivery time exceeds the deadline, and thus can be

represented as:

Cp = cp B max[(DR/V - TR), 0] (4)

where

cp : unit penalty cost, $/barge-hr

B : tow size, barges/tow

"T1 : expected remaining travel time, hr

TR = TA - TU-T (5)

TA : allowed delivery time, hr

62
Tu : used delivery time, hr

Tut : expected remaining lock transit time, hr

The fleet cost for the remaining delivery is assumed to be proportional to the remaining

delivery times and can be represented as follows:

Ca = c. B (D,/V + TR) (6)

where

c 8 : unit fleet cost, $/barge-hr

Therefore, Eq. 1 can be expanded as a constrained minimization problem:

minimize Cý = k V2 DR + cP B maxAID/V - Tv, 0] + ca B (DR/V + Tu) (7)

subject to V • Vm~x (8)

V k V•.W (9)

where

Vy•x :maximum speed limit

Vm: minimum speed limit

The maximum speed limits may be based on mechanical limitations or policy constraints.

For generality, minimum speed limits may also be imposed but that is usually unnecessary for

waterways. Therefore, the minimum speed is assumed to be zero in this study.

To solve we must consider whether penalties (1) should be paid for the sake of more

economical speeds or (2) should not be accepted. Assuming penalties are acceptable, without

considering the constraints of speed limits, the optimal speed V, is obtained by setting the

deriviative of Cq (Eq 7) equal to zero and solving for V:

V, = [(c, + cn)/(2 k)]' (10)

63
The above solution assumes some penalty is paid. However, it is necessary to check whether V,

is less than the critical speed Vcv, which represents the boundary speed between paying or not

paying penalties. Tows need not pay penalties when their speeds exceed Vt. The critical speed

Vat can be represented as follows:

Vcx - D/Tt 11

If penalties are not paid, and without considering the constraints of speed limits, the optimal

speed V2 can be obtained by differentiating the total cost function (Eq. 7), setting it equal to zero

and solving for V. Then the optimal speed is

V2 = [cd(2 k FL)]'0 (12)

It is also necessary to check whether V2 is greater than Vcp In addition, both Vt and V2 need

to satisfy the speed limits. Otherwise, tows operate at the nearest feasible speed limits.

4.3 Optimal Allowed Delivery Times

The optimal speeds derived in the previous section are for individual tows with a specific

allowed delivery time. Different allowed delivery times require different optimal speeds, and

hence, the associated total costs are different. This section discusses the model for finding the

optimal allowed delivery times for tows moving between the same origins and destinations. It

is noted that the costs in this model represent the optimal costs for each individual tow with a

certain delivery deaiiine. The total cost function can be represented as follows:

Cr =CPU + C +CFL (13)

where

Cr: total cost for tows connecting a given origin-destination pair

64
C.Fu : fuel cost for tows connecting a given origin-destination pair

C. : penalty cost for tows connecting a given origin-destination pair

CIL : fleet cost for tows connecting a given origin-destination pair

To estimate the total costs, it is necessary to know the operating speeds and delivery

times. This study assumes that each individual tow may experience different lock transit time

at the same lock, based on observed or simulated distributions. Therefore, the operating speeds

and delivery times are different for each individual tow. This assumption increases the difficulty

of estimating the total costs unless the movement of each tow is traced by simulation. To avoid

the time-consuming simulation, this study splits tows into several groups. The tows in each

group have similar delivery times. Then, optimal speeds and travel times are determined for each

group. It is noted that the tows in same group may experience different lock transit times at the

next lock. Therefore, after passing each lock, it is necessary to regroup the tows. Equivalently,

the service time distribution is divided into n intervals (n is usually 10 here) and tows are redistributed into those n intervals after each lock.

The total fuel costs for all tows in this service can be represented as

CF = Ti cf.(Vij) MK (14)

where

Vii : average speed of Speed Group j in Segment i, mi/hr

Mj : total barge miles for Speed Group j in Segment i, barge-mi/hr

Mj = Q•JFL BIL, (15)

Q) : number of tows in Speed Group j in Segment i, tows/mi

LI : length of Segment i, mi

65
The penalty costs are charged when tows are laze, resulting in total penalty costs C, for the entire

fleet of

Cp c. ;T~jQ4 B (16)

where

TO : number of average late hours/tow for tows in Speed Group j

Q4: number of tows/br in Speed Group j in the last Segment I

The fleet operating costs C, are represented as follows:

C. =cU c QZ B (TA + TDj) (17)

Therefore, the total costs in Eq. 13 can be expanded as follows:

Cr = I ; c&(VI) Qj FL B LI +ccp; TDj Q B + c. ;QU B (TA + TDI) (18)

The optimal allowed delivery time is that which minimizes the above total cosL

4.4 Lock Transit Times

To optimize speeds and allowed delivery times, we must consider lock transit times as

well as travel times between locks. For the same allowed delivery times, if the lock transit times

increase, the-allowed travel times decrease, thus increasing expected penalties.

The lock transit times include two components: (1) lock service times and (2) waiting

times. This model uses information on the distributions of lock transit times as well as the mean

("expected") lock transit times. The expected lock transit times are useful when the tow

operators want to adjust speeds for the remaining distance. At the time operators adjust their

speeds, the remaining lock transit times are still uncertain. They can adjust their speeds based

66
ca estimated probable transit times, which are in turn based on prior data. The distributions of

lock transit times are useful for estimating the lock transit time for each individual tow.

In this study the lock transit times are estimated by simulating a series of locks and

recording the transit times for each tow. Therefore, the simulation results can provide the total

transit times at each lock for each individual tow and may be used to estimate averages and

probability distributions. To adjust speeds for the remaining distance, it is necessary to know the

expected total transit times through all the remaining locks between the current locations and

destinations.

The distribution of lock transit times at each lock is specified in table form for several

groups (10 groups in this study). The average lock transit time and the probability table of lock

transit times for each group are also obtained from the simulation.

5. Case Study

A four lock section of the Ohio River around the Gallipolis Lock was selected for a case

study since that lock constitutes a relative bottleneck in the waterway capacity. Compared to the

other three locks nearest to it (Belleville, Racine, and Greenup), Gallipolis is the oldest and its

two chambers are the smallest.

The Stuart utility plant of the Dayton Power and light Co., located between the Greenup and

Meldahl locks, is chosen for this case study. It is 63.5 miles downstream from Greenup and 31.7

miles upstream from Meldahl.

There are five segments in this study. The segment characteristics are shown in Table

5-1 and the lock transit times are shown in Table 5-2.

67
Table 5-1 Segment Characteristics

Segment From To Length (mi)

1 Origin Belleville 21.1

2 Belleville Racine 33.6

3 Racine Gaflipolis 41.7

4 Gallipolis Greenup 61.8

5 Greenup Destination 63.5

Table 5-2 Lock Transit Tunes

Lock Transit Time (hr)

Lock V/C=0.45' V/C=0.80' V/C=0.95'

AVG STD AVG STD AVG STD

Belleville 0.83 0.37 0.92 0.50 0.96 0.54

Racine 1.10 0.98 1.57 2.09 1.75 2.49

Gallipolis 3.92 3.66 11.83 11.05 48.84 39.52

Greenup 0.79 0.36 0.92 0.52 0.97 0.58

I. The V/C ratio is measured at Gaflipolis Lock

The baseline values for the other variables are listed in Table 5-3.

Table 5-3 List of Variables and Baseline- Values

Variable Description Value

B aveage tow size, barges/tow 8.04

ca unit fleet cost, $Ibarge-hr 28.56'

cp unit penalty cost, $/barge-hr 50

k parameter for unit fuel cost 0.022961

Vm minimum speed limit, mi/hr 0

1. Source [81

The case study finds the optimal speeds and optimal allowed delivery times for different

combinations of speed limits and volume to capacity ratios (V/C). The associated costs also are

provided in this case study. Tables 5-4 lists the optimal costs for various allowed delivery times

and speed limits when the volume/capacity (V/C) ratio at the critical Gallipolis lock is 0.80.

68
Table 5-5 shows the optimal costs for various V/C ratios at Gallipolis Lock. The optimal costs

for various standard deviation of lock service times are listed in Table 5-6. In Table 5-6, the

ratio of standard deviation of lock service times is measured against the baseline case when V/C

ratio is 0.80 at Gallipolis Lock.

Figure 5-1 shows the relation between the total costs and the allowed delivery times for

different speed limits when the V/C ratio is 0.80. The total costs are U-shaped curves. They

first decrease and then increase as the allowed delivery times increase. In fact, when the allowed

delivery times are small, tows have higher speeds and fuel costs and, possibly, higher penalty

costs although fleet costs may decrease. At first, as allowed delivery times increase, the fuel and

penalty savings outweigh more than the extra fleet costs. However, beyond a certain range, the

marginal benefit decreases and is outweighed by the increased fleet costs associated with the

increased allowed delivery times. This graph also shows that as the speed limits increase, the

optimal allowed delivery times and associated total costs decrease until reaching a limit. When

speed limits are low, tows may have to operate below their most economic speeds. Therefore,

relaxation of speed limits may sometimes reduce total costs.

Figure 5-2 shows the optimal total costs for different V/C ratios. In this graph, the

optimal total costs increase as the V/C ratios increase. In Table 5-2, as V/C ratios increase, the

congestion becomes severe and the mean and standard deviation of lock transit times increase.

Especially when the V/C ratio is 0.95 at the Gallipolis Lock, the mtn and standard deviation

of lock transit times increase substantially. The standard deviation of lock transit times is an

indicator of lock reliability. The higher the standard deviation of lock transit times, the less

69
reliable the lock is. Therefore, as V/C ratios increase, the variance as wel as the mean of the

transit times increase, thus reducing service reliability and increasing delivery times and costs.

Figure 5-3 shows the optimal total costs for various standard deviations of lock service

times. In it, the optimal total costs increase as the standard deviations, and thus the unreliability,

of lock service times increase. As the standard deviation of lock service times increases, the

overall travel time reliability is reduced, and thus, the optimal total cost increases.

70
Table 5-4 Optimal Costs for Different Allowed Delivery Times and Speed Limits (V/C=0.80)

TA V1 MA CFU C? CIF c

(hr) (mi/br) (S/tow) (S/tow) (S/tow) (S/tow)

24 6.3 1988.57 10740.64 11645.99 24375.19

36 6.3 1988.57 5916.64 11645.99 19551.19

48 6.3 1988.57 2161.96 12256.79 16407.31

60 6.3 1988.57 788.72 14227.86 17005.15

72 6.3 1988.57 248.84 16674.95 18912.35

84 6.3 1988.57 .00 19288.28 21276.85

96 6.3 1988.57 .00 22043.75 24032.32

108 6.3 1988.57 .00 24799.22 26787.79

120 6.3 1988.57 .00 27554.69 29543.25

24 8.3 3532.40 7176.10 9609.93 20318.43

36 8.3 3467.30 2849.38 9893.97 16210.64

48 8.3 3237.95 1086.20 11642.31 15966.47

60 8.3 3212.40 39878 14005.13 17616.31

72 8.3 3193.23 .00 16532.81 19726.04

84 8.3 3193.23 .00 19288.28 22481.51

96 8.3 3193.23 .00 22043.75 25236.98

108 8.3 3193.23 .00 24799.22 27992.45

120 8.3 3193.23 .00 27554.69 30747.92

24 10.4 5527.33 5037.46 8388.33 18953.13

36 10.4 4754.28 1809.40 9299.94 15863.62

48 10.4 3466.02 842.99 11503.39 15812.41

60 10.4 3325.15 278.18 13936.24 17539.58

72 10.4 3193.23 .00 16532.81 19726.04

84 10.4 3193.23 .00 19288.28 22481.51

96 10.4 3193.23 .00 22043.75 25236.98

108 10.4 3193.23 .00 24799.22 27992.45

120 10.4 3193.23 .00 27554.69 30747.92

24 12.5 6226.45 4538.98 8103.60 18869.04

36 12.5 5000.28 1663.88 9216.81 15880.97

48 12.5 3548.95 784.30 11469.87 15803.12

60 12.5 3366.61 248.44 13919.25 17534.30

72 12.5 3193.23 .00 16532.81 19726.04

84 12.5 3193.23 .00 19288.28 22481.51

96 12.5 3193.23 .00 22043.75 25236.98

108 12.5 3193.23 .00 24799.22 27992.45

120 12.5 3193.23 .00 27554.69 30747.92

24 14.6 6226.45 4538.98 8103.60 18869.04

36 14.6 5000.28 1663.88 9216.81 15880.97

48 14.6 3548.95 784.30 11469.87 15803.12

60 14.6 3366.61 248.44 13919.25 17534.30

72 14.6 3193.23 .00 16532.81 19726.04

84 14.6 3193.23 .00 19288.28 22481.51

96 14.6 3193.23 .00 22043.75 25236.98

108 14.6 3193.23 .00 24799.22 27992.45

120 14.6 3193.23 .00 27554.69 30747.92

71
Table 5-5 Optimal Costs for Different V/C Ratios

V/c Cr

($/tow)

0.45 11902.66

0.80 15803.12

0.95 31737.85

Table 5-6 Optimal Costs for Different Standard Deviations of Lock Service Times

Ratio of S.D. Cr

($/tow)

0.95 15741.94

0.98 15760.91

1.00 15803.12

1.03 15816.03

1.07 15874.33

1.09 15942.45

1. Ratio of S.D.: Ratio of standard deviations of service times to the baseline case (V/C=0.80)

6. Conclusions

A methodology is developed for analyzing how tow operating costs are affected by

operator-controlled variables such as speeds and allowed delivery times as well as exogenous

factors such as congestion levels, lock service time means and variances, fuel cost rates and other

operating cost factors. Speeds and allowed delivery times, which are important decision factors

for carriers, may be optimized with this approach. The total cost function for optimizing speeds

and allowed delivery times includes fuel costs, penalty costs and fleet costs.

The optimization of speeds and allowed delivery times is conducted in two stages. The

first stage finds the optimal speeds associated with different allowed delivery times and speed

72
limits. The optimal speeds are determined for each individual tow after passage through each

lock. The second stage determines the optimal allowed delivery times based on the optimal

speeds obtained in Stage 1.

A four-lock section constrained by the Gallipolis Lock was selected for a case study. The

results show that the total costs are U-shaped functions of the allowed delivery times. As the

speed limits increase, the optimal allowed delivery times and costs decrease toward asymptotic

limits. In addition, as V/C ratios increase, the optimal allowed delivery times and costs also

increase.

This methodology is useful for evaluating tow operations with uncertain lock transit times.

It can be improved in the following aspects:

1. Improved cost functions should ae developed.

2. Penalty costs should be related to actual inventory or stock-out costs incurred by

shippers.

3. The effects of service reliability on demand should be analyzed.

References

1. Carroll, J.L and Bronzini, M.S., "Waterway Transportation Simulation Models: Development

and Application", Waterway Resources Research, 9, No. 1, pp. 51-63, 1973.

2. Dai, M.D.M., Delay Estimation on Congested Waterways, Ph.D. Dissertation, University of

Maryland, College Park, 1991.

3. Dai, M.D.M. and Schonfeld, P., "Metamodels for Estimating Waterway Delays Through Series

of Queues", forthcoming in Transportation Research Part B.

73
4. Dal, M.D.M. and Schonfeld, P., "Simulation of Waterway Transportation Reliability",

Transportation Research Record, No. 1313, pp. 98-105, 1991.

5. DeSz•_,•, J.S. and Lave, L.B., "An Analysis of Towboat Delays", J. Transpn Econ. Policy, 2,

pp. 232-241, 1968.

6. Fleming, M., Wood, D., and Goodwin, R., Lock Performance Monitoring System User's

Manual for Data Analysis. U.S. Army Corps of Engineers, Institute for Water Resources, Fort

Belvoir, Virginia, 1985.

7. Howe, C.W., et al. Inland Waterway Traw portation Studies in Public and Private

Managemem and Investment Decisions, Johns Hopkins Press, Baltimore, 1969.

8. Barge CostlService Analysis Model, User Reference Manual, Reebie Associates, July 1989.

9. Wilson, H.G., "On the Applicability of Queueing Theory to Lock Capacity Analysis", Transpn

Res, 12, pp. 175-180, 1978.

Appendix - List of Variables

B : tow size, barges/tow

CF: fleet cost for tows connecting a given origin-destination pair

Cvu :fuel cost for tows connecting a given origin-destination pair

Cp penalty cost for tows connecting a given origin-destination pair

Cr: total cost for tows connecting a given origin-destination pair

Ca fleet cost for the remaining delivery distance

Cf: fuel cost for the remaining delivery distance

Cp: penalty cost for the remaining delivery distance

74
q: total cost for the remaining delivery distance

c. unit fleet cost, $/barge-hr

ct.(V) : unit fuel cost when a tow is operating at speed V and fully loaded, $/barge-mi

cp : unit penalty cost, $/barge-hr

D• : remaining distance to travel, mi

Lj : length of Segment i, mi

Mj : total barge mile for Speed Group j in Segment i, barge-mi/hr

Q• :number of tows/hr in Speed Group j in the last Segment I

Qj number of tows in Speed Group j in Segment i, tows/mi

TA: allowed delivery time, hr

TD: number of average late hours/tow for tows in Speed Group j

Ta: expected remaining lock transit time, hr

Ti: expected remaining travel time, hr

Tu: expended delivery time, hr

V: speed, mi/hr

Vc : boundary speed between paying or not paying penalties

Vv.,,: maximum speed limit

Vw : minimum speed limit

Vii : average speed of Speed Group j in Segment i, mi/hr

V1 : unconstrained optimal speeds with penalties

V2 : unconstrained optimal speeds without penalties

75
30000"

- V<6.3

V <8.3 - V < 10.4

V < 12.5

_ V < 14.6

020000-

10000 , ,

20 40 60 80 100 120 140

Allowance(hrs)

Figure 5-1 Total Costs for Various Speed Limits (V/C=0.80)

76
40000

30000

20000

10000,

0.4 0.5 0.6 0.7 0.8 0.9 1.0

V/C

Figure 5-2 Optimal Costs for Various V/C Ratios

77

!- | .. ...... S- ... . • . . .
16000'

15900-

0

15800

15700-

0.9 1.0 1.1

Ratio of S.D.

Figure 5-3 Optimal Costs for Various Standard Deviations of Lock Service Times

78
PRIORITIZING AND SCHEDULING INTERDEPENDENT

LOCK IMPROVEMENT PROJECTS

by

David Martinelli and Paul Schonfeld

79
80
Prioritizing and Scheduling Interdependent

Lock Improvement Projects

David Martinelli', AN ASCE and Paul Schonfeld1 , AM ASCE

Abstract

Current methods of capital budgeting are quite satisfactory for analyzing

mutually exclusive projects and reasonably satisfactory for independent projects.

However, there is a void in existing methodologies in prioritizing and scheduling

projects that are interdependent. This is because project evaluation,

sequencing, and scheduling must be performed simultaneously in order to yield an

exact solution. The benefits (or cost reductions) associated with navigational

lock improvements are interdependent, i.e. the improvement benefits of a given lock are affected by the acceptance or rejection of other lock improvement

projects. Therefore, the problem is beyond the scope of conventional solution

techniques. In this paper, we develop a methodology for obtaining an optimal or

near optimal sequence and schedule of projects subject to a limited budget.

Given a model to compute capital and delay costs for interdependent lock

improvements (evaluation), the algorithm presented in this paper may be used to

search the solution space of possible project permutations of sequences

(prioritization) as well as obtain the start times for projects subject to a

budget constraint (scheduling). A validation of the algorithm is also provided

yielding promising results.

Keywords

capital budgeting, project prioritization, project scheduling, interdependent

projects

Summary

An algorithm for sequencing and scheduling interdependent lock improvement

projects subject to a limited budget is developeO. The problem representation

and solution method overcome mrny of the difficulties with conventional

approaches. A validation of the algorithm is also provided.

'Assistant Professor, Department of Civil Engineering, West Virginia

University

2

Associate Professor, Department of Civil Engineering, University of

Maryland3 81
1. Introduction

Capital budgeting is the process of determining which investments or

"projects" will be funded and pursued in order to meet prespecified goals and

objectives over a planning horizon. A set of projects to be implemented at

specific times constitute a capital investment program. In order to assist

decision makers in funding decisions, capital budgeting models have been proposed

for use in planning based on various quantifiable criteria. Techniques such as

present worth economics, risk analysis, "what if" financial models, and

mathematical programming have all been employed in capital budgeting. The

complete capital budgeting process may be divided into three components: project

evaluation, project selection, and project scheduling.

The project evaluation phase involves quantitatively assessing the benefits

and costs of each project under consideration for each period of the planning

horizon. The project sequencing phase uses measures of effectiveness (MOEs),

agency or firm priorities, budget, and other factors to determine the relative

priority of projects. The project scheduling phase assigns a start time to each

project. While this phase is conceptually simple, budgets and other constraints

may impose delays on project start times.

This paper reports the development of a method that accounts for system

effects in inland waterway improvement projects. Current methods of capital

budgeting are quite satisfactory for analyzing mutually exclusive projects and

reasonably satisfactory for independent projects. However, there is an obvious

void in analyzing projects that are interdependent. Interdependencies exist

whenever the benefits or costs of any one project may depend on the acceptance

of one or more other projects. It seeff, that overcoming this void requires three

tasks: 1) the development of a framework whereby application-specific evaluation

functions may be formulated for aggregating benefits and costa among

interdependent projects, 2) the development of a technique whereby the numerous

permutations of possible programs may be represented and searched, and 3) the

82
determination of efficient project implementation schedules. This paper focuses

on Tasks 2 and 3, while Task 1 is discussed in Martinelli and Schonfeld (1992).

The standard analysis techniques for capital investment projects do not

deal with the interdependencies among available projects. For this reason, it

is common practice to reduce all problems to either independent or mutually

exclusive sets of projects. This is usually done by amalgamating those projects

with strong interdependencies and ignoring any remaining interdependencies. The

extent to which this practice leads to good decisions has been obscured by the

lack of analysis techniques capable of adequately representing interdependencies

in specific instances. The objective of this paper is to introduce a heuristic

technique for the sequencing and scheduling of interdependent lock improvement

projects. While the context for the development for the technique is lock

improvements, it is likely that the technique could be applied to other networkbased capacity expansion programs.

2. Sequencing and Scheduling for Lock Improvements

The National Waterways Study identified a need for substantial investment

in the waterway infrastructure (USACE 1987). This need stems from 1) waterway

traffic projections that approach or exceed the capacity of some existing

facilities and 2) the age and physical deterioration of facilities. Currently

there are about 100 locks that have exceeded their 50 year design life.

Experience with aging locks indicates that lock closures or stalls and subsequent

navigational delays can be expected to increase as locks age. Also, aging locks

tend to have substantially longer tow processing times. Stalls and high

processing times can result in increased shipping costs, delayed shipments, loss

of cargo, higher logistics costs, and other adverse effects.

By far the most significant of these benefits is the reduction in trip

delays associated with expanded capacity resulting from reconstruction. However,

the delays at a given lock may depend significantly on conditions at various

83
other locks, thus introducing the difficulties associated with interdependent

project sequencing and scheduling (Martinelli 1991).

3. Problems with Interdependent Sequencing and Scheduling

Explicitly, the project sequencing process chooses a subset of n investment

projects from a set of N desirable projects in the most desirable order. The

problem confronting the decision analyst is to choose from among the NI possible

permutations of project sets, the one which yields the maximum return. One

possible method of selecting a set of projects might be to choose the highest

payoff set out of a complete enumeration of sets that satisfy the budget and

other constraints. However, as a practical matter, complete enumeration becomes

infeasible as a method of finding the optimal sequence. If one is to consider,

by complete enumeration, all of the possible sequences of 30 projects, then about

2.6 x 103 alternative sequences must be examined. Clearly, it becomes

prohibitively expensive to select and/or sequence interdependent projects through

complete enumeration of alternatives.

If interactions among projects are assumed to be limited to pairwise

interactions, the problem may be formulated as a 0-1 integer programming problem

where -he objective is to maximize the total net present value (TNPV) subject to

a budget constraint for each period. The decision variables y. indicate the

projects to be implemented and their appropriate start dates.

N T T T r

max TNPV=ZZ NPVi.yi.+ZZ di* yi.y) Eq. 1 i-I -1 joi s v-I

N r

s.t. • Ci, y. + D, - D,.1 (l+I) = 8, t1l..T Eq. 2

y6 6 (0,1) s=1.. .T, i-1.. .N Eq. 3

F yisl i=l.. .N Eq. 4

84
D0 * 0 Eq. 5

D, Z 0 t-l...T Eq. 6

In the above formulation, NPV6 is the net present value of project i with start

time a, while di represents the deviation (in present dollars) from linear

addition in the net return from two interacting projects i and j with start times

a and v, respectively. This deviation may be more explicitly stated as

difference between the deviation in benefits and the deviation in costs

di* =bi* - cij Eq.?7

where b, and cO are the is the deviation in present value of benefits and costs,

respectively, for two interacting projects.

In the above formulation, C, is the required expenditure in period t, for

project i, starting in period 9, D, is the unspent budget in period t, I is the

interest rate, and B, is the limit on expenditures in period t. N is the total

number of all projects considered, while T is the number of time periods in the

planning horizon. The variable D, is included to allow unspent portions of the

budget in each period to be "rolled over" into the budgets of succeeding periods.

The decision variable y= is 1 if project i is to start in period a, and 0

otherwise.

There are significant shortcomings with this formulation. First, only

paired interactions are represented. Depending on the application, three, four,

or more projects may be simultaneously dependent. Second, the number of integer

variables is excessive. For example, a problem with 30 projects and a planning

horizon of 50 years (time periods) could have 1,500 binary integer (y6) decision

variables. This same problem would also require approximately 2.25 million

interaction coefficients (dij.'s) as well as 75,000 cost parameters (Cw's). In

85
general, the interaction variables are quite difficult to estimate. While many

problems may be smaller than this example, mixed-integer programming packages

have serious difficulties with problems of this size. Also, interdependencies

are only considered to exist among those projects that are actually implemented.

There is a need to formulate the problem in a manner that is not as

computationally expensive and does not require excessive estimation of

interaction parameters.

4. Methodology

4.1 Project Sequencing Procedure

The proposed approach for searching the solution space of possible project

permutations represents the solution space in two dimensions and applies a

heuristic search algorithm in selecting the preferred sequence. Given a system

cost evaluation function for interdependent projects g(X,Y), the selection and

sequencing problem may be represented in two dimensional space. The function

g(X,Y) incorporates both benefit and cost factors into a generalized cost while

accounting for project interdependencies where X is a vector of delay variables

and Y represents a particular combination of projects.

Assuming that each set of projects may be viewed as a system generating a

common time-dependent output, then a two dimensional representation is quite

feasible. For the lock rehabilitation problem, the costs associated with a given

combination of projects in a given time period t, may be written as

(SC), = Ci + g(XY), OW Eq. 8

where Ci is the total capital cost of construction for project i. The term

g(X(;),Y), represents the delay, and corresponds to the function(s) obtained from

some interdependent evaluation, e.g. from a simulation model, while Ow is the

opportunity cost of delay. Evaluating SC at different levels of output for a

combination of projects Y, defines a curve with annual system costs SC, on the

vertical axis and output level, X, on the horizontal axis. Repeating for

86
different values of Y (i.e. different project combinations) produces a family of

curves. By always choosing the lowest cost curve for any given output level

1, i.e. by choosing the "lower envelope" of the curves in Figure 1, a sequencing

and scheduling decision path is defined. Because the output is assumed to be

time dependent, the horizontal axis may also represent time periods, e.g. years.

Output and time may be linked through a demand function, 1(t).

Consider an example with interdependent projects A, B, and C. Figure 1

shows a family of system cost (SC) curves corresponding to the possible

combinations of these three projects. Note that in general, combinations

involving only one project are preferable (lower SC) for low levels of volume

(thus earlier in the horizon stage), and become less preferable as volume

increases. Under this representation, one combination is preferred to another

at a given output level (or time period) if its corresponding curve lies above

the other. 3

In the example depicted in Figure 1, the selection and sequence of projects

is dictated by the lower "envelope" fined by the curves. This lower envelope

corresponds to the minimization of the time integral of the system cost for

feasible expansion paths. Here, all three projects would be accepted if the

volume level is expected to eventually exceed Q2. We see also that the sequence

of projects should be A, B, C; this is because Curve A lies below B and C, and

AB lies below AC in the relevant regions. Project A is preferred up to volume

level Q, at the same time Project B should be implemented since Curve AB falls

below Curve A. At volume level Q2, Project C should be added to A and B, thus

implementing Combination ABC.

3

Although the convex and monotonically increasing properties of the curves

in Figure 1 are likely to occur for costs with a delay component, they are not

a prerequisite for the methodology.

87
AC AS a6C

A

L)0 w'I

E

(/)

0

4

J

01 Q2 Volume

T1 T2 Time

Figure I Plot of System Cost for Three Interdependent Projects (Case 1)

A second case involving the three projects A, B, and C is shown in Figure

2. Here, because Curve AB lies completely below Curve ABC, only Projects A and

B are included in the program. The sequencing decision would be the same as in

the previous example, with the intersection of Curves A and AB indicating that

the implementation of Project B should be timed at t,.

Unfortunately not all such families of curves can be interpreted as easily

as Cases 1 and 2. Consider a third case shown in Figure 3 where Curves A and AS

are unchanged but the others are different. Here, Curves AB and AC intersect

each other before intersecting Curve ABC. It cannot be stated a priori weather

Combination AB or AC should be selected on the expansion path between A and ABC.

One would expect that if Area I is greater than Area 2, then Combination AS is

preferred to AC and Project B should precede Project C on the expansion path.

Areas 1 and 2 correspond to the difference savings when integrating over Paths

88
A-AB-ABC and A-AC-ABC, respectively.

8 ABC

AB

AC

A I

E

4)

0

01 Volume I II

T1 Time

Figure 2 Plot of System Cost for Three Interdependent Projects (Case 2)

4.2 Scheduling

Under the assumption that the benefits associated with a given combination

of projects in some period vary only with the output of the system in that

period, the start dates of the projects do not affect the system costs. Thus the

SC curves for a project combination depend only on the presence, rather than

start times, of particular projects in that combination. The implications in the

context of waterways are that the capital cost of construction, operating and

maintenance costs, and benefits from reduced delays are not affected by the age

of the locks at any given time (i.e. by project start dates) but only by the

volume of traffic using the locks. This assumption is very reasonable for the

capital costs, but somewhat simplifies the operating and maintenance costs. The

assumption is also reasonable for delay benefits although it neglects the effect

of long term economic changes induced by the presence and performance of waterway

investments.

89
AB AC ABC

A

"LO0

I-)

VolIume

T Ime

Figure 3 Plot of System Cost for Three interdependent Projects (Case 3)

4.3 Incorporating a Budget Constraint

The representation of project combinations proposed thus far has not

incorporated the effects of a budget constraint. In structuring the budget

constraint, it will be assumed that funds not spent in a given period will be

available in subsequent periods. This assumption is expressed by Equation 2 and

is true for the Inland Waterway Trust Fund. For example, if $5 million is

available and nothing is implemented in Period 1, then the $5 million is added

to the budget limit for Period 2. Under this assumption, budget limitations have

the effect of delaying the earliest feasible start date of a given project

combination, just as they limit the earliest start of an individual project.

Consider the small example of two projects A and B. In constructing the Curves

A, B, and AB, the infeasible portion must not be included. Figure 4 illustrates

that Combination A is not financially feasible until time T, corresponding to

90
output Q1. Combination AB is not feasible until time T,. The three possible

expansion paths are then as follows:

1. start A at time T, and B when Curves A and AB intersect

2. start B immediately and A when Curves B and AB intersect.

A E3 AB

0

ci)

V)

T1 T2

0 Time

Figure 4 Incorporating a Budget Constraint

5. Sequencing and Scheduling Algorithm

The algorithm for selecting, sequencing and scheduling a subset of projects

from a set of candidate projects searches for the minimum cost expansion path on

a two dimensional plot of system cost functions for various combinations. The

algorithm begins with an initial ranking of all projects based upon an evaluatiun

where interdependencies among projects are neglected. This initial ranking is

then modified iteratively to account for the interdependencies among the

projects. The ranking of projects represents a sequence of search steps.

Beginning with the null alternative, the next ranked project in the sequence is

added to the provisional expansion program at each search step. At each

91
implementation step, consideration is given to swapping two (or more) of the

projects in the sequence. This is done by comparing the total system cost

savings in a situation represented in Figure 5.

Total system cost curves represent the sum of delay and capital costs at

all lock sites. It is computed for any set of independent as well as

interdependent projects. The costs are then totaled across all interdependent

projects and added to the costs of the independent projects to obtain the total

system cost.

At each search step, four system cost curves are plotted. At the first

search step (shown in Figure 5), the four combinations (project subsets) are C,)

no projects (null), C.) null plus first ranking project, CA null plus second

ranking project, and C4 ) first and second ranking project, Figure 5. These four

curves define two possible paths, 1) null-first-(first & second) or 2) nullNULL 1st

2nd 1st + 2nd

LO)

0

t t

1 2

Tfme

Figure 5 Plot for First Implementation Step

92
second-(first & second). The first path represents no change from the

independent ranking, while the second path represents a swap between the first

and second ranked projects. The selection between the two paths is based on the

relative cumulative costs.

2nd le

2ndd

2nd lo *r

S• Area 2

I--

Lfl

-J I-I

P •Area 1'

0

TIME

Figure 6 Plot for Second Implementation Step

If a swap to a higher ranking is found to be desirable for a project, then

the additional swaps to successively higher rankings are considered for that

project, until no further swap is desirable. Following all possible swaps, a new

ranking is established. Subsequent implementation steps plot Cl) corresponding

to the current implementation set, C2) corresponding to the addition of the

current first ranked among remaining projects, C3) corresponding to the addition

of the current second ranked among remaining projects, and C4) corresponding to

the addition of both the current first and second ranked projects. For example,

if a swap occurred during the first search step, then Figure 6 shows the

combinations that would be plotted during the second search step. The start

times for the projects are determined directly from the plots at each

93
implementation step. The procedure is described in more detail in the

subsections that follow.

5.1 Initial Ranking of Projects

The initial sequence is based on a relative evaluation of the projects

assuming that they are independent. The evaluation may be made on the basis of

the BCR. The benefits of improvement are a reduction in delay associated with

the increased capacity, while the costs are the capital cost of construction.

The BCR of a project i, that is independent from all others, may be written as

BCR, = L_ ((I(t) - I|(t)) X(t)i Ow(t)/Ki Eq. 15 t-0

where I(t) S T.

The function Io(t) is the delay before improvement, while I(t) is the delay

function after improvement. I(t) may be determined analytically or

experimentally. It should be noted that the project index on 1(t), implies that

volume may vary with time. Also, the opportunity cost of delay may be projectspecific and be a general function of time. The constant T refers to a

congestion tolerance in hours per tow. Tows will divert to other modes or

waterways when delays reach T + e. After computing BCRý for all projects, the

projects are ranked according to decreasing BCR. This is the initial ranking

(sequence) of the projects. It should be noted, that any project that has a BCR

less than 1.0 should be eliminated from the expansion program at this stage of

the analysis. This is because the presence of interdependence will only tend to

lower a project's BCR (Martinelli and Sconfeld 1992).

5.2 A Routine for Modifying the Initial Sequence

The routine described in this section for modifying the initial sequence

may be either coded algorithmically or performed interactively through a highlevel programming environment. The routine begins with an initial

94
ranking/sequence of projects, described by the n dimensional vector R,, e.g.

R,(2)=A indicates that project A is second in the initial sequence. Project

combinations are represented by a vector C of indices referring to a subset of

projects in the vector R,. For example, if Rý=(A,B,C,D), then the combination ABD

would be expressed by the vector C-(1,2,4). The combination corresponding to the

null alternative is denoted by C=(0). Finally a scheduling vector T is defined

as the vector of start times corresponding to the projects in the sequence vector

R. For example, T(3)=20 indicates that the third project in the sequence begins

in time Period 20. The final sequence and schedule is represented by the vectors

R. and T,, respectively.

Following an initial ranking, the first project in the initial sequence is

then tested for a possible swap with the second. A possible swap of the first

project with the third project may also be considered. However, the

computational requirements of the routine increase with the number of possible

swaps considered at each iteration. In the interest of clarity, the description

in this section is for consideration of one swap at each step.

The following are the steps for sequencing and scheduling a set of

interdependent projects in which two or more are interdependent. The number of

iterations in the routine is equal to one less than the total number of projects

and is indexed by i.

Steo) 1

Compute the benefit cost ration (BCR) according to Equation 15 for each

lock, assuming locks are independent. Rank in descending order. Let the initial

sequence vector, k, equal this ranking. To is initialized to all zeros. Let

i=O. Go to Step 2.

Sten 2

If i=O, then "plot" the interdependent system cost (TSC) curves for

95
combinations C1 -C(O), C12=C(l), C3=C(2), C,=C(l,2) versus time. If i ? 1, then plot

the curves for C1-C(l,..,i), C-=C(l,..,i,i+l), C(1,..,i,i+2),

C-=C(l,..,i,i+l,i+2). The curves begin at a time corresponding to the

availability of revenues equal to the sum of capital costs of the projects

contained in a given combination.

r"

B(t) - L.- KN /crf Eq. 16

jE C

The purpose of the plotting in this step is to locate the values of t for

which five intersections may take place. These intersections are, t,: C, and C.,

t 2: Ct and C , 3t: C and C3, t4: C and C4, and t,: C3 and C4. The intersections help

define areas for combination comparison such as Areas 1 and 2 in Figure 3 While

these intersections may be determined visually if plotted, they may also be

determined numerically without plotting. Numerical procedures for efficiently

locating the intersections of convex functions currently exist.

The budget constraint is initially considered in this step of the routine.

Revenues for major rehabilitations of lock facilities come primarily from the

Inland Waterway Trust Fund and the federal matching share. Unspent funds

accumulate according to a specified account interest rate. Because the start

times for projects are on a continuous rather than discrete scale, the effect of

the budget constraint is to delay the earliest possible start time for each

combination. Therefore the times corresponding to the five intersection points,

t1 , t

1 2 ... ,to, will be replaced with t',, t' 21 ... ,t',. Because a budget limitation

may delay the start of a combination, combination curves that would otherwise

intersect, will have no intersection. Therefore, times t' 1 , t, ... ,t'5 do not

correspond to intersections but rather to the adjusted earliest start time for

a combination.

96
C2 C3 C4

0

V)

I I I I I i a

III I

t t t t t

Ti me

Figure 7 Quantities obtained in Step Two

Figure 7 illustrates an example of the quantities obtained in this step.

First, combination C, represents the current subset whose sequence ha. already

been determined in previous steps. Combination C2 represents the resulting

combination if the current sequence is maintained, while C3, represents the

resulting combination if a swapping were performed at this step. Combination C,

represents the implementation of the following project in the current sequence.

Ir' this example, the budget constraint has delayed the earliest start time for

combination C2 to tl,. Therefore Project R(i+2) cannot start at t,.

Go to step 3.

Evaluate to determine if Project R(i+l) should be swapped with R(i+2) in

the sequence. This corresponds to a comparison between C., and C,. The comparison

is based on relevant areas defined by the system cost curves and the intersection

points found in Step 2. If a swap is made, then a swap between project R(i+2)

and R(i) is considered by redefining C, from C(l. ..i,i+l) to be C(l... .i). If a

second swap is made, then an additional swap between R(i+2) and R(i) is

97
considered by setting C. to C(l...i-1). Swaps are iteratively considered until

a comparison is made where no swap is necessary.

In each case, unless one curve lies completely above the other, there will

be an area that will favor R(i+l) and an area that will favor R(i+2). The

evaluation is divided into four cases. The following tests may be used in making

the correct comparisons for evaluation:

a. If the capital costs for combination C, are greater than for C, and the minimum

capacity in C. is less or equal to that of C3, then R,(i÷l) is swapped with

R1(i+2).

b. If the capital costs for combination C. are less than for C and the minimum

capacity in C, is greater or equal to that of C3, then Rý(i+l) is not swapped with

R•(i+2).

However, if a budget limitation delays the earliest start for Combination

C. to t', then, if t', > t 2 , the following condition must be tested for a possible

swap between R1 (i+l) and R.(i+2):

f TSC(t),i dt + f TSC(t), 2 dt > f TSC(t),i dt

St*' to

A LS

+ TSC(t), dt-+ TSC) dt Eq. 17

L t4

In addition, if the budget delays the start of Combination C, to t,, then the

third term on the right hand side (RHS) of Equation 17 is omitted and t, replaced

with t's in the integration limits.

Figure 8 is an illustration of the effects associated with a change in the

earliest start time of Combination 2 due to the budget constraint. Without the

98
budget constraint, path C,-Cý-C, starts at 0 with C,, switches to C. at t, and

switches to C. at ts. .,.wever, since funds are not available to begin C. at t,,

the system must ron-inue under C, until t',. Because t', > t,, path C,-C:-C, is no

longer entirely below path C,-C3-C.. Therefore, the test of Equation 17 is

necessa:y to determine which path yields the lower cumulative system costs.

c. If the capital costs for combination C, are less than that of C3 afd the

minimum capacity in C- is less than or equal to that of C3 , then if the condition

of Equation 18 holds (Area 1 < Area 2), R,(i+l) jj swapped with P,(i+2).

C 0

f (TSC(t)•, - TSC(t),2 )dt + f(TSC(t)L - TSC(t),o)dt

ý (s~ ) -. 1 (S~t. -

f (TSC(t)a - TSC(t)o)dt + f(TSC(t), - TSC(t)(J)dt Eq. 18

L3 14

C C 3 2 C4

Tn V)

V)

0

0 t t t t t

1 2 3 45

TIME

Figure 8 Illustration of Budget Effects (Case 1)

99
If the budget constraint alters t1 and/or t2, then the LHS of Eq. 18 (Area

1) in decreased by an amount given in Eq. 19 and increased by an amount given in

Eq. 20.

f (TSC(t),| - TSC(t),z)dt Eq. 19

"(2

J (TSC(t),1 - TSC(t), 3 )dt Eq. 20

Figure 9 is an illustration of the effects of the budget constraint if the

earliest start time of C3 is delayed from t 2 to t',. Here, path C,-C3-C 4 follows

C, for an additional period of time (t',-t,) yielding a higher cumulative cost

than the same path without the budget constraint. This causes an increase in the

size of Area 1.

C C3 C

8

22

"LU Area 2

>-.n

_J

S, Area 1I I0ITIME

Figure 9 Illustration of Budget Effects (Case 2)

100
d. If the capital costs for comb. nation C2 are greater than that of C3 jand the

minimum capacity in C, is greater or equal to that of C., then if the condition

of Eq. 19 holds, Rý(i+1) is swapped with R•(i+2).

f (TSC(t),1 - TSC(t)•)dt + f(TSC(t)a2 - TSC(t)l)dt

f (TSC(t)a - TSC(t)a)dt + f(TSc(t), - TSC(t), Li~ 2 )dt Eq. 21 L

If the budget constraint alters t, and/or t,, then the LHS of Equation 21

(Area 1) is decreased by an amount given in Equation 22 and increased by an

amount given in Equation 23.

f (TSC(t)., - TSC(t)a)dt Eq. 22

Ut'f (TSC(t)}1 TSC(t)4•)dt Eq. 23

In general, TSC(t) is not an integrable function. However, as an

approximation, the integrals may be replaced with summations and the limits

rounded to the nearest integer value of t. Even better approximations are

available through the rule of trapezoids or other numerical techniques. If the

combination curves are plotted graphically, it should be possible to compare the

sizes of Area 1 and Area 2 visually.

Go to Step 4.

The sequence vector, P, is updated for iteration i+1. If Projects R(i+l)

and R(i+2) were not swapped in Step 3, then Rý÷1 = -•. If the projects were

101
swapped then let

R÷1,(i+l) - Rý(i+2) Eq. 24

and

R,.I(i+2) = Rý(i+1). Eq. 25

Go to Step 5.

Step 5

The scheduled start time for the next project in the sequence is obtained

directly from one of the intersection points found in Step 2. The following

rules apply when assigning the start time for the next project in the sequence:

a. if Project R(i~l) was not swapped with R(i+2), then T(i+l) - t,

b. if Project R(i+l) was swapped with R(i+2), then T(i+l) = t 2.

If i=n-1, then stop. Current sequence and schedule are final, otherwise

increment i and return to Step 2.

5.3 Summary of Routine

The flow chart in Figure 10 provides a summary of the sequencing and

scheduling routine described in this section. First, the user specifies the

relevant data for the problem. These include 1) lock characteristics such as

capacity, volume, distance from the previous lock, and growth rate, 2) project

data such as construction costs and capacity improvements, and 3) other

information such as planning horizon, interest rates, and congestion tolerance

factor. This information is then employed to establish an initial ranking based

upon an independent evaluation.

102
i Aputs:

- lock characteristics

- project oata

- other information

Establish Initial

rlicing - a

C inaeoaow evetatIorc

SpecIfy ,• 4 4 :fy Project I"o ect cMInat ¢€ t~o ions .4 ................ ........................

C11- C C1 I

C2 - C CI 1 i .*•

C3-C CI.. i, 1-23

04 - C CI ... I. 1-1 1.2)

caput. system Co~t for

all Iour cofmlIt Ion

Plot aI four camln f net+ I on*

eve I usltSe

- Swell R(I 1) arnd I - 2) ?

- Road Start time from Intersection

Wdete

Figure 10 Flow Chart of Sequencing and Scheduling Routine

At the first iteration, the null alternative and three combinations

involving the first two projects in the sequence are specified. Step 2 of the

routine describes this specification step for subsequent iteration.. Next, the

103
total (delay and construction) system cost is computed for each combination for

all time periods. The cost of each of the four combinations is plotted versus

time on a single graph. Using the plot, the user determines whether the path ClC2-C4 or C,-4-C 4 is less costly. If path C1-C3-C4 has the lower cumulative cost,

while considering the budget constraint, then projects R(i+l) and R(i+2) aze

swapped in the sequence, and the sequence is updated accordingly. An additional

swap between R(i+2) and R(i) is then considered, followed by additional

comparisons if swaps are continually made. Also, if a swap occurred, then the

start time for project R(i+2) is read from the intersection of C, and 4, else the

start time for project R(i+l) is read from the intersection of C1 and C2. If the

end of the project list L. not reached, the user specifies the next set of four

combinations, and repeats the plotting and evaluation steps.

6. Validation of Sequencing Algorithm

The algorithm presented is not theoretically guaranteed to yield the

optimal sequence of projects. Therefore it is necessary to conduct an empirical

validation of the algorithm. For validation, two experiments, one with four

locks and one with six locks were conducted. In each experiment parameters of

lock systems and proposed projects were randomly generated. The algorithm was

then applied to each case to obtain a project sequence. Associated with the

sequence is a cumulative system cost over the 40 year planning horizon. This

sequence is compared to the optimal sequence, i.e. the sequence with the minimum

cumulative system cost. The optimal sequence is determined through exhaustive

enumeration of possible sequences. There are ni possible sequences, 24 and 750

for'four and six lock cases respectively. There were 30 cases tested for four

lock systems and 20 cases tested for six lock systems. The evaluation function

used in the experiment is a one-directional metamodel (i.e. a model estimated

statistically from simulation results) obtained from a related study (Dai and

Schonfeld 1991).

Cases of locks and projects were randomly generated according to a uniform

104

J
distribution. Ranges of variables were set in such a way as to guarantee a

significant amount of interdependence. For example, the distance between locks

for four lock systems is fixed at 10 miles and is between 5 and 20 miles for six

lock systems. These low distances are not very common for the actual inland

waterway system, but they yield a higher level of interdependence for a more

challenging test of the sequencing algorithm. Table 1 shows the range of the

randomly generated problem parameters.

The results of the experiments are tabulated in Appendix III. For each of

the two experiments, there is a table providing the inputs for each case and a

table providing the outputs for each case. The tables of inputs sho%: the

capacities before and after improvement, the capital cost of improvement for each

lock, the initial volume level, growth rate, opportunity cost of delay, and

distance between locks. Finally, the resulting BCR for the independent

evaluation is given for each project. The output tables contain the sequences

based on independent evaluation, Lhe sequences based on the algorithm, and the

optimal sequence obtained from exhaustive enumeration. Also given are the

cumulative system cost of the algorithm sequence and the optimal sequence. From

these costs an error may be computed for each case.

Table I Range of Problem Parameters for Sequencing Validation

Lower Upper

Limit Parameter Limit

5 Initial Volume 35

.3 Initial Utilization 7

1% Annual Growth Rate 5%

1.5 1,/ /o 2

$100 Opportunity Cost $500

5 Distance 20

105
While the number of cases analyzed in each experiment is not

extraordinarily high, the results suggest that the algorithm is likely to be

effective in yielding an efficient project sequence. As the results in Appendix

III indicate, the four lock experiment had two cases in which the algorithm did

not successfully yield the optimal sequence. This is a success rate of 93.3% for

the 30 cases. The two suboptimal cases had cumulative costs that were 0.8% and

2.3% higher than optimal. It appears that even when the algorithm does not yield

an optimal solution, it does yield a reasonably good sub-optimal solution.

Similar results were obtained for the six lock experiment. Specifically, the

algorithm failed to yield the optimal sequence in one of the 20 six lock cases.

This is a success rate of 95%. The error associated for the one suboptimal case

was 4.1%.

By examining the cases were the algorithm is in error, ideas for

improvements in the algorithm may be obtained. For example, Case 12 of the six

lock experiment yielded an incorrect sequence resulting in a cost error of 4.1%.

A close examination of this case reveals where in the algorithm the error was

made. The algorithm yielded a sequence of 1,4,3,5,6,2 while the optimal sequence

is 1,4,3,2,5,6. Applying the algorithm by hand on this case revealed that a swap

between Projects 2 and 6 was considered, but the swapping criterion was not

satisfied. If we go against the algorithm and consider a swap of Projects 2 and

5, we find that the swappin" criterion is satisfied. However, the algorithm on

its own does not attempt a swap between 2 and 5 because no swap was performed

between 2 and 6.

There may be some adjustments in the algorithm which might eliminate such

errors with little cost in computation time. For example, in a given iteration,

swaps beyond the first unsuccessful swap may be considered. In other words, if

a lower ranked project is not swapped with its predecessor, a swap between it and

the next highest project may be cois.:.•ed. In order to control the amount of

additional computations associated with this change, some maximum percentage cost

106
difference may be specified for the initial comparison as a condition for

considering the next highest swap. In applying this change to Case 12, a swap

between Projects 2 and 5 would be considered if the cost difference in

considering the swap between Projects 2 and 6 is within a specified amount.

7. Conclusions

A mathematical programming model was introduced as a conventional

formulation for capital budgeting of interdependent projects. This formulation

was shown to be inadequate for the prioritization and scheduling of inland

waterway improvement projects. An alternative formulation that addresses the

shortcomings of the conventional has been developed and shown to be effective for

the application to inland waterway lock improvements. The method is conceptually

and computationally straightforward, and has been show to yield promising results

based on an experiment involving systems of four and six locks.

Appendix I References

Akileswaran, V., Hazen, G. B., and Morin, T. L., "Complexity of the

Project Sequencing Problem," Operations Research, Vol. 31, No. 4,

Jul.-Aug., 1983 pp. 772-778.

Clark, J. J., Hindelang, T. J., and Pritchard R. E., Capital Budgeting:

Planning and Control of Capital Expenditures, 2nd ed., Englewood

Cliffs, N.J.: Prentice Hall, Inc., 1984.

Dai, D. M. and Schonfeld, P., "Reliability Analysis of Coal Inventories on

the Ohio River", Interim Working Paper, U.S. Army Corps of

Engineers, Institute for Water Resources, Feb. 1989.

Dai, D. M. and Schonfeld, P., "Metamodels for Estimating Waterway Delays

Through Series of Queues", Working Paper #9122, University of

Maryland, College Park, MD, Mar. 1991.

Erlenkotter, D. and Rogers, J. S., "Sequencing Competetive Expansion

Projects", Operations Research, Vol. 25, No. 6, Nov.-Dec. 1977, pp.

937-950.

Janson, B. N., "Method of Programming Regional Highway Projects, Journal

of Transportation Engineering, ASCE, Vol. 114, No. 5, Sep. 1988, pp.

584-605.

Martinelli, D. R., "Investment Planning of Interdependent Waterway

Improvement Projects", Unpublished Ph.D. dissertation, University of

Maryland, 1991.

U. S. Army Corps of Engineers, Institute for Water Resources, "National

Waterways Study," 1987.

107
Appendix I1 Notation

b deviation in present value of benefits

B Budget

BCR Benefit Coat Ratio

c deviation in present value of costs

C project capital costs (conventional formulation)

C combination of projects to be implemented d deviation from net present value

D unspent budget

g9 generalized cost

I() delay function assuming independence K project capital costs (new formulation) O opportunity cost of delay

NPV Net Present Value

R project ranking SC System Costs (capital and delay) T planning horizon; congestion tolerance

TNPV Total Net Present Value

y 0,1 decision variable for project implementation

II( output level (arrival rate at lock)

Appendix III Results of Sequencing Validation

Table 1 Outputs for Four Lock Experiment

Case SEQ SEQ SEQ Cost cos

INDEP. ALG. OPT ALG. OPT. Emr

1 3,2,4,1 4,2,3,1 4,2,3,1 2.80x10 2.SOx1ot 0

2 3,2,4,1 3,2,4,1 3,2,4,1 4.27xlot 4.27x10' 0

3 1,3,2,4 3,1,4,2 3,1,2,4 6.16x10' 6.13xl0 0

4 1,4,2,3 1,4,2,3 1,4,2,3 2.07x10' 2.07xl0' 0

5 1,2,3,4 1,2,3,4 2,1,3,4 2.56x10' 2.56xiO 0.8%

6 3,2,4,1 3,2,4,1 3,2,4,1 1.55xlO' 1.55xlO 0

7 3,1,4,2 3,4,1,2 3,4,1.2 2.54x00 2.54x10 0

a 4,3,2,1 4,2,3,1 4,2,3,1 2.91xl0 2.91x11? 0

9 4,2,3,1 4,3,2,1 4,3,2,1 7.49x0I 7.49x1iO 0

10 4,3,2,1 3,2,4,1 3,2,4,1 8.20x10' 8.28x10 0

11 2,4,1,3 2,1,4,3 2,1,4,3 2.83x10' 2.8301Wl 0

12 3,1,4,2 3,1,4,2 3,1,4,2 2.43x10 2.43xl0 0

13 2,4,3,1 4,2,3,1 4,2,3,1 4.59x10' 4.59xl0 0

14 3,2,1,4 3,2,1,4 3,2,1,4 2.70xl0 2.70x101 0

Is 3,2,1,4 3,2,1,4 3,2,1,4 2.96x1ir 2.96x10' 0

108
16 2,4,3,1 4,2,3,1 4,3.2,1 3.O0xlO' 2.94x10' 2.3%

17 1,3,2,4 1,3,4,2 1,3,4,2 2.37x10 2.37x10 0

18 3,1,4,1 1,4.2,3 1,4,2,3 6.99xlU' 6.99x10O 0

19 4,2,1.3 4,2,1,3 4,2,1,3 1.86xl f 1.86xi 0

20 2,1,3,4 2,1,4,3 2,1,4,3 1.0oxIo' 1.00xio0o 0

21 3,1.4,2 3,4,1,2 3,4,1,2 3.38x10 3.38x10m 0

22 3,1,2,4 1,3,4,2 1,3,4,2 2.83x10' 2.83x10' 0

23 1,3,4,2 1,3,4,2 1,2,4,2 1.94x10' 1.94xl0 0

24 2,4,1,3 2,4,1,3 2,4,1,3 3.9xlg 3.9xI01 0

25 2,4,1,3 4,2,1,3 4,2,1,3 1.73x0' 1.73xi0' 0

26 1,2.3,4 2,4.1,3 2,4,1,3 4.99x10' 4.99"x0 0

27 4,2,3,1 4,2,3,1 4,2,3,1 3.62xl0 3.62x109 0

28 1,2,3,4 1,2,3,4 1,2,3,4 2.0sxl(' 2.05x10' 0

29 1,2,3,4 2,1,3,4 2,1.3,4 7.33x10' 7.33x lO 0

30 2,1,3,4 1,3,2,4 1,3,2,4 5.22x10V 5.22xl1" 0

"109
Table 2 Outputs for Six Lock Experisent

INDEP ALG OPT ALG OPT

CASE SEQ SEQ SEQ COST COST ERROR

1 241635 421356 421356 5.17x10O 5.17x10' 0

2 364152 345162 345162 3.73x10' 3.73x10' 0

3 432165 432165 423165 2.08xl0 m 2.08x10' 0

4 214365 132465 132465 5.87x109 5.88x10' 0

5 135462 153426 153426 5.31x10' 5.31x10' 0

6 453612 421356 345126 1.24x10°' 1.24x10'° 0

7 452631 425136 425136 6.22x10' 6.22x10' 0

8 415236 154236 154236 6.18x109 6.18x10' 0

9 231456 321546 321546 6.31x109 6.31x10' 0

10 142563 241536 241536 7.95x109 7.95x10' 0

11 526134 521346 521346 6.71xlO 6.71x10 0

12 134652 143562 143256 7.38x109 7.SlxlO 4.1%

13 142356 514236 514236 8.94x10' 8.94x10' 0

14 134562 153642 153642 5.00xl0 5.00xlO' 0

15 526431 524631 524631 3.22x10'0 3.22x10'0 0

16 451326 154236 154236 5.23x109 5.23x109 0

17 236145 321465 321465 1.12x10'0 1.12x1010 0

18 423165 234153 234156 8.60x109 8.60x109 0

19 153642 351462 351462 7.04x10' 7.04x109 0

20 132465 123465 123465 6.47x10' 6.47x10' 0

110
APPROXIMATING DELAYS AT INTERDEPENDENT LOCKS

by

David Martinelli and Paul Schonfeld

111
112
ApproZiRating Delays At Interdependent Locks

David Martinelli, AM ASCZ' and Paul Schonfeld, AX ASCE1

Abstract

As with much of the nation's infrastructure, the inland waterway

system is in critical need of expansion and repair. Many of the

inland waterway lock and dam facilities have become major

constraints to navigation due to increased traffic and facility

deterioration, leading to costly delays. Because funds for lock

and dam improvements are severely limited, comprehensive analysis

methods are necessary to ensure efficient allocation of resources

among the many proposed improvement projects. Unfortunately, locks

and dams are often treated as independent facilities with regards

to operations when in fact, there are likely to be significant

interdependencies among many of them. There is a need for an

analysis technique that accounts for interdependencies between

locks when considering lock improvements. In this paper, a method

is developed whereby the delays of a set of interdependent locks

may be calculated. By incorporating interdependencies into benefit

calculations of lock improvement projects, a more comprehensive

assessment of improvement priorities can be established.

Keywords

inland waterways, interdependence, queues, simulation, metamodel

Summary

A method for computing delays for interdependent locks is developed

through the formulation of evaluation functions based on the

results of a simulation experiment. The method could help lead to

more comprehensive project evaluation and prioritization of lock

rehabilitation projects.

'Assistant Professor, Department of Civil Engineering, West

Virginia University, Morgantown, WV 26506-6101,

drm@wvnvaxa.wvnet.edu

2Associate Professor, Department of Civil Engineering,

University of Maryland, College Park, MD, 20742

113
I. Introduction

Inland waterways are an important part of the nation's

transportation network. Approximately 16 percent of the intercity

freight in the U.S. moves by waterway. Coal, petroleum products,

and grains are the top three tonnage commodities, accounting for

about 60 percent of the inland waterway commerce. The National

Waterways Study (NWS) forecasts an increase in total U.S.

waterborne traffic from 1,915 million tons in 1977 to a 2,890

million tons by 2003 (USACE 1987). Locks and dams are essential

for creating stepped navigational pools with reliable depths for

navigation. However, many of these facilities have become major

constraints to inland navigation, due to increased traffic and

facility deterioration, leading to costly delays.

Prediction of lock delays is important for, among other

things, evaluating rehabilitation and reconstruction alternatives.

Each lock, however, is part of a system of locks, and in general,

its delays may be dependent on operations at one or more other

locks. These interdependencies limit the value of available tools

for predicting lock delays. Specifically, the queuing process at

locks cannot be modeled as an M/M/1, or even M/G/l, process.

Moreover, locks typically have more than one chamber and do not

operate under a first-in-first-out queue discipline. Therefore,

queuing theory does not provide any satisfactory analytic solution

for predicting lock delays.

114
The objective of the research described in this paper has been

to develop a technique for approximating average delays at inland

waterway locks while considering their interdependencies. The

ultimate use of such a technique is to evaluate the benefits

associated with rehabilitating given combinations of locks for

investment planning. While a microsimulation model for capturing

such delays is available (Dai and Schonfeld 89), a significant

amount of computer time is required for variance reduction, when

obtaining reliable predictions of delays. Furthermore, for

investment planning purposes, it is necessary to execute such a

model for numerous combinations of reconstruction projects.

Therefore there is a need to develop a functional simplification

that will adequately substitute this simulation and perform

satisfactorily as an evaluation tool for investment planning.

2 Defining Lock Improvement Interdependencies

If locks are independent, then however a lock operates, it

does not affect the performance of any other locks upstream or

downstream. Then the total delay of a system of locks is no

different from the sum of the delays of each lock acting in

isolation. On the other hand, if interdependencies between two or

more lccks in a system exist, then the total delay of the system

will be different from the sum of the isolated delays. If the

total system delay, S, is equal to the total isolated delays, I,

then the 'ocks may be considered independent. Therefore, a ratio

of system delay to isolated delay, S/I, may serve as a theoretical

115
measure of the total interdependence in the system. If S/I = 1,

then there are no interdependence effects, and if S/I < 1 then

interdependence effects exist. It may be noted that S/I cannot

exceed 1. It should also be noted that we are concerned only with

assessing the total effects of interdependence and not an

itemization of all component interdependencies.

It is possible to establish a theoretical lower bound for S/I.

Consider a one directional system of two identical locks A and B.

Also, assume the spacing between vessels to be unchanged when

traveling between Locks A and B and the lock service times are not

random variables. Although there may be delays, WA at Lock A,

there will not be any delays at Lock B, due to a metering effect

from Lock A. In other words, tows will arrive at Lock B at

intervals that exactly correspond to service times from A.

However, if each lock is considered in isolation, then the delays

at B will equal the delays at A, yielding a total delay of 2 WA for

the two lock system. Therefore the lower bound on S/I for a two

lock system is 1/2. In general, for similar systems of n locks

which represents a lower bound on S/I on S/I (or an upper bound on

the amount of interdependence in the system). For the more

realistic stochastic case, S/I will be larger than 1/n due to

randomization opportunities for arrival patterns and service times

at Lock B.

S 1 - = -1 Eq. 1 I n

116
3. Possible Factors Affecting Lack Interdependence

There are numerous factors that might be related to lock

interdependence. For example, a Pennsylvania State University

study classified locks as dependent or independent according to the

linehaul distance separating them (Carrol 1972). Queuing theory is

helpful in identifying the most relevant factors. Using an M/G/l

queuing model, the average wait time, W, has been reported in

(Whitt 84) to be

1 p

2 + XZ a2

W=- + Eq. 2

A 2X(I-p)

where g and X are the mean service and arrival rates respectively,

p is lock utilization, and a is the standard deviation of the

service time distribution. In examining this queuing model, as

well as work by others, several possible factors relating to delay

interdependence may be identified. In a series of locks, if the

arrivals at a particular lock are Poisson distributed, then it is

likely that the effects from the previous locks have been diluted.

In other words, interdependence is related to the opportunity (or

lack thereof) for vessels to "randomize" between locks. Linehaul

distance, speed (mean and variance), volume, passing opportunities,

and network geometry are all likely to affect this opportunity.

Similarly, factors may be identified from the service process such

as utilization, relative utilizations, tow size distributions,

queue discipline, size and number of chambers, and lock

reliability.

117
4. SBiaulation Ixperizent

The stochastic nature of the lock delay problem under

assumptions of generalized arrival and service distributions may

limit the scope of an analytic model. It is indeed quite difficult

to develop expressions for average delays under these conditions

while capturing the effects of interdependencies. For this reason,

a simulation model proves to be a viable approach to obtaining

realistic relationships between certain factors and interdependence

among locks. Simulation models are appropriate when a complete

mathematical formulation of the problem does not exist or

analytical methods require excessively restrictive assumptions.

The Transportation Studies Center at the University of

Maryland has developed a microscopic waterway simulation model to

analyze the relationships between tow trips, travel times, delays,

and lock operations [Dai and Schonfeld 89]. The model traces the

motion and records the characteristics of each tow (e.g. number of

barges, commodity type, speed, origin, destination, direction, and

arrival times), while allowing for variability in many of the lock

queuing factors such as capacity, volume level, etc.

The model is event scanning, i.e. the status is updated by the

occurrence of one of five events 1) trip generation, 2) tow

entrance at locks, 3) tow arrival at destinations, 4) lock stall,

and 5) end of inventory period. The model places no restrictions

on the number of locks, chambers, cuts, waterway links, tows, O-D

118
pairs, and time periods. One limitation of the model, however, is

that it currently has only been validated for series network

geometry. The validity of the model has been tested by comparing

the model predictions with actual data along five Ohio River locks.

Traffic volumes were predicted quite accurately by the model, with

an average deviation of 1.53%. The waiting times at locks were

predicted within a 10% error. The estimates of these quantities

were made without any systematic bias. Therefore, the simulation

model, which is based on real (obtained from the Lock Performance

Monitoring System) data, can act as a surrogate in an experiment

involving variables believed to be important in lock

interdependence.

As is common with micro-simulation, the model developed by Dai

& Schonfeld (1989) requires excessive computer time to evaluate

numerous combinations of projects. In order to evaluate a

combination of projects, it is necessary to compute the total

average delays both at current capacity levels and improved

capacity levels. Also, for variance reduction purposes, it is

desirable to perfo:-m numerous runs (e.g. 30) for each observation.

Because the number of possible combinations of projects may be

large, a naive selection and sequencing technique would require the

complete evaluation of numerous combinations. For most project

proposal sets, using a micro-simulation model alone would be

prohibitively expensive.

119
An alternative to direct application of micro-simulation is to

employ the simulation model in an experiment to assess, in

functional form, the degree to which certain explanatory factors

contribute to lock interdependence. Specifically, an experiment

was done to explore the extent to which delays at a particular lock

are affected by changes in the characteristics of other locks. The

data generated from the experiment were employed to estimate

functions to be used as a substitute for the simulation model. A

model estimated from simulated data is termed a metamodel (Law and

Kelton 1982).

Because queuing delays are the source of interdependence, an

experiment was designed involving a system of locks that have the

total delay in the system as the response variable. Factor

variables were then chosen on the basis of queuing theory and the

opportunities for vessels to randomize between locks. The factor

variables found relevant for the simulation experiment are:

Distance, D - is the linehaul distance between locks and is

likely to be a strong indicator of the opportunity for the spacing

between vessels to randomize. The larger this distance, the

greater the vessel spacing randomization and the smaller the degree

of interdependence.

Critical utilization, p, - is the maximum volume to capacity

ratio in the system and is a measure of the extent to which traffic

120
is "metered" through a critical lock. Poisson distributed traffic

(a condition for independence) may be distorted by the metering of

traffic at a lock. Therefore a high critical utilization is likely

to give rise to interdependence.

Relative utilization, U - is the ratio of the utilization of

a given lock to that of the critical lock and measures the extent

to which the delays at a given noncritical lock may be dominated by

the delays at the critical lock.

Table 1 shows the values associated with each level of the

factor variables used in the simulation experiment. The range of

values included are derived from typical values observed in the

inland navigation system. A combination of values of the factor

variables represents a simulation case. For each case, the

simulation model must provide the data necessary to compute S/I.

The experiment involved the simulation of various systems of

two locks, labeled Lock 1 and Lock 2. Larger systems would not

allow for as large a range of factors and levels. The basic system

of locks simulated is shown in Figure 1. First, two locks with

given levels of the factor variables were simulated as a system,

where the interdependence is captured and included in the resulting

average delay, (Fig. 1 top). Second, two independent locks with

identical levels of the factor variables were each simulated as a

one-lock system (Fig. 1 bottom). Specifically, the average total

121
delay must be obtained for the locks acting as a system and for

both locks acting independently. The ratio of these two totals is

S/I.

To achieve the various levels of the factors for a two lock

system, Lock 1 was considered critical, i.e. it always had the

larger utilization. The capacity of Lock 1 was fixed at 60.6 tows

per day and the system volume was adjusted to yield the desired

utilization. For example, to achieve a critical utilization, pc,

of .89, the volume level used in simulation is 53.93 tows per day,

since 53.93/60.6 = .89. The desired utilization of the second lock

is obtained by the given level of U. For example, if U-.633 and p,

= .89, then the utilization of the second lock is .560. The

utilization of the second lock is achieved by adjusting its volume.

In this case, the volume of Lock 2 would be 33.94 because

33.94/60.6-.56. A summary of the utilizations and volumes of Lock

2 necessary to yield desired combinations of U and p, is provided

in Table 2.

An assumption of this simulation experiment for estimating

evaluation functions is that the only geometric configuration

considered is a series of locks. This assumption is reasonable

given the near tree-structure of the inland waterway network. A

possible expansion of this methodology would include simulation of

the junction points. A conservative number of independent

simulation runs per observation was found to be 30 in order to

122
sufficiently reduce the simulation variance. A conservative number

of tows required per simulation run to achieve this objective is on

the order of 13,000. Typically in the simulation runs, the first

1000 tows were deleted to allow the system to stabilize and results

were based on the next 12,000 tows in each run.

5. Experizent Results

Table 3 provides a sample of the data for utilization ratio of

1.00 and critical lock utilization of 0.89. Specifically, mean

isolated and system waiting times are shown for each of the two

locks for all three distance levels for directions 1 and 2. Also

included are the total system and isolated delays for both locks

over both directions. Finally, the corresponding value for the

interdependence coefficient, S/I, is computed from the total system

and isolated delays from traffic in both directions.

Some of the data are plotted in Figure 2 with S/I on the

vertical axis and utilization of the critical lock on the

horizontal axis. Each point in the plots represent the average of

30 runs. This and other plots were helpful in making first

assessments of the functional form of the interdependence

coefficient. It appeared from these plots, that the upper bound on

S/I is slightly less than 1.0 and that it decreases at an

increasing rate with the utilization of Lock 1. There was a

noticeable relation between the level of interdependence and both

the critical utilization and distance. The exploratory analysis

123
suggested that a functional model for S/I should include relative

utilization, critical utilization, and distance between locks as

variables.

The simulation output suggests that as the distance between

two locks increases, the amount of interdependence among those

locks also increases (S/I decreases). This result is consistent

with earlier studies, e.g. (Carrol 1972). In Section 3, it was

mentioned that lock interdependence is inversely related to the

opportunities for the intervals of vessels to randomize.

Intuitively, the distance between locks tend to increase the

randomization of traffic. On the other hand, distance may not be

the predominate influence on interdependence as some studies

suggest. For example, in Carrol (1972), distance was the only

variable considered. Based on these results, it appears that the

critical utilization has a more significant effect on

interdependence than distance.

6. Functional Estimation for S/I

In this section, a mathematical function that is reasonably

consistent with the compiled simulation data is derived. The

experiment results suggest that the functional form of the

interdependence coefficient is nonlinear in terms of the factor

variables with an upper bound of 1.0. However, since a lower bound

of 1/2 is based on a deterministic system, it may not be used in

establishing a functional form. A functional form that yields 1.0

124
at a volume level of 0 and decreases faster than linearly with

critical utilization would seem to closely fit the data as plotted.

One tractable mathematical form expressing such a relation is given

in Eq. 3:

S1 1 a0 D. Eq. 3

Expanding the p, and U terms for locks labeled 1 and 2 we obtain:

S= 1-a (max(p 1 ,p 2))' [min(p 11 p2),max(p1 ,p 2)]7 D26 Eq. 4

This relation may be interpreted to have an upper bound of 1.0 with

the second term representing a quantity of interdependence to be

subtracted. Conceptually, a max(p,)O represents the maximum

interdependence that may be possible, while LP and D' are

multipliers that determine the portion of the possible

interdependence that may be realized. The interdependence

coefficient may then be used to compute the total system delay of

a two lock system, S,2

S

S12 - (1 + 12). Eq. 5

In Eq. 5, I and 12 are the delays of the first and second locks

acting in isolation, respectively. Later it will be shown how the

values for I, and 12 were estimated.

125
While the functional form for S/I is nonlinear, it is

exponential and subject to logarithmic transformation.

S

Log(l--•-) =log a + #log p, + 7log U + 6log D Eq. 6

The estimation results for this model are shown as numerical

parameters in Eqn. 7. Converting the transformed variables to

their original form yields the following estimated model for the

interdependence coefficient of a two- lock system

s 1 - 0.713 (p) 2

. 455 U09" D4"-9 Eq. 7

7. Queuing Nature of Independent Locks

In a one-directional series of n isolated locks, the first

lock is independent of any previous lock. Therefore, the waiting

time at the first lock may be described by either an M/G/1 queuing

model, as shown by Burke's Theorem, or by a model estimated from

simulation results for isolated locks. All other locks in the

series will have a significantly more complex G/G/1 arrival

process. If an M/G/1 process is assumed for the first lock, then

the waiting time is expressed in Eq. 2. It may be more accurate to

estima%' a function for the delay at an independent lock from the

simulation data. A comparative plot between the delays for an

M/G/1 lock and the simulated data revealed that while the M/G/1

function is reasonable, an estimated function would be more

accurate. One functional form that includes the same parameters as

the theoretical expression for M/G/1 is the following:

126
1 = X6 ( 1 -p)b ac Eq. 8

Like the theoretical expression for M/GIl, this function is

asymptotic to the capacity as shown by the (l-p) term. The

parameters for this function were estimated using the simulated

data. Because this model is exponential, it also is subject to

logarithmic transformation.

log W = a log X + b log p + c log a 2 Eq. 9

The estimated values for the parameters of the isolated delay

model yields the following model

a 2.95

I = . Eq. 10 X.413 ( -p ) 1.9

a. Expanding from Two Locks to n Locks

The model for the interdependence coefficient, S/I, is

sufficiently consistent with the results of the simulation

experiment. Thus, for two lock systems, a satisfactory function

has been developed to evaluate interdependent projects without the

use of simulation (Eq. 5). However, it is likely that groups of

locks in a series of three or more, are interdependent. In this

section, a procedure for expanding the functional relationship for

127
interdependent 2-lock systems to n-lock systems, such as shown in

Figure 3, is described. Note that the assumption of a one-way

system is necessary in developing the expansion.

One possible method is to sum the interdependevrce among the

successive pairs in the system. To illustrate, Y12 is substituted

in place of the second term in the expression for the

interdependence coefficient and a subscript added to denote the

number of locks in the system. The variable Y may be referred to

as the interdependence variable since it represents the amount of

interdependence among locks.

Y12 = C (p.)o (U12 )7 (D12)' Eq. 11

S

(+)2 = 1 - Y12 Eq. 12

It follows that the coefficient for the three lock system would

include a term for the interdependence between Locks 2 and 3.

S

3 = 1 - (Y12 + YM) Eq. 13

However, this way of summing interdependence has some

shortcomings. First, the technique does not incorporate the

variance in lock service times. Second, the interdependence among

nonadjacent locks is not accounted for. To illustrate the

technique used to overcome such shortcomings, the three lock system

128
in Figure 4 is employed. The figure indicates that there is a

variance, utilization, arrival process, and departure process

associated with each lock. Distance D,2 separates Locks 1 and 2

while Dn separates Locks 2 and 3.

Note that Lock 1 has an independent arrival process, while the

arrival processes at the remaining locks are related to the

departure processes from the previous locks. Thus, the delay for

Lock 1 may be determined by using Eqn. 10. Enclosing Locks 1 and

2 in the figure establishes an effective lock, e2, which may act as

a proxy for the delays of Locks 1 and 2 combined. Associated with

the effective Lock e2 is also a variance, utilization, arrival

process, and departure process are also associated with effective

Lock e2. The arrival process of Lock e2 is the same as that of

Lock 1, while its departure process is the same as that of Lock 2.

Note that Locks e2 and 3 constitute a two lock system, i.e.

the 2-lock model for the interdependence coefficient estimated

directly from the simulation experiment may be applied. If pa and

ae. are found, then in terms of total delay, the three lock system

will be successfully converted to an equivalent two lock system.

Because the arrival process for Lock e2 is independent, Pa2 is the

utilization that yields the total system delay for Locks 1 and 2,

S,2 , but as an independent lock. Therefore Eq. 10 may be applied to

compute the delay for Lock e2, Su.

129
S a a (1-pz)b (a,)c Eq. 14

However, the delay for Lock e2, S,2, is the same as the total system

delay for Locks 1 and 2, S12.

Sa - S12 = 1 - a p.0 XP D6 = ( 1 -p.)b (G.)e Eq. 15

Next, an expression for the combined service time variance of

Locks 1 and 2, (a,2)2 is obtained. If it is assumed that the

service times of any group of n locks are independent, then the

variances may be added linearly to yield the system variance, (a") 2 .

a

2 2

a a Eq. 16

i-1

Therefore (a,)c may be replaced with (ao2 + a22) d2 in Eqn. 15. The

utilization of the effective lock, pa may now be solved directly

from 15. Solving for p. we have

Pa = 1 - (S12 k" (al2 + a2 2)-)1/b Eq. 17

The interdependence coefficient may now be computed for a three

lock system by adding the interdependence variable between Locks e2

and 3 to that of Locks 1 and 2, as illustrated in Figure 5. This

is equivalent to replacing Y¥ in Eqn. 10 with Ya,

130
S(,2 + ¥.) Eq. 18

where

Y,= a max(pa,p3)0 (Ud)7 (D23)' Eq. 19

and

U• : max r=in (p,2, (pa, ppO) 3 ) Eq. 20

The coupling technique may now be applied in succession to

yield the S/I for an N-lock system, (S/I)N. This is done by first

computing YN starting with Y2 = Y12. The following is an algorithm

for applying the technique in computing (S/I)K.

Ste2 1

Begin with the two locks farthest upstream and compute the

interdependence index between them.

Y2 = a max(plpP2 )0 (U12)' (D12)' Eq. 21

Using Y2, compute the interdependence coefficient for the first 2

locks.

(-)2 = 1 - Y2 Eq. 22

Set n= 3.

131
Compute the system delay for the first n-i locks.

S= (•-)•i I Eq. 23

Compute the combined standard deviation of service time for the

first n-1 locks.

2

a (a )1' Eq. 24

i-I

Steip4

Compute the effective utilization for the first n-1 locks.

P.i 1 - (S., X'4 (a.,))1/b Eq. 25

Compute the interdependence index between Lock en-i and Lock n.

YM = a max(pm- 1 p3.)O (U.)" (D.,.)j Eq. 26

Step 6

Compute the interdependence index for the n lock system.

Y' = Y.- + Y. Eq. 27

$teD 7

Compute the interdependence coefficient for the first n locks.

(S/I). = (S/I)., -Y, Eq. 28

If n=N, then stop, else increment n and go to step 2.

132
The above derivation for (SII), was performed using the

statistically estimated formula for the delay at the first lock in

the series, Eqn. 10. If an M/G/1 process is assumed for the first

lock in the series, then p. is determined from Eqn. 29 where WI,,

is replaced with S, 2. The remaining steps in the technique are

unchanged.

pa = .5(2 + 2XS 12)-((2 + 2XS12)2 + 4(X2(aa)c-2XS)]ln Eq. 29

The results from the simulation experiment have now been

expanded from incorporating systems with only two locks, to series

with any number of locks. The expansion technique first assesses

the interdependence of the first two locks estimated directly from

the simulation results. Next, an additional factor of

interdependence is added for the third lock. This additional

factor is not based on the interdependence between Lock 3 and Lock

2 only, as Eqn. 13 would suggest, but rather is based on the

interdependence between Lock 3 and the system composed of Locks 1

and 2. In a similar manner, terms for interdependence associated

with additional locks are added one at a time as suggested by the

iterative nature of Eqns. 21 through 28.

9. Validation of Lock Coupling

The coupling of locks can be used to obtain an S/I ratio that

reflects the interdependence within a system of more than two

locks. That ratio can then be used to calculate the total system

133
delay among the locks. Therefore, the effectiveness of the

coupling technique may be measured by the ability to yield values

of system delay that are within acceptable deviations from

simulated values.

To perform a test of the coupling technique, an experiment was

conducted to provide simulated results for a three lock system.

Comparisons of the total delay between the simulation model and the

coupled metamodel provides some measure of effectiveness for

coupling from a two lock system to a three lock system. Because

the systems simulated were bidirectional, the experiment also

provides an indirect validation of the one-directional assumption

employed by the coupling technique.

The experiment involved three lock systems with the same

utilization levels as the two lock simulation experiment, namely

.890, .750, .660, and .320. A total of 40 three-lock combinations

with these utilizations were simulated. The simulations were

conducted at a constant volume level of 30 tows/day and the

capacities of the locks were adjusted accordingly to yield the

desired utilizations. The distance between locks was 20 miles in

all cases. A range of standard deviations of service time, a,, were

considered by holding the coefficient of variation, a/p, constant

at 0.5. As with the two lock simulation experiment, there were 30

runs per each of the 40 observations and approximately 13,000 tows

per run.

134
Appendix 2 summarizes the numerical results of the experiment.

Tabulated are 1) the average delay from simulation observed at each

lock, 2) the variance of the service time at each lock, 3) the

computed delay of each lock in isolation, 4) the level of

interdependence as measured by the S/I ratio for each three lock

system, 5) the computed total simulated delay, 6) the computed

total delay, and 7) the percent deviation from simulated results.

The average deviation from simulated results is 10.06%.

Although there does not appear to be a systematic bias in the %

(not absolute) errors, the errors tend to be larger for systems

involving lower utilizations. For example, the system with all

three locks having a utilization of .320 has an error of 77.4%.

However, systems consisting of low-utilization locks account for a

significantly smaller amount of delay. This observation may be

illustrated by computing the absolute value of the deviations for

each system. The total of the absolute value of deviations for all

systems is only 7.24% of the total simulated delay for all systems.

10. Observations Concerning Lock Interdependenae

The generalized model for lock delay interdependence may now

be utilized to explore the nature of lock interdependence which

ultimately would reveal the impacts of interdependence on the

benefits associated with lock capacity improvements. This may be

done by 1) plotting the interdependence coefficient for different

values of the relevant variables, e.g number of locks, distance,

135
and capacity, and 2) performing a sensitivity analysis on the

expressions for S/I.

First, a plot of S/I for both a two lock system and three lock

system provides a first look at how the coefficient changes with an

inclusion of an additional lock in the system. In Figure 6, S/I is

plotted versus volume for a two lock system with each lock having

a capacity of 18 tows/day, a variance of 1.2, and distance of 20

miles. Also plotted are curves representing the inclusion of a

third lock which is identical in every respect except that its

capacity is 18, 30, or 100. Note that the curves do not shift

uniformly, but rather, the decrease in S/I associated with

increasing the size of the cluster from two to three locks

increases with volume. Also, as the capacity of the third lock

increases, the change in S/I associated with it decreases.

Next, the same system of two locks is plotted with curves

representing three different three-lock systems in Figure 7. The

three lock systems differ in their distance to the second lock,

which is 100, 20, or 5 miles, but are identical in all other

respects. This plot reveals that the farther away the third lock

is from the two lock system, the smaller its contribution to the

total interdependence. However, even very large distances, e.g.

100 miles, show some change in interdependence at very high

utilizations. It is clear that the addition of a third lock

reduces the value of S/I and the size of that reduction depends on

136
the values of various lock and system variables.

11. Summry and Conclusions

Inland waterway navigation continues to be a vital component

to our nations freight transportation system. With many inland

navigation locks in need of repair, it is important to have

reliable and efficient means of modeling the delays at locks for

evaluation and investment planning. This paper has shown the

development of an approximation to microsimulation that allows for

an evaluation of lock delays that incorporate interdependencies.

A simulation experiment was performed to obtain the data

necessary to calibrate the first step in the metamodeling approach.

The results of rimulation for both the system and isolated cases as

well as S/I have been presented. It has been shown that the model

of the factor variables for S/I adequately fits the data obtained

from the simulation model. The metamodel was expanded to a system

of iterative equations to incorporate systems of more than two

locks. A validation of the lock coupling technique showed that the

deviation from simulation for three lock systems averaged 10.1% for

systems involving utilizations ranging from .320 to .890.

Using the metamodel for S/I, some observations concerning lock

interdependence were made. It was found that the interdependence

of the system increases with system size. The size of the increase

depends on various lock characteristics. In addition to

137
discovering some aspects of lock interdependence, the model may be

used as part of an overall planning framework that includes project

sequencing and scheduling.

AcEnovledgaents

The authors wish to thank the U.S. Army Corps of Engineers,

Institute for Water Resources, Navigation Division for their

sponsorship of this research.

Appendix I: Referenoes

Albin, S.L., and Kai, S.R. (1986). "Approximation for the

Departure Process of a Queue in a Network. Naval Research

Logistics Quarterly. 33(1), 129-143.

Bertsimas, D. (1988). "An Exact FCFS Waiting Time Analysis for a

General Case of G/G/s Queuing Systems," Queuing Systems Theory

and Applications, 3, 305-320.

Carroll, J.L. (1972). "Analysis of Waterway Systems," Final

Report, U.S. Army Corps of Engineers, North Central Division.

Dai, D. M., and Schonfeld, P. (1989). "Reliability Analysis of

Coal Inventories on the Ohio River" Interim Working Paper,

U.S. Army Corps of Engineers, Institute for Water Resources.

138
DeSalvo, J.S., and Lave, L.B. (1968). "An Analysis of Towboat

Delays," Journal of Transportation Economics and Policy, 2,

232-241.

Law, A., and Kelton, W. (1991). Simulation Modeling and Analysis,

2nd Edition, McGraw-Hill, Inc., New York.

National Waterway Study (1983). U.S. Army Corps of Engineers,

Water Resources Support Center, Fort Belvoir, Virginia.

A Statistical Survey of Vessel Performance in Configuration

Characteristics on Inland Waterways (1983). Research Report

83-R-1, U.S. Army Corps of Engineers, Institute for Water

Resources, Fort Belvoir, Virginia.

Whitt, W. (1984). "Approximation for Departure Processes and

Queues in Series," Naval Research Logistics Quarterly, .1(4),

499-521.

139
Appendix II. Notation

The following symbols are used in this paper:

a,b,c Parameters for estimation

Dii Distance between locks i and j

I, Total isolated delay for lock i

n Number of locks in a given system

S Total system delay

S/I Ratio of system to isolated delay (interdependence

coefficient)

U Relative utilization P/pc

V/C Volume to capacity ratio

W Average wait time at locks with an M/G/1 queuing process

W, Estimated wait time for isolated locks

Y4 Amount of interdependence between locks i and j

Yq Amount of interdependence between the set of locks e and

lock j

a,06,7,6 Parameters for estimation

X Mean arrival time

Mean service time

p Lock utilization

Pej Effective lock utilization for the first j locks in the

system

PC Maximum of "critical" utilization in the system

a Standard deviation of service times

aoi Standard deviation of service times for first j locks in

the system

140
appendix III. Simulation Results for Three Lock Systtma

Volume level, X = 30 tows/day

Distance - 20 miles

p - utilization

a2 - variance of lock service time

I - computed isolated delay

(S/I) 3 - interdependence coefficient for the three locks

TW, - total delay obtained from simulation

TW, - total delay obtained from the meta model

Lock p W a2 I (S/I)3 TM, TMN. %ERR

1 0.890 24.349 1.560 34.25

2 0.890 22.454 1.561 34.27

3 0.890 24.443 1.552 33.99 0.751 71.246 76.990 8.06

1 0.890 25.801 1.560 34.25

2 0.750 2.477 1.113 4.27

3 0.890 25.330 1.552 33.99 0.770 53.608 55.853 4.19

1 0.890 25.742 1.560 34.24

2 0.660 0.900 0.784 1.42

3 0.890 25.276 1.552 33.99 0.781 51.918 54.396 4.77

1 0.890 26.825 1.560 34.24

2 0.320 0.199 0.213 0.06

3 0.890 26.098 1.552 33.99 0.823 53.122 56.202 5.80

1 0.750 2.846 1.113 4.27

2 0.890 23.750 1.561 34.27

3 0.890 25.205 1.552 33.99 0.770 51.802 55.863 7.84

1 0.750 3.018 1.113 4.27

2 0.750 2.612 1.113 4.27

3 0.890 26.497 1.551 33.98 0.802 32.127 34.098 6.13

1 0.750 3.167 1.113 4.27

2 0.660 0.994 0.784 1.42

3 0.890 26.042 1.551 33.98 0.813 30.204 32.230 6.71

1 0.750 3.030 1.113 4.27

2 0.320 0.227 0.213 0.06

3 0.890 27.050 1.551 33.98 0.844 30.307 32.316 6.63

1 0.750 2.547 1.113 4.27

2 0.890 31.306 1.561 34.27

3 0.750 2.961 1.107 4.24 0.790 36.814 35.586 3.33

141
Lock p W a2 1 (S/I) 3 TM. TM %ERR

1 0.750 3.463 1.113 4.27

2 0.750 2.903 1.113 4.27

3 0.750 3.171 1.107 4.24 0.820 9.538 10.474 9.82

1 0.750 3.434 1.113 4.27

2 0.660 0.902 0.784 1.42

3 0.750 3.214 1.106 4.23 0.830 7.550 8.238 9.11

1 0.750 3.547 1.113 4.27

? 0.320 0.190 0.213 0.06

3 0.750 3.218 1.106 4.23 0.862 6.955 7.375 6 05

1 0.660 0.869 0.784 1.42

2 0.890 25.865 1.561 34.27

3 0.890 25.595 1.551 33.98 0.781 52.329 54.407 3.97

1 0.660 0.938 0.784 1.42

2 0.750 2.956 1.113 4.27

3 0.890 26.622 1.551 33.98 0.813 30.516 32.232 5.62

1 0.660 0.992 0.784 1.42

2 0.660 1.131 0.784 1.42

3 0.890 27.017 1.551 33.98 0.828 29.140 30.490 4.63

1 0.660 1.010 0.784 1.42

2 0.320 0.223 0.213 0.06

3 0.890 27.730 1.551 33.98 0.856 28.963 30.345 4.77

1 0.660 0.939 0.784 1.42

2 0.890 33.315 1.561 34.27

3 0.750 3.204 1.107 4.24 0.800 37.458 33.091 11.66

1 0.660 1.185 0.784 1.42

2 0.750 3.530 1.113 4.27

3 0.750 3.318 1.106 4.23 0.830 8.033 8.240 2.58

1 0.660 1.289 0.784 1.42

2 0.660 0.938 0.784 1.42

3 0.750 3.303 1.106 4.23 0.846 5.530 5.989 8.31

1 0.660 1.289 0.784 1.42

2 0.320 0.236 0.213 0.06

3 0.750 3.368 1.106 4.23 0.874 4.892 4.993 2.06

1 0.660 1.152 0.784 1.42

2 0.890 29.154 1.561 34.27

3 0.660 0.971 0.784 1.42 0.813 31.277 34.875 11.51

142
Lock p W 02 I (S /I) 3 TM, TMO %ERR

1 0.660 1.422 0.784 1.42

2 0.750 2.801 1.113 4.27

3 0.660 0.993 0.784 1.42 0.842 5.216 5.988 14.80

1 0.660 1.486 0.784 1.42

2 0.660 0.811 0.784 1.42

3 0.660 1.025 0.784 1.42 0.857 3.321 3.658 10.12

1 0.660 1.505 0.784 1.42

2 0.320 0.165 0.213 0.06

3 0.660 1.169 0.784 1.42 0.885 2.839 2.568 9.54

1 0.320 0.222 0.213 0.06

2 0.890 26.768 1.561 34.27

3 0.890 27.338 1.551 33.98 0.823 54.328 56.215 3.47

1 0.320 0.226 0.213 0.06

2 0.750 2.732 1.113 4.27

3 0.890 26.044 1.551 33.98 0.844 29.002 32.318 11.43

1 0.320 0.256 0.213 0.06

2 0.660 1.215 0.784 1.42

3 0.890 26.814 1.551 33.98 0.856 28.285 30.345 7.28

1 0.320 0.259 0.213 0.06

2 0.320 0.267 0.213 0.06

3 0.890 28.284 1.551 33.98 0.890 28.809 30.348 5.34

1 0.320 0.195 0.213 0.06

2 0.890 33.286 1.561 34.27

3 0.750 3.442 1.107 4.24 0.843 36.922 33.079 10.41

1 0.320 0.209 0.213 0.06

2 0.750 3.751 1.113 4.27

3 0.750 3.461 1.106 4.23 0.862 7.421 7.378 0.59

1 0.320 0.216 0.213 0.06

2 0.660 0.976 0.784 1.42

3 0.750 3.448 1.106 4.23 0.874 4.640 4.993 7.60

1 0.320 0.225 0.213 0.06

2 0.320 0.300 0.213 0.06

3 0.750 3.848 1.106 4.23 0.913 4.373 3.969 9.23

1 0.320 0.190 0.213 0.06

2 0.890 31.413 1.561 34.27

3 0.660 1.026 0.784 1.42 0.856 32.629 34.993 7.25

143
Lock p w a2 I (S/I)3 T4, TM, %ERR

1 0.320 0.246 0.213 0.06

2 0.750 2.859 1.113 4.27 3 0.660 1.202 0.784 1.42 0.874 4.306 5.023 16.64

1 0.320 0.249 0.213 0.06

2 0.660 0.913 0.784 1.42

3 0.660 1.210 0.784 1.42 0.885 2.372 2.568 8.29

1 0.320 0.246 0.213 0.06

2 0.320 0.181 0.213 0.06

3 0.660 1.215 0.784 1.42 0.922 1.641 1.417 13.70

1 0.320 0.175 0.213 0.06

2 0.890 27.238 1.561 34.27 3 0.320 0.247 0.212 0.06 0.905 27.660 34.464 24.60

1 0.320 0.256 0.213 0.06

2 0.750 2.974 1.113 4.27 3 0.320 0.250 0.212 0.06 0.919 3.480 4.326 24.30

1 0.320 0.275 0.213 0.06

2 0.660 1.084 0.784 1.42 3 0.320 0.200 0.212 0.06 0.928 1.559 1.449 7.04

1 0.320 0.277 0.213 0.06

2 0.320 0.246 0.213 0.06

3 0.320 0.202 0.212 0.06 0.956 0.725 0.164 77.36

144
Table I Values of Factor Variables Used in Siaulatioa Experiment

Linehaul Critical Utilization

Distance UtLlization Ratio

(Miles)

Level 1 5 .320 .053

Level 2 20 .660 .369

Level 3 30 .750 .633

Level 4 80 .890 .845

Level 5 1.000

145
Table 2 -umary of Volume and Utilizations for Lack 2

U \ pcI 0.890 0.750 0.660 0.320

12 p2 12 p2 12 p2 12 p2

1.000 53.93 .890 45.45 .750 40.00 .660 19.39 .320

0.845 45.45 .SO 38.36 .633 33.94 .560 16.36 .270

0.633 33.94 .560 28.78 .475 25.33 .418 12.24 .202

0.369 20.00 .330 16.36 .270 14.54 .240 7.27 .120

0.053 2.85 .047 2.18 .039 2.12 .035 1.03 .017

146
Table 3 Results of SJimlation for U-1.00 and pco.89

U - 1.00

Lock 1: V/C - 0.89

Lock 2s V/C - 0.89

Lock 1 Lock 2

Mean I Mean 8 Mean I Mean S Tot. I Tot. S /S/ I ~II

Dir 1 47.184 33.362 47.184 34.336 1

5 DiU 2 46.856 33.337 46.856 34.006 I 94.04 67.52 0.7180

DLr 1 47.184 38.008 47.184 37.824 1

20 Dir 2 46.856 37.414 46.8S6 41.562 1 94.04 77.40 0.8231

Dir 1 47.184 39.477 47.184 40.046 1 1

30 Dir 2 46.856 40.721 46.856 38.928 I 94.04 79.59 0.8463

Dir 1 47.184 43.751 47.184 44.038

80 Dir 2 46.856 45.177 1 46.856 4S.091 1 94.04 89.03 0.9467

147
1 i2

System - S

1 2

Isolated Lock 1 Isolated Lock 2

ifoqure I system and Isolated Configurations for Simulation

148
1.1

W

I •U=.053

0.9 sU=.369

U=.633

.8U=.845

. U=1.O0 0.7

S 0.6

0.5

0.325 0.66 0.75 0.89

Utilization of Lock 1

Fiqure 2 Simulation Results for D=S

149
a, a 2 a.

0 12 D- c Lock I --- gLock 2 ON000 Lok

Pi P2 Pa

FigurO 3 Series of n Interdependent Locks

150
ao, 2

a, a'2 0'3

Independentc 1 Lock 2D Lock 3

Arrivals

P1 Lock e2 P2 P3

Pe2

Pigure 4 Conceptual System for Lock Coupling

151
Lock< 1 y 1 Lock 2 y e3ýLock 3

Lock e2

Pigure 5 Adding Interdependence Variables for a 3 Lock Systea

152
S 11

-

.• 0 9

2 Lock System

S• - Cap=30

- Cap= Is

08 .

• 0.6

S 0.5

04 I I

3.000 6.600 10.200 13.800 17.400

Volume (Tow/Day)

Figure 6 B/I for a 2 Look and 3 Lock Systesm of Capacity 100,

30 and 18

153
S0.9

0. 8 --- 2 Lock System D=100

'm 0.7

• -- D=5

' 0.6

0.5

0.4 I I

3.000 6.600 10.200 13.800 17.400

Volume (Tows/Day)

Figure 7 8/1 for a 2 Lock and 3 Lock Systems of Distance 100, 20,

and S

154
APPROXIMATE DELAYS DUE TO LOCK

SERVICE INTERRUPTIONS

by

Venkatesh Ramanathan and Paul Schonfeld

155
156
Approximate Delays Due to Lock Service Interruptions

by Venkatesh Ramanathan and Paul Schonfeld

March 3, 1993

Transportation Studies Center

University of Maryland

College Park, MD 20742

157
ABSTRACT

A model is developed in the form of one relatively simple equation to estimate

tow delays due to one lock service interruption. This model is developed by

combining deterministic queuing theory and an adjustment factor estimated

statistically from simulation results. The model provides accurate estimates of

delays more quickly and inexpensively than simulation

158
I.INTRODUCTION

A model is developed to estimate the delay due a single lock service

interruption in a waterway lock. The existing simulation model[ 1,2] was modified

to estimate the actual delay due a single stall. This delay can now be estimated

quite accurately with the proposed model. The model combines non-probabilistic

queuing relations based on uniform continuous arrival and service rates with a

factor derived from simulation results that accounts for discrete probabilistic

arrivals and service times.

2. PROPOSED MODEL

The effect of a single stall based on uniform continuous flow is shown in

Fig.1. The service interruption would reduce the normal capacity c to a partial

capacity p in a lock with multiple chambers. If the lock has a single chamber, the

partial capacity will probably drop to zero, in which case, the equation becomes

even simpler.

Let:

c = normal lock capacity ( tows/ hr)

p = partial lock capacity (tows/ hr)

d = duration of partial capacity condition or stall = queue growth time (hours)

v = traffic volume = tow arrival rate from both directions ( tows / hr)

s = queue dissipation time (hours)

159
FIG 1 EFFECT OF ONE STALL BASED ON UNIFORM CONTINUOUS FLOW

ARRIVAL RATE

(TOWS/HR) c I

v v

p ............ ... ...............

TIME (HOURS)

<d-- dTOWS IN QUEUE

L=,d (v-p)

• d • s TIME (HOURS)

160
Dd= deterministic delay with continuous flow (tow hours)

L = maximum queue length (tows)

The queue dissipation time s is given by:

= d(v - p) _ d(v - p) (1) -(v- c) (c -v)

The delay Dd for the tows is:

Dd = L(d + s) = dv ) (c~ - v) + ) (2)

If the partial capacity p is zero, then the delay Dd in equation 2 simplifies to:

Dd- dv (cdv) + d) (3)

Example:

If v = 30 tows/hr, c = 40 tows/hr, d = 2 hours and p = 1 tow/hr, then the delay Dd

will be, using equation 2:

Dd (( 1) (2(3- +2) 12 tow hours

The Eqs. 2 and 3 are general enough to apply to one or two way traffic and

to single or multiple chamber locks. Since their derivation is based on uniform

continuous arrival rates and service rates, they are only approximations for a real

waterway with probabilistic arrivals and service times.

161
3. SIMULATION RESULTS

The simulation model was modified to estimate the delay due to a single stall.

The delay due to a single stall is the difference between the delay with a stall and

the delay without a stall. The delay due to stall increases for the duration of the

stall and then decreases and finally becomes negligible. The results were recorded

for a sufficiently long period to insure that the full effects of the stall were

captured. The results from the simulation experiment were then compared with

those obtained using Eq.3.

The comparison was conducted for a variety of volume/capacity ratios ranging

from 0.4 to 0.95. To reduce the variance of the delay, the final result used for

comparison was obtained by averaging the output from 40 independent simulation

runs. To insure that the systems reach steady states before the stall occurs, the first

12,000 tow waiting times are discarded from each simulation run.

The results are shown in Table 1. Fig.2 shows that the discrepancy between

the simulated and theoretical delay approaches zero as the stall duration increases.

Fig 3 shows that similar results are obtained as the delay increases. Fig.3 is

consistent with Fig.2, if we remember that delay Dd varies roughly with the square

of stall duration d in Eq.3.

162
TABLE 1 COMPARISON OF SIMULITED DELAY WITH DETERMINISTIC

QUEUING DELAY

a) P = 25tows/day ; X = l0tows/day ; V/C = 0.4

Stall Delay(D,) Delay (Dd) Discrepancy(%) Std. Std. t-test

Dur (Simulated) (Determ.) (DS-Dd) Dev. Error Value

(Days) (tow-days) (tow-days) (Simul) (Simul)

Dd

0.3 0.985 0.75 31.3 0.41 0.07 3.625

1 9.597 8.33 15.2 2.25 0.36 3.560

2 36.630 33.33 9.9 9.31 1.47 2.242

3 79.740 75.00 6.3 12.34 1.95 2.430

4 138.400 133.33 3.8 21.26 3.36 1.508

6 306.160 300.00 2.0 39.85 6.30 0.978

8 541.920 533.33 1.6 71.21 11.26 0.762

10 843.500 833.33 1.2 86.50 13.68 0.745

12 1208.400 1200.00 0.7 116.23 18.38 0.457

b) p = 16.67tows/day ; X = l0tows/day ; V/C = 0.6

Stall Delay(D,) Delay (Dd) Discrepancy (%) Std. Std. t-test

Dur (Simulated) (Determ.) (D,-Dd) Dev. Error Value

(Days) (tow-days) (tow-days) (Simul) (Simul)

Dd

0.3 1.556 1.125 38.3 0.71 0.11 3.85

1 15.198 12.500 21.6 6.88 1.09 2.48

2 57.390 50.000 14.8 15.75 2.49 2.96

3 123.870 112.500 [0.1 19.61 3.10 3.69

4 213.470 200.000 6.7 33.46 5.29 2.55

6 465.890 450.000 3.5 63.15 9.98 1.59

8 818.600 800.000 2.3 86.42 13.66 1.36

10 1269.050 1250.000 1.5 92.10 14.56 1.30

12 1819.600 1800.000 1.0 139.96 22.12 0.89

163
TABLE 1 COMPARISON OF SIMUnATED DELAXY WITH DETRKMINISTIC

QUEUING DELAY

c) p = 12.5tows/day ; X. = 10tows/day ; V/C = 0.8

Stall Delay(D,) Delay (Dd) Discrepancy(%) Std. Std. t-test

Dur (Simulated) (Determ.) (D,-Dd) Dev. Error Value

(Days) (tow-days) (tow-days) (Simul) (Simul)

Dd

0.3 3.253 2.25 44.5 1.71 0.27 3.70

1 31.920 25.00 27.7 12.28 1.94 3.56

2 117.890 100.00 17.9 28.06 4.44 4.07

3 252.950 225.00 12.4 51.21 8.09 3.45

4 437.900 400.00 9.4 69.29 10.95 3.45

6 956.100 900.00 6.2 110.40 17.46 3.21

8 1672.110 1600.00 4.5 199.29 31.51 2.28

10 2586.610 2500.00 3.4 300.90 47.58 1.82

12 3682.85 3600.00 2.3 348.40 55.09 1.50

d) p = 10.53tows/day ; X I lOtows/day; V/C = 0.95

Stall Delay(D,) Delay (Dd) Discrepancy(%) Std. Std. t-test

Dur (Simulated) (Determ) (D,-Dd) Dev. Error Value

(Days) (tow-days) (tow-days) (Simul) (Simul)

Dd

0.3 13.29 9 47.7 5.21 0.82 5.20

1 132.06 100 32.0 41.56 6.57 4.87

2 489.25 400 22.2 126.30 19.97 4.46

3 1044.00 900 16.0 135.23 21.39 6.73

4 1787.59 1600 11.7 283.29 44.79 4.18

6 3902.84 3600 8.4 509.11 80.50 3.75

8 6798.16 6400 6.2 736.77 116.49 3.42

10 10493.12 10000 4.9 922.00 126.81 3.38

12 14907.31 14400 3.5 1196.53 189.19 2.68

164
FIG 2 VARIATION OF STOCHASTIC DELAY

ADJUSTMENT FACTOR WITH STALL

DURATION

1. 0 .- v/c = 04

-- vic = 0.6

- v/c = 0.

1. 40 ' vlc=O.95

STOCHASTIC

ADJUSTMENT 1. 30

FACTOR

Fs = (De/Dd)

1. 20

1. 10-

1. 00,

0 2 4 6 8 10 12 14

STALL DURATION (DAYS)

165
FIG 3 VARIATION OF STOCHASTIC DELAY

ADJUSTMENT FACTOR WITH

SIMULATED DELAY

-mO vic = 0.4

-mO v/c 0.OA

1m c" v/c = 0.4

1.4- ","o, v/c = 0.95

1.3

STOCHASTC

ADJUSTMENT

FACTOR

Fs = (De/Dd) 1.2

1.1

,.o.; .-.... ... ;o.. "-o......... ...... o- o

.1l 1 100 1000 10000 100000

DELAY (Ds) (TOW - DAYS)

166
4. STOCHASTIC DELAY ADJUSTMENT FACTOR

A stochastic adjustment factor (F,) was derived which, when multiplied with

the results of the Eq.3, produces results approximating the simulation output. This

factor was estimated statistically, using simulation results as data.

Let:

Dd = deterministic delay with continuous flow

Ds = simulated delay

De = estimated delay = Fý Dd

De

F, = stochastic adjustment factor -

Dd

Then:

Fs- Ds (4)

Dd

The stochastic adjustment factor which accounts for the probabilistic arrivals and

service times decreases with the stall duration and increases with the

volume/capacity ratio(Fig.2). The factor is large for smaller stall durations and

negligible at longer stalls. When the arrival rate is large, the factor is very small

even at smaller stall durations.

167
Eq.4 can be rearranged as:

Ds = [Fs] [Dd]

Substituting the statistically estimated relation for the stochastic adjustment factor

F, and Eq.2 for Dd we obtain the following:

Ds= [I + (0.6 ( v) eo-0-32 d)] [(dv P) ( d(V A) + d) ] 09

(5)

A comparison of the simulated delay D, and estimated delay De is given in

Table 2. It is observed that the deviation between the simulated delay and the

estimated delay is less than 2% for larger stall durations when the volume/capacity

ratios are less than 0.6. The deviation is also below 2% for stall durations less than

4 days when the volume/capacity ratios are above 0.6.

5. CONCLUSION

The model developed in this paper provides a good approximation for the

estimation of delay due to a single stall in a real waterway with probabilistic

arrivals and service times. The deviation between the estimated delay and the

simulated delay is less than 2% for larger stall durations when We volume/capacity

ratio is small and for smaller stall durations when the volume/capacity ratio is

large. Thus, it becomes a very fast and inexpensive substitute for simulation.

168
TABLE 2 COMPARISON OF SIMULILTD DELAY WITH ESTIMATED DELAY

a) p = 25tows/day ; X = lOtows/day ; V/C = 0.4

Stall Simulated Estimated % Deviation

Duration(d) Delay(D,) Delay(D,) 100 x (D.-D.)/D,

(Days) (tow-days) (tow-days)

0.3 0.985 0.907 7.91

1 9.597 9.621 -2.50

2 36.630 36.696 -1.80

3 79.740 79.875 -1.69

4 138.400 139.009 -0.43

6 306.160 305.370 0.26

8 541.920 537.16q 0.80

10 843.500 835.829 0.90

12 1208.400 1201.614 0.56

b) p = 16.67tows/day ; X = 10tows/day ; V/C 0.6

Stall Simulated Estimated % Deviation

Duration (d) Delay (D,) Delay (De) 100 X (De-Ds) /D.

(Days) (tow-days) (tow-days)

0.3 1.556 1.480 4.88

1 15.198 15.412 -1.40

2 57.390 57.850 0.80

3 123.870 123.581 0.23

4 213.470 212.780 0.32

6 465.890 462.105 0.80

8 818.600 808.800 1.19

10 1269.050 1255.875 1.03

12 1819.600 1803.600 0.87

169
TABLE 2 COMPARISON or SIMZULTED DELAY WITH ESTIMATED DELAY

c) p = 12.5tows/day ; k = l0tows/day ; V/C = 0.8

Stall Simulated Estimated % Deviation

Duration(d) Delay(D,) Delay(De) 100 x (De-Ds)/Ds

(Days) (tow-days) (tow-days)

0.3 3.253 3.195 1.78

1 31.920 32.775 -2.60

2 117.890 120.200 1.95

3 252.950 254.540 -0.60

4 437.900 434.000 0.89

6 956.100 931.500 2.57

8 1672.110 1624.000 2.87

10 2586.610 2515.959 2.73

12 3682.850 3609.684 2.01

d) p = 10.5tows/day ; k = l0tows/day ; V/C = 0.95

Stall Simulated Estimated % Deviation

Duration(d) Delay(D,) Delay(De) 100 x (D.-D,)/Dý

(Days) (tow-days) (tow-days)

0.3 13.290 13.500 1.58

1 132.060 137.000 -3.74

2 489.250 496.000 -1.36

3 1044.000 1039.500 0.40

4 1787.390 1761.610 1.45

6 3902.840 3753.360 3.80

8 6798.160 6514.560 4.17

10 10493.120 10075.000 3.98

12 14907.310 14445.000 3.09

170
REFERENCES

1. Dai M.D.M. and Schonfeld P. (1989) Simulation of Waterway Transportation

Reliability. Transportation Research Record.

2. Dai M.D.M. (1991) Delay Estimation on Congested Waterways. Dissertation

Series UMCP-TSC-DS-91-2. University of Maryland.

171
172
COMPUTATIONAL CHARACTERISTICS OF A NUMERICAL

MODEL FOR SERIES OF WATERWAY QUEUES

by

Chung-Hung Wei, Melody D. M. Dai, and

Paul Schonfeld

Published in Transportation Research Record

Volume No. 1333, Paper No. 920798

173

m " -~i 'i i• i •| i " Si " • i =- iiI
174
COMPUTATIONAL CHARACTERISTICS OF A NUMERICAL MODEL

FOR SERIES OF WATERWAY QUEUES

Second Revision March 1992

7650 word equivalents

by

Chien-Hung Wei

Melody D.M. Dai

Paul M. Schonfeld

Transportation Studies Center

University of Maryland

College Park, MD 20742

Tel (301) 405-1954

Fax (301) 314-9867

Paper No. 920798 accepted for publication in the Transportation Research Record

¶It I _ 175
Wei, Dai, & Schonfeld

ABSTRACT

A numerical method has been developed for estimating delays on congested waterways

represented by series of G/G/1 queues, i.e., with generally distributed arrival and service times

and one chamber per lock. It is based on a metamodelling approach which develops simple

formulas to approximate the results of simulation models. The functional form of the

metariiodels is derived from queueing theory while their coefficients are statistically estimated

from simulation results. The algorithm scans along a waterway and sequentiaUy estimates at each

lock the arrival distributions, departure distributions, and delays. It can be applied to systems

with two-way traffic through common bi-directional servers as well as to one-way traffic systems.

Computational results are presented in this paper to illustrate the speed and convergence

properties of the algorithm and to investigate some of its variants. The algorithm works

satisfactorily and flexibly with different convergence criteria and scanning processes. For an

illustrative 20-lock system, parameter estimates converge within five iterations and less than three

seconds of CPU time to differences lower than 0.1 percent between successive iterations. The

computation time is found to increase only linearly with the number of locks in the system, thus

allowing the analysis of very large systems of interdependent queues.

176
Wei, Dai, & Schonfeld 2

INTRODUCTION

Inland waterway transportation is quite important in the U.S. and elsewhere, especially for heavy

or bulky commodities, since it is inexpensive, energy efficient and safe. Most U.S. waterways

consist of stepped navigable pools formed by dams across natural rivers. The lock structures

used to raise or lower vessels between adjacent pools constitute the major bottlenecks in the

waterway network (1) and generate extensive queues. Some locks have only one chamber, while

others may have two parallel chambers whose characteristics may differ. The service time

distributions at locks depend heavily on chamber size and tow size distributions. The lock

service time distributions would be affected by the chamber assignment discipline at locks with

two dissimilar chambers.

The waterway locks constitute a series of queueing stations. In queueing terms, locks are

the servers and tows are customers waiting to be served by locks. Tows from both directions,

upstream and downstream, share the same lock servers while in most other queueing systems

servers are exclusively one-directional. Hence, the term "two-way traffic operations"

characterizes the lock system analyzed below.

Arrival and service time distributions at locks are fairly complex. Carroll (2) and Desai (3)

found that service times are not exponentially distributed, and arrivals are not Poisson distributed.

Other standard distributions have been tested for the present study, without consistent success.

Thus, empirical distributions (specified for 50 intervals) are used here for simulation while

general tabular ditributions, described usually only by their means and variances, are used for

queueing models. Although locks with a single chamber may be modeled as G/G/1 queueing

systems (i.e., "general arrival/general service times/l server per station"), locks with two parallel

177
Wei, Dai, & Schonfeld 3

chambers may not be treated simply as G/G/2 queueing systems unless these chambers are

identical.

Considerable interdependence may exist among locks in a series. The departure distributions

differ from the arrival distributions since the service time distributions change the tow headways.

Departures from one lock usually affect arrivals at the next lock. Interdependence among locks

increases the difficulty in estimating systemwide delays since the interarrival time distributions

from adjacent locks must be identified at each lock. Two-way traffic operation through common

servers complicates the interdependence of lock delays and precludes the use of some otherwise

interesting queueing models.

Random failures (called "stalls") contribute significantly to the difficulties in estimating

delays. Stalls, which interrupt lock operations and thereby increase delays, are relatively rare

compared to other events and quite difficult to predict. Thus, Kelejian's efforts to model stall

frequencies and durations have not yet yielded strong results despite the rigorous statistical

methods employed (4).

The following special problems are encountered in estimating delays of waterway queues:

1. Arrival and service time distributions are too complex for analytic solutions and do not

match known statistical distributions.

2. Parallel chambers are not identical.

3. Service time distributions are affected by the chamber assignment discipline.

4. Considerable interdependence exists among a series of locks.

5. Two-way traffic operates through bi-directional chambers.

6. Arrival distributions depend on distances and speed distributions between locks, as well as

178
Wei, Dai, & Schonfeld 4

departures from adjacent locks.

7. Stalls increase the means and variances of delays.

Delay estimation for a realistic lock queueing system has been undertaken by Dal and

Schonfeld (5,6,7) using several approaches, including queueing theory, simulation, and numerical

methods. Their simulation model does deal with all seven problems listed above and is quite

efficient for analyzing particular system configurations. However, when large numbers of system

alternatives must be evaluated for investment scheduling, a much faster numerical method, which

approximates the results of simulations, becomes preferable. The primary purpose of this paper

is to assess the computational characteristics of the numerical method developed for this role.

In particular, the number of iterations and the computation time required to reach convergence

using various criteria and scanning procedures are investigated. The effects of system size (i.e.,

number of locks) on computational requirements are also examined.

LITERATURE REVIEW

The available analytic solutions for estimating delays in G/G/1 queues are quite inadequate.

Kleinrock (8) suggested an approximation solution for a G/G/1 queue with heavy traffic, which

is a useful upper bound for average waiting times in G/G/i queues. Bertsimas (9) derived an

exact solution for mixed generalized Erlang distributed arrivals and service times. However,

without a departure function this result is difficult to extend to a series of locks.

Exact solutions for networks of queues are still limited to Markovian networks. For more

general networks of queues, approximation methods are employed in Whitt (10) and Albin (11)
Wei, Dai, & Schonfeld 5

for system performance analysis. The underlying concept is to decompose the network into

individual queues that are analyzed independently, and then recombine the results. Their efforts

are quite valuable but employ unreasonable coefficients of variation (standard deviation divided

by mean) and are not applicable to bi-directional servers.

System simulation models to analyze lock delays and tow travel times were developed by

Howe (12) and by Carroll and Bronzini (13). These two models, which did not account for

stalls, required considerable data and computer time. However, simulation models can, in

principle, represent the complexities of traffic on waterway networks much better than analytic

queueing models.

A new waterway simulation model was developed by Dai and Schonfeld (5). This model

accommodates generally (i.e., arbitrarily) distributed trips and service times. It can also evaluate

stall effects. This simulation model requires only a few seconds to a few minutes on a PS/2

computer for each run, depending on traffic volumes, simulation period durations, network size,

etc. Still, it is hardly affordable for direct application in large combinatorial network investment

problems.

To avoid the computational expense of simulation, a metamodelling approach (14) was

developed. This approach consists of 1) developing and validating a simulation model to

represent waterway networks with queues a, locks, 2) formulating functions developed from

queueing theory for delays through series of locks, 3) statistically estimating the parameters of

these functions using simulation results, and 4) employing an iterative sequential scanning

procedure to estimate interarrival and interdeparture time distributions lock by lock until results

converge at each lock. Thus, relatively simple equations may serve as a proxy for the simulation

180
Wei, Dai, & Schonfeld 6

model.

SIMULATION MODEL

The simulation model developed for this work is documented in Dai and Schonfeld (5). Only

a brief description is provided below.

The simulation model was developed using the PMS (lock Performance Monitoring System)

data base, which includes very detailed information on traffic through the locks as well as

physical aspects of lockages (15). The simulation model is programmed in Fortran-77, which

provides great flexibility in modelling. Basically, it is a stochastic, microscopic and eventscanning simulation model which can handle any distributions for trip generation, travel speeds,

lock service times and tow sizes. Currently, tabular distributions based on empirical observations

are used for most input variables. A FIFO (First-In-First-Out) service discipline is currently

employed. This model simulates two-way traffic through common servers and accounts for stalls.

The validation results (5,6,7) show that the overall mechanism of the simulation model is

correct, and that the simulated average waiting times for each lock and for the entire series of

locks are closely similar to those observed. Dai (6) documents the statistical methods used in

developing, validating, and applying the simulation model.

NUMERICAL METHOD

Overview

A numerical method has been developed for estimating delays through a series of queues with

bi-directional servers. A brief description of the method follows, while details of its development

181
Wei, Dai, & Schonfeld 7

and validation are provided elsewhere (6,7).

The method consists of three major modules, namely arrival processes, departure processes,

and delay functions (as summarized in FIGURE 1), which are applied in that sequence at each

jock. The basic concept is to decompose the waterway system into locks (which remain

interdependent since they are affected by inflows from adjacent locks), identify the parameters

of the interarrival and interdeparture time distributions for each lock, and then estimate the

implied waiting times. The structure of the equations used in each module is based as much as

possible on queueing theory, while the parameters in those equations are statistically estimated

based on simulation results. Currently, the following assumptions are used in the numerical

method:

1. Interarrival times and service times are generally distributed.

2. Each lock has one chamber.

3. Inflows and outflows occur only at the two end nodes of a series of locks.

4. The average upstream volumes are equal to the downstream volumes in the long run.

5. The long-run volume to capacity (V/C) ratio is less than 1.0 at every lock.

It should be noted that Assumptions 2, 3, and 4 are only applicable to the numerical method.

The simulation model is not limited by those assumptions. The numerical method can provide

a quick and inexpensive analysis of lock delays. However, Assumptions 2, 3, and 4 limit fairly

sigi•ificantly the applicability of the currently developed numerical method and necessitate the

substitutira of the simulation model when significant deviations from those assumptions must be

considered. With some extensions to the numerical method, Assumptions 2 and 3 may be

182
Wei, Dal, & Schonfeld 8

eliminated. Assumption 4 could be relaxed fairly easily even though it is usually realistic for

waterways. Assumptions I and 5 should be kept since they reflect realities rather than analytic

limitations.

Structure of Numerical Method

To estimate delays in a queueing system, we need to know the means and variances of the

interarrival, interdeparture and service time distributions. For series of G/G/i queues and bidirectional servers, a difficulty arises in identifying the variances of interarrival and interdeparture

times. Because the interarrival times at each lock depend on departures from both upstream and

downstream locks, the variances of interarrival times cannot be determined from one-directional

scans along a series of queues. To overcome such complex interdependence, an iterative

scanning procedure is proposed. The core concept is to decompose the system into individual

locks and then sequentially analyze each of those locks. At each lock, the tow arrivals from both

directions are first combined into an overall arrival distribution and then split into two directional

departure distributions.

The algorithm is initiated by scanning along waterways from either direction, sequentially

estimating the interarrival and interdeparture time distributions for each lock. Initially assumed

values for the variances of interdeparture times from the opposite direction must be provided for

the first scan. Then, the scanning direction is reversed and the process is repeated, using the

interdeparture time distributions for the opposite direction estimated in the previous scan.

Alternating directions, the scanning process continues until the relative difference in the

preselected convergence criteria stays within preset thresholds through successive iterations.

183
Wei, Dai, & Schonfeld 9

Waiting times at locks can be computed in every iteration (and then used as convergence criteria)

or just once after all iterations are completed.

Arrival Processes

The mean and standard deviations of interarrival times are estimated in two steps. First, the

means and standard deviations of directional interarrival times at a particular lock are estimated

from the interdeparture time distributions of the adjacent locks. If flows are conserved between

locks and if the V/C ratio is less than 1, such relations are represented in Eq. 1 (variables are

defined in FIGURE 1):

'q# =- 'dk "k=i+ k=i-l, 1, if j=- if j=2

Because speed variations change headway distributions between locks, Eq. 2 was developed

to estimate the standard deviation of directional interarrival times at one lock.

Sa k+.0 2 5 11n(l+ o) ik-i-1, ifj=l

Q1 O 1 k=i+1, if j=2 (2)

(.002)

R2 = 0.999954 n = 107 s, = 0.0586 p = 5.1685

This suggests that, theoretically, the standard deviation of directional interarrival times should

be equal to the standard deviation of directional interdeparture times plus an adjustment factor

depending on the speed distribution and distance.

Second, the overall mean and coefficient of variation of interarrival times for this lock are

184
Wei, Dai, & Schonfeld 10

estimated based on the coefficients of variation of directional interarrival times.

- _ tj,* tgau* (3)

MA - -

Cj2 = 0.179+0.41(Can 2 + Caa 2) (4)

(0.027) (0.014)

R2 = 0.9188 n = 79 se = 0.0059 la = 0.988

In Eq. 4, the coefficients of variation of upstream and downstream interarrival times carry

the same weight in estimating the overall variance of interarrival times, since the mean

directional trip rates are equal (Assumption 4).

Departure Processes

The departures module estimates the mean and coefficient of variation of interdeparture times.

Based on the flow conservation law, if capacity is not exceeded, the average directional

interdeparture equals the corresponding interarrival time:

t+- (5)

The coefficient of variation of interdeparture times is estimated in two steps. First, the

coefficient is estimated for combined two-directional departures. Departure processes with

generally distributed arrivals and service times are analyzed using Laplace transforms (8). Some

analytic relations obtained are shown in Dai (6). The following metamodel was eventually

developed to bypass the difficulties of determining the variances of the lock idle times:

185
Wei, Dai, & Schonfeld 11

CD2 = 0.207 +0.795(CA2(1-p)+p)+ 1.001(CS2p2 -p) (6)

(0.065) (0.066) (0.0046)

R2 = 0.9984 n = 79 s. = 0.0058 p = 0.8311

Next, the coefficient of variation of directional interdeparture times is estimated. The

following metamodel was developed for this purpose:

CO 2 = 0.518+0.491C, 2 Ci 2 (7)

(0.0056) (0.0068)

R2 = 0.9710 n = 158 s. = 0.013 p = 0.9164

Delay Function

The delay function is intended to estimate the average waiting time at a lock. By applying

Marshall's formula for the variance of interdeparture times (16), an exact solution for the average

waiting time W, was obtained as follows:

A2 +20s2-o2

w -D A (8) 2 tA(l - p)

In this delay function, the average waiting time increases as the variance of interarrival and

service times increase and decreases as the variance of interdeparture times increases. The

average waiting time approaches infinity as the V/C ratio approaches 1.0.

Comparison of Simulated and Numerical Results

To validate the numerical method, its results were compared to the results of the previously

validated simulation model. Various system configurations were compared, including the

186
Wei, Dai, & Schonfeld 12

relatively large 20-lock system shown in TABLE 1.

The parameter values for this test system (e.g., means and standard deviations of input

distributions and distances between locks) were obtained from random number generators, except

for traffic volumes, which were assumed to be 10 tows/day in each direction throughout the

system. TABLE I shows the input parameters and a comparison of waiting times, which are the

output variable of greatest practical interest. It can be seen that the numerical model estimates

aggregate waiting times within 7.85% of the simulated ones. At individual locks the percent

error can be considerably greater, especially when absolute errors are very small (e.g., in

comparisons with zero waiting times). The comparisons of intermediate outputs, e.g., the

parameters of directional interarrival and interdeparture time distributions, show that differences

below 10% are achieved. The detailed validation results are presented in Dai (6).

COMPUTATIONAL TESTS

A number of computational tests have been conducted to investigate the speed, accuracy and

convergence properties of the numerical method. Some of the results obtained are presented

below. All these results were obtained with the two-directional iterative algorithm (coded in

Fortran-77) compiled and executed on an IBM PS/2 model 70 personal computer with an 80386

processor and an 80387 math coprocessor.

Any variable that is computed in every iteration of the algorithm may be used to check for

convergence and stop the algorithm when further changes between iterations become arbitrarily

small. The most interesting candidate variables for convergence criteria are the variances in the

interdeparture times from each lock (which affect error propagation) and the waiting times in

187
Wei, Dai, & Schonfeld 13

queues (which are the output variables of greatest practical economic interest).

The convergence threshold may be specified as a relative change in the value of a variable

from one iteration to the next (i.e., a ratio or percentage change) or an absolute difference. The

ratios may be very large if and when some variable values approach zero even though absolute

differences may be insignificant.

Convergence may be sought based on aggregate or systemwide outputs (e.g., total delay per

tow through a series of locks) or may be based on localized outputs (e.g., delay at each lock).

In principle, it should be easier to reduce changes between iterations to x percent for a

systemwide variable than for every single location in that system.

The original algorithm used the squared coefficients of variation of directional interdeparture

times (VARDEP) as the convergence criteria. In this work, the individual lock waiting times

(LOCWAIT) and system weighted waiting times (SYSWAIT) are also tested as convergence

criteria. Waiting times must then be computed in every iteration rather than just at the end.

The required inputs for the algorithm include the inflow rates, V/C ratio and service time

variance at each lock, distances between locks, means and standard deviations of tow speed

distributions, and the choice of convergence criterion. We generally used 0.001 as the

convergence threshold, i.e., results were considered sufficiently accurate and additional iterations

were deemed unnecessary when the variables chosen as convergence criteria changed by less than

0.1% from the previous iteration.

3-Lock Systems

The first test concerns the eight 3-lock -ystcms an.Jyz,..d in Dai (6). These eight systems

188
Wei, Dai, & Schonfeld 14

(described in TABLE 2) were originally used to show the performance of various algorithms.

The distances and speed distributions between locks were kept equal within each of these eight

systems. Using VARDEP, LOCWAIT, and SYSWAIT as convergence criteria, the estimated

individual lock delays and system delays, and number of iterations required are listed in TABLE

3. Also included are the simulated waiting times. Generally, the three criteria perform equally

well for each of the eight systems in terms of number of iterations required for convergence. The

SYSWAIT criterion produces slightly faster convergence than the others.

While assessing the differences in the number of iterations required with various criteria in

System 1, we found that delays at low V/C ratios are so small that relative differences may be

large and unstable even for very small changes in the absolute magnitudes of delays.

Consequently, more iterations are required to satisfy a relative threshold. If, instead, we set an

absolute threshold for delay, e.g., less than 0.001 hr/tow difference between successive iterations,

System 1 converges at the fourth iteration for both LOCWAIT and SYSWAIT.

We also sought to check whether the convergence was monotonic, i.e., whether the changes

always decrease through successive iterations. We found that relative changes decrease

monotonically for all systems when SYSWAIT, but not VARDEP or LOCWAIT are the

convergence criteria. However, the magnitudes of various criterion variables change

monotonically through successive iterations for all systems, as shown in TABLE 4. It seems that

monotonic convergence is more difficult to achieve for local variables when the algorithm scans

along the series of locks in alternating directions. When an iteration is defined as a two-way

scan (e.g., first upstream, then downstream, and only afterwards compare results to previous

iterat~in) monotonic convergence is achieved for the local variables LOCWAIT and VARDEP.

189
Wei, Dai, & Schonfeld 15

It is achieved without two-way iterations for the aggregate variable SYSWAIT which,

incidentally, requires 3-12% less CPU time than the local criteria.

The algorithm was also allowed to run for 100 iterations to check the convergence and CPU

times for various criteria. The results were quite satisfactory since no system ever diverged in

this experiment. This is illustrated in FIGURE 2 using System 6 as the example.

20-Lock Systems

To further check the behavior of the algorithm, we randomly generated parameter values for a

20-lock system in which the values of the V/C ratio were uniformly distributed between 0 and

1, and the coefficients of variation of service time were uniformly distributed between 0.2 and

1.0. This test system was assumed to have equal mean inflow rates in the two directions, as well

as identical tow speed distributions and distances between any pair of locks. TABLE 5 describes

this 20-lock system.

The aggregate results for the 20-lock system are summarized in TABLE 6. We found that

the number of iterations required for convergence within 0.001 is almost identical to the numbers

in TABLE 3, even though this 20-lock system is more than six times larger. This suggests that

the algorithm may be applicable for very large systems. Comparisons of CPU times required for

convergence again confirm that the aggregate criterion SYSWAIT saves iterations compared to

the local criteria LOCWAIT and VARDEP and reaches convergence with approximately 25% less

CPU time. As in 3-lock systems, the 20-lock system never diverges and the monotonic

properties with various criteria are similar. With the LOCWAIT criterion a single violation of

monotonic convergence was found at lock 2 in the 4th iteration. Consequently, one more scan

190
Wei, Dai, & Schonfeld 16

is desired to bring the entire system into convergence. Such violations were never found when

the aggregate convergence criterion SYSWAIT was used or when iterations were defined to

consist of two scans in alternate directions.

The relation between system size and computational requirements was also examined using

the 20-lock system and arbitrarily chosen subsets of that system. The CPU times and number

of iterations required for convergence in various system sizes are shcwn in FIGURE 3. It again

seems very promising that the number of iterations does not change much for different criteria

and system sizes. The CPU times seem roughly proportional to system sizes in all cases.

FIGURE 3 demonstrates the apparently linear relations. We sought to statistically estimate the

relations between CPU time and the number of locks in the system, using the following structural

form:

CPU, = K, (9)

In Eq. 9 CPU, is the central processing time using convergence criterion i, K, and P, are

statistically estimated parameters associated with criterion i, and N is the number of locks in the

system. The P, parameter was expected to be very close to 1.0, based on the nearly linear

relations shown in FIGURE 3, and indeed turned out to be nearly 1.0, confirming the essentially

linear relation. The value of Pi was, therefore, fixed at 1.0 and the remaining parameter K,- was

estimated as shown in TABLE 7.

The small standard errors and high R2 again confirm that CPU time is essentially linear with

respect to the number of locks in the system. Among the three criteria, the aggregate criterion

SYSWAIT has the smallest standard error and highest R2, suggesting it yields not only the fastest

191
Wei, Dai, & Schonfeld 17

but also the most predictable computer times. The structural form of Eq. 9 forces the computer

time function through the origin, since Eq. 9 has no intercept. When an intercept A, is provided

in Eq. 10 (presumably to reflect the fixed times required for setup or input and output functions),

even better fits were obtained, as shown in TABLE 7.

CPUL = Ai + Kj N (10)

The best fit is again obtained for the SYSWAIT criterion. Thus, based on our very small sample,

the best estimate of CPU time (in seconds to reach convergence within 0.001) for N-lock systems

is obtained with the SYSWAIT criterion as:

CPU = 0.107 + 0.0853N (11)

TABLE 6 shows that convergence to within 0.1% difference between successive iterations

is reached in 1.75 seconds of CPU time for the 20-lock system and SYSWAIT criterion. The

corresponding time for the simulation model to analyze the same 20-lock system on the same

computer is 53 minutes per replication, i.e., 1,590 minutes or 95,400 seconds for 30 replications.

Thus, in this case simulation requires 54.514 times more CPU time than the numerical method.

However, it should be noted that our simulation runs were designed to extract very precise

estimates for estimating new metamodels. We usually simulated 22,000 tows, discarded the first

10,000 of those, and replicated the simulation 30 to 80 times for each "data point". For practical

application, the simulation would require 10r to 105 times more CPU time than the numerical

method.

192
Wei, Dai, & Schonfeld 18

Double Scanning Versus Single Scanning

In the baseline algorithm an iteration consists of scanning the waterway from one end to the

other, i.e., in one direction. The next iteration would then scan in the opposite direction. The

results obtained so far suggest that a smoother convergence may be obtained by double scanning,

i.e., checking for convergence only after two full scans in opposite directions are completed.

With such double scanning, the changes in variables are always found to decrease (or at least not

increase) with each successive convergence check, which is performed every second iteration by

comparing iteration i with iteration i-2 (instead of i-i).

However, double scanning imposes a computer time penalty by increasing the number of

iterations required for convergence to a specified threshold. That is shown in TABLE 8 where

the convergence threshold is still 0.001. There are two reasons for the penalty. First, an even

number of iterations is required in double scanning, even when convergence is reachable with

one less iteration. Second, a larger change may be expected after two iterations than after one,

making the same threshold (e.g., 0.001) harder to satisfy.

Thus, it seems that double scanning provides added reassurance that the algorithm converges

in a smooth and well behaved way. However, since convergence seems so assured regardless

of scanning procedure, it seems preferable to opt for the computation savings of single scanning.

CONCLUSIONS

A numerical method has been developed to estimate waterway travel times through a series of

lock queues. This numerical method was estimated from simulation results. It can approximately

duplicate simulation results for complex systems of interdependent queues, while requiring

193
Wei, Dai, & Schonfeld 19

l104~10 times less computer time than simulation. The basic approach used in this numerical

method and several of its components (or "metamodels") should lead to numerical analysis

methods for other types of queueing networks with greater complexity.

This paper focused on the main computational characteristics of the baseline numerical

method and some of its variations. The main computational findings are as follows:

1. Variables other than the original interdeparture time variance VARDEP are suitable as

convergence criteria. In particular, the aggregate waiting time SYSWAIT yields convergence

faster than the other variables considered. Not surprisingly, more iterations may be needed

if a specified convergence threshold (e.g., 0.1%) is to be satisfied at every location and in

every direction rather than for an aggregate criterion.

2. Convergence to within 0.1% of values in the previous iteration is achieved relatively quickly

(typically in 4-6 iterations), even when that 0.1% threshold must be satisfied everywhere in

a 20-lock system.

3. Convergence is achieved smoothly and, with rare exceptions, differences in the variable

values decrease with each successive iteration. The exceptions are all traceable to scans in

alternating directions, and can be avoided by double scanning before convergence checks or

by always scanning in the same directions. However, since convergence seems always

assured, the single scanning in alternating directions seems preferable to save computer time.

4. The computer time required by the algorithm seems to be linear with respect to the number

of locks in the system. It also seems to be quite predictable. Thus, the numerical method

should analyze efficiently relatively large systems of interdependent queues.

"194
Wei, Dai, & Schonfeld 20

ACKNOWLEDGEMENT

The authors wish to thank Dr. George Antle and the Institute for Water Resources of the U.S

Army Corps of Engineers for advice and support in this work.

REFERENCES

1. National Waterways Study - A Framework For Decisi'on Making - Final Report. Water

Resource Support Center, Institute for Water Resources, U.S. Army Corps of Engineers, Fort

Belvoir, Virginia, 1983.

2. J.L. Carroll, M.S. Bronzini and J.C. Hayward. Analysis of Waterway Systems: Su-nmary

Report. Pennsylvania Transportation and Traffic Safety Center, University Park,

Pennsylvania, 1972.

3. K.J. Desai. Analysis of Waterway Systems: A Study of Tow Arrival Processes at Inland

Waterway Locks. Pennsylvania Transportation and Traffic Safety Center, University Park,

Pennsylvania, 1972.

4. H. Kelejian. A Study of Lockage Stall Probabilities. Institute for Water Resources, U.S.

Army Corps of Engineers, Fort Belvoir, Virginia, 1989.

5. M.D.M. Dai and P.M. Schonfeld. Simulation of Waterway Transportation Reliability.

Forthcoming in the Transportation Research Record 1330, TRB, Washington, D.C., 1992.

6. M.D.M. Dai. Delay Estimation on Congested Waterways. Ph.D. Dissertation, University of

Maryland, College Park, Maryland, 1991.

7. M.D.M. Dai and P.M. Schonfeld. Metamodels for Estimating Waterway Delays through

Series of Queues. Forthcoming in Transportation Research. Part B, 1991.

195
Wei, Dai, & Schonfeld 21

8. L. Kleinrock. Queueing Systems Volume I: Theory. John Wiley & Sons, Inc., New York,

1975.

9. D. Bertsimas. An Exact FCFS Waiting Time Analysis For a General Class of G/G/s

Queueing Systems. Queueing Systems Theory and Applications, Vol. 3, 19F8, pp. 305-320.

10. W. Whitt. The Queueing Network Analyzer. The Bell System Technical Journal, Vol. 62,

1983, pp. 2779-2815.

11. S.L. Albin. Approximating a Point Processes by a Renewal Process, II: Superposition Arrival

Processes to Queues. Operations Research, Vol. 32, 1984, pp. 1133-1162.

12. C.W. Howe, et al. Inland Waterway Transpcrtation Studies in Public and Private

Management and Investment Decisions. Johns Hopkins Press, Baltimore, 196&..

13. J.L. Carroll and M.S. Bronzini. Waterway Transportation Simulation Models: Developme'nt

and Application. Waterway Resources Research, Vol. 9, No. 1, 1973, pp. 51-63.

14. A. Law and W. Ke!ton. Simulation Modeling & Analysis, 2nd Edition. McGrawHill, Inc.,

New York, 1991.

15. M. Fleming, D. Wood, and R. Goodwin. Lock Performance Monitoring System User's

Manual for Data Analysis. Institute for Water Resources, U.S. Army Corps of Engineers,

Fort Belvoir, Virginia, 1985.

16. K.T. Marshall. Some Inequalities in Queuing. Operations Research, Vol. 16, 1968, pp. 651-

665.

196
Wei, Dai & Schonfeld

TABLE I VALIDATION OF NUMERICAL METHOD FOR 20-LOCK TEST CASE

TABLE 2 PHYSICAL CHARACTERISTICS OF 3-LOCK SYSTEMS

TABLE 3 COMPUTATIONAL COMPARISON FOR VARIOUS CRITERIA IN 3-LOCK SYSTEMS

TABLE 4 CONVERGENCE PROPERTIES FOR VARIOUS CRITERIA IN SYSTEM 2

TABLE 5 RELEVANT DATA FOR THE 20-LOCK SYSTEM

TABLE 6 COMPUTATION RESULTS FOR THE 20-LOCK SYSTEM

TABLE 7 PARAMETERS FOR CPU TIME VS. SYSTEM SIZE

TABLE 8 ITERATIONS REQUIRED FOR VARIOUS SCANNING PROCESSES

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Wei, Dai & Schonfeld

TABLE 1 VALIDATION OF NUMERICAL METHOD FOR 20-LOCK TEST CASE

Lock OA& Cb (o CD as Cs V/C Distc

1 1.21 1.01 0.92 0.77 0.52 0.56 0.78 7.04

2 1.18 0.98 1.17 0.98 0.10 0.70 0.12 49.04

3 1.20 1.00 0.91 0.76 0.74 0.69 0.90 46.05

4 1.19 0.99 1.05 0.88 0.69 0.76 0.75 47.74

5 1.20 1.00 1.02 0.85 0.81 0.78 0.86 105.56

6 1.19 0.99 0.90 0.75 0.61 0.60 0.84 71.76

7 1.19 0.99 1.05 0.88 0.91 0.84 0.90 39.91

8 1.21 1.01 1.20 1.00 0.13 0.57 0.19 91.12

9 1.19 0.99 0.94 0.79 0.65 0.65 0.83 60.55

10 1.17 0.97 0.99 0.83 0.75 0.74 0.85 22.44

11 1.21 1.01 1.08 0.90 0.56 0.71 0.66 53.38

12 1.22 1.02 1.20 1.00 0.23 0.67 0.28 89.78

13 1.22 1.02 1.19 0.99 0.28 0.69 0.34 103.77

14 1.23 1.02 0.89 0.74 0.57 0.57 0.83 125.02

15 1.21 1.01 1.16 0.97 0.35 0.71 0.41 105.41

16 1.22 1.02 1.18 0.99 0.37 0.80 0.39 80.29

17 1.21 1.01 1.02 0.85 0.45 0.57 0.66 99.98

18 1.20 1.00 0.95 0.79 0.62 0.64 0.81 65.54

19 1.18 0.99 1.13 0.94 0.45 0.74 0.51 2.38

20 1.22 1.02 0.96 0.80 0.71 0.70 0.85 96.75

Estimated Waiting Time, hrs/tow

Lock Numerical Simulation Difference

1 2.15 2.04 0.11 5.31

2 0.00 0.01 -0.01 --d

3 6.91 6.37 0.54 8.48

4 1.91 1.78 0.12 6.88

5 4.71 4.20 0.50 11.96

6 3.39 2.67 0.73 27.19

7 7.76 7.23 0.53 7.38

8 0.00 0.04 -0.03 --

9 3.30 2.83 0.47 16.48

10 4.21 3.73 0.49 13.02

11 1.10 1.08 0.01 1.04

12 0.08 0.10 -0.03 -27.02

13 0.13 0.17 -0.04 -22.84

14 3.33 3.20 0.13 4.19

15 0.22 0.26 -0.04 -16.41

16 0.21 0.26 -0.05 -19.91

17 0.95 0.99 -0.04 -4.26

18 2.80 2.74 0.06 2.29

19 0.42 0.45 -0.03 -7.54

20 4.34 4.27 0.08 1.81

System 47.92 44.44 3.49 7.85

lai: Standard deviation of interarrival time, interdeparture time, and

service time distributions, respectively. bCi: Coefficients of variation of interarrival time, interdeparture time,

and service time distrif)utions, respectively. cDist: Distance to the next lock, in miles.

'Not applicable.

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Wei, Dai & Schonfeld

TABLE 2 PHYSICAL CHARACTERISTICS OF 3-LOCK SYSTEMS

Two-way Tow Speed Variance of

Flow Rate Distance miles/day Service Time

System Lock tows/day V/C miles 14a FVb hr 2 /tow2

1 1 6.0 0.01 5 270 85 0.0007

2 6.0 0.07 5 270 85 0.0360

3 6.0 0.17 5 270 85 0.1897

2 1 12.0 0.15 5 325 102 0.0309

2 12.0 0.34 5 325 102 0.1620

3 12.0 0.25 5 325 102 0.0915

3 1 18.0 0.22 5 108 34 0.0309

2 18.0 0.03 5 108 34 0.0006

3 18.0 0.50 5 108 34 0.1618

4 1 24.0 0.50 5 162 51 0.1883

2 24.0 0.29 5 162 51 0.0646

3 24.0 0.67 5 162 51 0.3330

5 1 27.0 0.75 10 108 34 0.2271

2 27.0 0.57 10 108 34 0.1279

3 27.0 0.89 10 108 34 0.3167

6 1 27.0 0.75 20 216 68 0.1616

2 27.0 0.57 20 216 68 0.0909

3 27.0 0.89 20 216 68 0.2259

7 1 28.5 0.60 5 325 102 0.1557

2 28.5 0.05 5 325 102 0.0011

3 28.5 0.80 5 325 102 0.2738

8 1 28.5 0.35 60 162 51 0.0645

2 28.5 0.60 60 162 51 0.1882

3 28.5 0.80 60 162 51 0.3332

a'v: Average tow speed.

bag: Standard deviation of tow speeds.

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Wei, Dai & Schonfeld

TABLE 3 COMPUTATIONAL COMPARISON FOR VARIOUS CRITERIA IN 3-LOCK SYSTEMS

Estimated Waiting Time, hrs/tow

VARDEP LOCWAIT SYSWAIT

System Lock Wsim" Wvb Dvc WL DL WS Ds

1 1 0.0003 0.0001 -0.0002 0.0001 -0.0002 0.0001 -0.0002

2 0.0153 0.0175 0.0022 0.0176 0.0023 0.0176 0.0023

3 0.0989 0.0990 0.0001 0.0990 0.0001 0.0990 0.0001

Total 0.1145 0.1166 0.0021 0.1167 0.0022 0.1167 0.0022

Required Iterations 5 7 6

2 1 0.0334 0.0290 -0.0044 0.0290 -0.0044 0.0290 -0.0044

2 0.2316 0.2289 -0.0027 0.2289 -0.0027 0.2289 -0.0027

3 0.1139 0.1099 -0.0040 0.1099 -0.0040 0.1099 -0.0040

Total 0.3789 0.3678 -0.0111 0.3678 -0.0111 0.3678 -0.0111

Required Iterations 5 5 5

3 1 0.0542 0.0528 -0.0014 0.0528 -0.0014 0.0528 -0.0014

2 0.0008 0.0001 -0.0007 0.0001 -0.0007 0.0001 -0.0007

3 0.4621 0.4660 0.0039 0.4660 0.0039 0.4659 0.0038

Total 0.5171 0.5189 -n'8 0.5189 0.0018 0.5188 0.0017

Required Iterations 5 5 4

4 1 0.4355 0.4404 0.0049 0.4404 0.0049 0.4404 0.0049

2 0.0962 0.0999 0.0037 0.0999 0.0037 0.0999 0.0037

3 1.2028 1.1844 -0.0184 1.1844 -0.0184 1.1844 -0.0184

Total 1.7345 1.7247 -0.0098 1.7247 -0.0098 1.7247 -0.0098

Required Iterations 4 4 4

5 1 1.3926 1.4693 0.0767 1.4693 0.0767 1.4693 0.0767

2 0.3901 0.4127 0.0226 0.4127 0.0226 0.4127 0.0226

3 4.9837 4.7980 -0.1857 4.7980 -0.1857 4.7980 -0.1857

Total 6.7664 6.6800 -0.0864 6.6800 -0.0864 6.6800 -0.0864

Required Iterations 4 4 4

6 1 1.2203 1.3038 0.0835 1.3038 0.0835 1.3038 0.0835

2 0.3286 0.3416 0.0130 0.3416 0.0130 0.3416 0.0130

3 4.4608 4.2983 -0.1625 4.2983 -0.1625 4.2983 -0.1625

Total 6.0097 5.9437 -0.0660 5.9437 -0.0660 5.9437 -0.0660

Required Iterations 4 4 4

7 1 0.5430 0.5900 0.0470 0.5899 0.0469 0.5899 0.0469

2 0.0012 0.0001 -0.0011 0.0001 -0.0011 0.0001 -0.0011

3 2.0874 2.0906 0.0032 2.0906 0.0032 2.0906 0.0032

Total 2.6316 2.6807 0.0491 2.6806 0.0490 2.6806 0.0490

Required Iterations 4 3 3

8 1 0.1372 0.1405 0.0033 0.1405 0.0033 0.1405 0.0033

2 0.6381 0.6592 0.0211 0.6592 0.0211 0.6592 0.0211

3 2.3165 2.3146 -0.0019 2.3146 -0.0019 2.3146 -0.0019

Total 3.0918 3.1143 0.0225 3.1143 0.0225 3.1143 0.0225

Required Iterations 4 4 4

"Wsim: Waiting time estimated from simulation.

'Wi: Waiting time estimated when criterion i used.

cDi: Difference between numerically estimated waiting time at a given iteration

and simulated waiting time = Wi - Wsim.

200
Wei, Dai & Schon'eld

TABLE 4 CONVERGENCE PROPERTIES FOR VARIOUS CRITERIA IN SYSTEM 2

Criterion: VARDEP

Magnitude, Dir 1 Relative Difference, Dir 1

Iter Lock 1 Lock 2 Lock 3 Lock 1 Lock 2 Lock 3

1 0.9930 0.9629 0.9715 -- & -- --

2 1.0027 0.9724 0.9715 0.0098 0.0098 0.0000

3 1.0027 0.9778 0.9802 0.0000 0.0056 0.0090

4 1.0030 0.9780 0.9802 0.0003 0.0002 0.0000

5 1.0030 0.9782 0.9804 0.0000 0.0002 0.0002

Magnitude, Dir 2 Relative Difference, Dir 2

Iter Lock 1 Lock 2 Lock 3 Lock 1 Lock 2 Lock 3

1 0.9456 0.9214 0.9889 -- -- --

2 0.9884 0.9705 0.9889 0.0453 0.0533 0.0000

3 0.9884 0.9715 0.9907 0.0000 0.0011 0.0018

4 0.9897 0.9725 0.9907 0.0013 0.0010 0.0000

5 0.9897 0.9726 0.9907 0.0000 0.0000 0.0000

Criterion: LOCWAIT

Magnitude Relative Difference

Iter Lock 1 Lock 2 Lock 3 Lock 1 Lock 2 Lock 3

1 0.0177 0.1994 0.1065 -- -- --

2 0.0287 0.2254 0.1065 0.6167 0.1304 0.0000

3 0.0287 0.2283 0.1098 0.0000 0.0126 0.0312

4 0.0290 0.2288 0.1098 0.0110 0.0023 0.0000

5 0.0290 0.2289 0.1099 0.0000 0.0004 0.0007

Criterion: SYSWAIT

Magnitude Relative Difference

Iter System System

1 0.3236 --

2 0.3606 0.1142

3 0.3667 0.0171

4 0.3676 0.0023

5 0.3677 0.0004

aNot applicable.

201
Wei, Dai & Schonfeld

TABLE 5 RELEVANT DATA FOR THE 20-LOCK SYSTEM

Lock V/C CS Capb 9, as

1 0.5625 0.2482 48 0.5000 0.0154

2 0.2473 0.2591 109 0.2198 0.0032

3 0.4505 0.3725 60 0.4004 0.0223

4 0.4098 0.2942 66 0.3643 0.0115

5 0.9865 0.8953 27 0.8769 0.6163

6 0.2148 0.2328 126 0.1909 0.0020

7 0.8315 0.5422 32 0.7391 0.1606

8 0.7088 0.3447 38 0.6300 0.0472

9 0.8563 0.9832 32 0.7612 0.5601

10 0.5989 0.8641 45 0.5324 0.2116

11 0.2065 0.6823 131 0.1836 0.0157

12 0.0510 0.5392 529 0.0453 0.0006

13 0.9894 0.8309 27 0.8795 0.5340

14 0.5051 0.4834 53 0.4490 0.0471

15 0.6715 0.6363 40 0.5969 0.1442

16 0.6728 0.7805 40 0.5980 0.2179

17 0.9475 0.6943 28 0.8422 0.3419

18 0.8662 0.4078 31 0.7700 0.0986

19 0.9074 0.4017 30 0.8066 0.1050

20 0.8711 0.9968 31 0.7743 0.5957

Inflow Rate of Direction 1 (tows/day) 13.5

Inflow Rate of Direction 2 (tows/day) 13.5

Convergence Threshold 0.001

Tow Speed (miles/day) 213.48

Standard Deviation of Speed (miles/day) 67.68

Distance between Locks (miles) 20.0

"aC : Coefficient of variation of service time distribution.

bCap: Lock capacity, tnws/day.

'g,: Mean of service t.Lme distribution, hrs/tow.

d'a: Variance of service time distribution, hrs 2/tow2 .

202
Wei, Dai & Schonfeld

TABLE 6 COMPUTATION RESULTS FOR THE 20-LOCK SYSTEM

VARDEP LOCWAIT SYSWAIT

Required Iterations

for Convergence

Within 0.001 4 5 4

CPU Time (seconds) 2.15 2.36 1.75

Total Waiting Time

(hrs/tow) 151.2056 151.2044 151.2056

100 Iterations

Divergence None None None

CPU Time (seconds) 31.25 34.38 27.14

Total waiting time

(hrs/tow) 151.2043 151.2043 151.2043

203
Wei, Dai & Schonfeld

TABLE 7 PARAMETERS FOR CPU TIME VS. SYSTEM SIZE

Standard Standard Error

Criterion Kj Error of K. of CPU Estimate R2 Ai

Eq. 9

VARDEP 0.1129 0.0026 0.0774 0.9867

LOCWAIT 0.1245 0.005 0.1482 0.9537

SYSWAIT 0.0925 0.0019 0.0578 0.9885

Eq. 10

VARDEP 0.106 0.0055 0.0696 0.9919 0.102 LOCWAIT 0.108 0.0084 0.1074 0.9817 0.242 SYSWAIT 0.0853 0.0024 0.0315 0.9974 0.107

204
Wei, Dai & Schonfeld

TABLE 8 ITERATIONS REQUIRED FOR VARIOUS SCANNING

PROCESSES

3-Lock System 1 VARDEP LOCWAIT SYSWAIT

Single Scan 5 7 6

Double Scan 6 8 8

20-Lock System

Single Scan 4 5 4

Double Scan 6 6 6

205
Wei, Dai & Schonfeld

FIGURE 1 STRUCTURE OF NUMERICAL METHOD

FIGURE 2 CONVERGENCE FOR 3-LOCK SYSTEM WITH VARIOUS CRITERIA

FIGURE 3 RELATIONS BETWEEN SYSTEM SIZE AND COMPUTATION SPEED

206
Wei, Dai & Schonfeld

Arrival Process Departure Process

1"-0, •."4 k-i., ((J.2 fJi0 (1) 4--1= t, (5)

_Di°A i f J- ( 2 - C,1 207+.795(CA2(1-p)+p)÷1.O01(Cp 2-p2 o,41=o41;÷.02511n(l+ k- = +, 1, *if J 2 (2 ) (6)

,- ti* t. 2 (3) C- 2 0 o.s18.o.491C.' c7,m (7)

C 1 o.•.o.41(C . 7 ,+ C 2) (4)

Delay Function

W = AS+2 2-0 (8) 2

tA(l -p)

Notation:

CA coefficient of variation of interarrival times at Lock i

C,0 :coefficient of variation of directional interarrival times for Direction j and Lock i

Cm : coefficient of variation of interdeparture times for Direction j and Lock i

COp : coefficient of variation of directional inteneparure times for Direction j and Lock i

Cs coefficient of variation of service times

Dot :distance between Locks i and k

i :index of currently scanned lock

J •direction index (1 = downstream, 2 = upstream)

k index of adjacent locks

S :mean interarrival time at Lock i

tel mean interarrival time for Direction j and Lock i

t :mean interdepanture time for Direction j and Lock k

mean tow speed between Locks i and k

o, :standard deviation of interarrival times for Direction j and Lock i

standard deviation of interdeparture times for Direction j and Lock k

standard deviation of tow speeds between Locks i and k

FIGURE 1 STRUCTURE OF NUMERICAL METHOD

207
Wei, Dai & Schonfeld

CRTERION: VARIANCE OF INTERDEPARTURE TIME

7

LOCK S--e- I (DIR 1)._. LOCK 2 (DIR 1) -- LOCK 3 (DIR 1)

5 - --x- LOCK 1 (DI 2) +LOCK 2 (DIR 2) LOCK 3 (DIR 2)

4

'3

2

1 2 3 4 5 6 100

# ITERATIONS

CRITERION: WAITING TIME

10

A9

7 - LOCK I .. LOCK 2 --- LOCK 3 -x-SYSTEM

6-

5-

'4-

3

2

1 2 3 4 5 6 100 # ITERATIONS

FIGURE 2 CONVERGENCE FOR 3-LOCK SYSTEM WITH VARIOUS CRITERIA

208
Wei, Dai & Schonfeld

2.4-

2.2-

S2

S1.8

01.6

1.4

S1.2-

1I # ITERATIONS

0.8- 0 LOCKS VARDEP LOCArT :YSWArr

0.4- 12

0.2 16 4 5 5

20 4 5 4 I I I

12 16 20 #LOCKS

0 VARDEP + LOCWAIT < SYSWAIT

FIGURE 3 RELATIONS BETWEEN SYSTEM SIZE AND COMPUTATION SPEED

209
210
A METHODOLOGY FOR PLANNING EFFICIENT

INVESTMENTS ON INLAND WATERWAYS

by

Melody D. M. Dai, Paul Schonfeld,

George Antle

Published by Transportation Research Record

Paper No. 930698

211
212
A Methodology for Planning Efficient Investments

on Inland Waterways

David Martinelli

Assistant Professor

West Virginia University

Morgantown# WV

Melody D.M. Dai

Associate Professor

National Chang Mung University Tainan, Taiwan

Paul Schonfeld

Associate Professor

University of Maryland

College Park, Maryland

George Antle

Director

Navigation Division

Institute for water Resources

Ft. Belvoir, VA

Subaitted for Publication and Presentation to

The Transportation Research Board

Washington, DC

January, 1993

Abstract

This paper presents a methodology that addresses the analytic

complications associated with making investment planning decisions

for inland waterway improvements. These complicatons include

interdependencies between locks, bidirectional traffic, stalls,

dual chamber facilities, and budget limitations. The methodology

address most steps of the investment planning process for locks,

namely project evaluation, sequencing, and scheduling.

213
1. Introduction

The national waterway study, and other navigation studies

identified a need for substantial investment in the waterway

infrastructure derived from several trends and observations. The

first trend is that lock conditions are deteriorating, giving rise

to an increase in tow delays. Currently there are about 100 locks

that have exceeded their design life. The second is that traffic

levels are consistently increasing for many locks in the system.

Also, prospects for increased grain exports are improving.

Currency reform, the grain export enhancement program, reduction in

worldwide carryover stocks of grain, and other factors have

contributed to increases in exports. A third trend is an increase

in tow sizes. While this tends to increase overall transport

efficiency, large tows must be disassembled into several pieces to

move through the chamber and must later be reassembled. The fourth

observation is that additional funding sources for major lock

rehabilitation projects is not likely. The major sources of

funding for such projects are the Federal matching share and fuel

tax receipts. The Federal share of 50 percent and the fuel tax

rate of 20 cents beyond 1995 are not likely to increase in the near

future.

The trends identified by these studies present interesting but

challenging opportunities for developing a more comprehensive

methodology for inland waterway planning and operations analysis.

The following are the primary analytical needs in developing such

214
a methodology:

1. more reliable forecasting methods,

2. more reliable techniques for predicting delays at locks,

3. identification and assessment of the benefits of lock

rehabilitation, and

4. more efficient techniques for sequencing and scheduling lock

improvement projects.

This paper is an overview of a methodology designed to address many

of the analytical needs resulting from trends in conditions,

traffic levels, and funding sources for waterway locks. Particular

emphasis is placed on satisfying items 2 and 4 above. It is the

product of several research projects conducted over the last four

years through the Institute for Water Resources and consists of the

following components:

1. exploratory data analysis and characterization of problems,

2. a microsimulation model of waterway traffic and lock operations,

3. statistically estimated functions ("metamodels") to approximate

the results of the simulation model,

4. an algorithm for prioritizing and scheduling proposed lock

improvement projects, and

5. a computer program for cash flow analysis for the Inland

Waterway Trust Fund.

215
2. Background

There are numerous analysis tools available to assist in

modeling lock operations and investment parameters. These include

benefit-cost analysis, mathematical programming, queuing theory,

and simulation. There exist some significant works on the

application of these tools to waterway problems. However

extensions to the previously available methods were necessary to

meet the analytical demands of current U.S. waterway transportation

problems.

2.1 Detezrnining Delays at Locks (Analytic Models)

Two single-lock models based on.the application of queuing

theory have been found for estimating lock delays. DeSalvo and

Lave [6) represent the lock operation as a simple server queuing

process with Poisson distributed arrivals and exponentially

distributed service timeb. However, these assumed distributions do

not adequately fit the physical system of locks on waterways (5].

Wilson [15] improved on this model by treatinq the service

processes as general distributions rather than exponentially

distributed, which is far more realistic [5]. However, this was

for single chamber locks only and the Poisson arrivals assumption

is not realistic for all locks. Two other deficiencies exist in

both of the above models. First, neither of these models accounts

for stalls. Stalls cause service interruptions at locks, thus

reducing lock capacities or increasing delays. Their occurrence is

216
very difficult to predict. Secondly, both models were developed to

analyze delays at a single isolated lock. Since the delays at

adjacent locks may be highly interdependent, it is desirable to

analyze lock delays for entire systems.

Queuing theory offers some solutions for more general queuing

systems, i.e. those beyond Poisson arrivals and exponential service

times (M/M/i). In special cases combining Poisson arrivals with

general service times (M/G/l) and general arrivals with exponential

service times (G/M/l), closed-form solutions for the mean waiting

time have been obtained (15]. G/G/1 queues are difficult cases in

queuing theory and the available techniques for handling them are

incomplete. Solving G/G/M queues is even more difficult than

solving G/G/1 queues. The methods of approximations and bounds

have been proposed to solve G/G/M queues (11,12]. These can be

accurate and efficient under heavy traffic conditions. However, the

methods are difficult to extend to the a series and networks of

queues found in waterways.

2.2 Determining Delays at Looks (SBiulation Models)

An early microscopic simulation models to analyze lock delays

and tow travel times was developed by Howe (7]. In that model,

service times were based on empirically-determined frequency

distributions. To avoid some troublesome problems and errors

associated with the requirement to balance long-run flows in Howe's

model, Carrol and Bronzini (4] developed another simulation model.

217
It provided detailed outputs on such variables as tow traffic

volumes, delays, processing times, transit times, average and

standard deviations of delay and transit times, queue lengths, and

lock utilization ratios. Each of these models simulates waterway

operations in detail, but requires considerable amounts of data and

computer time, which limit their applicability for problems with

large networks and numerous combinations of improvement

alternatives. They both assume Poisson distributions for tow-trip

generation, which is not always realistic. Moreover, service

failures ("stalls") which are very different in frequency and

duration from other events and have significant effects on overall

transit-time reliability, are not accounted for. Hence a waterway

simulation model that explicitly accounts for stalls is desirable

for evaluating and scheduling lock improvement projects.

2.3 Benefit-Cost Analysis for Interdependent Improvement

Projects

The delays at locks have been shown to be interdependent, i.e.

the delays at one lock are related to the delays at one or more

other locks [10]. That is because the departure process from one

queuing station (e.g. a lock) in a network effects the arrival

process at the next queues in that network. Interdependence not

only yields difficulties in predicting lock delays, but also in

conducting benefit-cost analysis. Current methods of benefit-cost

analysis are quite satisfactory for analyzing mutually exclusive

projects, and reasonably satisfactory for independent projects.

218
However, there is a void in analyzing projects that are

interdependent.

In evaluating and sequencing mutually exclusive projects, the

net present value and benefit-cost ratio methods as discussed in

(2] can be used if the benefits and costs are quantifiable and can

be accurately assessed over the planning horizon. This is because,

in such cases, the benefits and costs of projects are not dependent

on the project set selected. When working with independent

projects, we can use an integer programming approach where the

objective is to maximize the sum of net present values subject to

a set of budget constraints (8].

However, for interdependent projects, the estimates of

benefits and costs must be performed simultaneously with project

selection. Therefore it may be necessary to enumerate all possible

project combinations when selecting a set of projects and all

possible permutations when sequencing a set of projects. However,

as a practical matter, complete enumeration becomes infeasible as

a method of finding optimal combinations and permutations of

projects as the number of projects becomes even modestly large. An

alternative to complete enumeration is an augmented integer

programming formulation. Such methods are discussed in (8]. To

capture some of the interdependence, the objective function

includes "interaction terms" for pairs of projects. These terms

represent the deviation from linear addition when summing the net

219
present values for two interdependent projects. For example, if

two Projects A and B are independent the net present values may be

summed linearly:

NPVA8 = NPVA + NPV,. (1)

Alternatively, for interdependent projects an interaction term is

added:

NPVAB = NPVA + NPV, + dA8. (2)

There are significant shortcomings with such an approach. First,

only paired interactions are represented; depending on the

application, three, four, or more projects may be simultaneously

dependent. Second, the number of integer variables and interaction

terms is excessive. The estimation of interaction terms is quite

complex for most applications. While many problems may be smaller

than this example, most integer programming algorithms have serious

difficulties with problems of this size.

There is a need to formulate the selection and sequencing of

interdependent projects in a manner that is not computational

intractable and does not require excessive estimation of

interaction terms. It seems that overcoming these voids requires

the development of a method whereby the numerous permutations of

possible programs may be efficiently represented and searched

220
(without complete enumeration) and the determination of efficient

project implementation schedules.

3. Components of the Methodology

3.1 Sinulation Model

In light of the many shortcomings and difficulties associated

with analytic methods of estimating delays at locks, a simulation

model has been developed to analyze tow operations along waterways.

The model may be used to determine the relations among delays, tow

trips, distributions of generated travel times, and coal

consumption and inventories. The model can account for the

stochastic effects and seasonal variations and can estimate the

following: tow delays at each lock, interarrival and

interdeparture-time distributions for each lock and for each

direction, tow travel times along the waterway, inventory levels

and expected stock-out amounts for commodities delivered by

waterway, and many other variables of interest to waterway users

and operators. Development of the model was based on the Lock

Performance Monitoring System (LPMS) data. The model is event

scanning, with four types of events initiating a status update: 1)

stochastic generation of tow trips; 2) tow entrances at locks as

determined by arrival times, chamber availability, and chamber

assignment discipline; 3) the arrival of a tow at its destination;

and 4) the occurrence of stalls at a chamber.

There are several features of this model that lend itself well

221
to waterway operations. The simulation model is micros-opic, i.e.

it traces the movement of each individual to and records its

characteristics, including the number of barges, commodity types,

speed, origin and destination, travel direction, and arrival time

at various points. Any distributions for trip generation, travel

speeds, lock service times and tow size may be handled by the

model. These distributions can be specified for each interval in

tables or by standard statistical distributions. Tows are allowed

to overtake other tows and the model simulates two-way traffic

through common servers and accounts for stalls. The size of

waterway systems that the model can handle is limited only by

computer and compiler capacity. Further, the model has been

developed with "dynamic dimensioning," for additional increases in

flexibility in modeling various waterway systems.

The main simplifying assumptions in the current version of the

simulation model are as follows:

1. The tows maintain a constant size through the entire trip.

2. The service discipline is First-In-First-Out as are

operations on the Mississippi and Ohio Rivers.

3. The queue storage area is unlimited.

4. The tow speeds are normally distributed and constant for

each round trip.

5. The time intervals between two successive stalls and the

stall durations are exponentially distributed.

These assumptions are not seriously restrictive, but can still be

222
easily modified.

The simulation model consists of five operation routines and

one scheduler routine. The operation routines are associated with

the five types of events and are invoked by the s..hedu>lr. Figure

1 is a chart of the flow of data as dictated by the model.

To check the logic of this simulation model, its results were

first compared to theoretical (but very well established) results

from queuing theory. The results of the model were then compared

with observed data to demonstrate how closely the model represents

real systems and verify its ability to simulate the special

features of waterways.

A partial validation of the model is possible by comparing the

model to theoretical results for the special case of Poisson

arrivals and generally distributed service times. The waiting

times predicted by the simulation model at a single lock were

compared with those obtained from queuing theory for this special

case. A validation has been conducted for a variety of

volume/capacity (V/C) ratios ranging from 0.0471 to 0.8934. To

reduce the variance of the output each result was obtained by

averaging the output from 30 independent simulation runs. To

insure results were compared for a steady state, each simulation

run discarded the first 10,000 tow waiting times and collected the

next 12,000 values for computing the average waiting time.

223
INPUT:

Link & Lock Characteristics

Traffic Demand

Probability Distributions

Inventories & Consumption

PROCESS:

Origin Nodes: generating tow trips

Destination Nodes: updating: cumulative deliveries

cumulative consumption

inventory levels

Locks: assigning chamber

determining number of cuts

determining lock service times

calculating queuing times

Links: determining traveling times

determining arrival times to next locks or

destinations

OUTPUT:

Average waiting time per tow at each lock

Average waiting time per tow at each lock for each O-D pair

Means and variances of interarrival and interdeparture times at

each lock

Cumulative deliveries, cumulative consumption, inventory levels

Average speed

Total number of tow trips for different O-D pairs

Total queuing times for different locks

To'al lock service times for different locks and chambers

Total tow travel times & distances

Figure I Structure and Elements of The simulation Model

224
The results, shown in Table 1, confirm that the simulated and

theoretical average waiting times are extremely close. Such

results verify that the overall mechanism of the simulation model

is correct. They also show that generally distributed service

times are generated satisfactorily in the simulation model. That

is reassuring since the same logic is also used to generate

generally distributed interarrival times for G/G/1 queues and,

ultimately to develop metamodels for series of G/G/1 queues.

Table 1. Comparison of Theoretical and Simulated Results for a

Single Lock Queue (M/G/1)

V/C TA*" Ts 1 WAn* W4 Devia."5

Avg Var Avg Var

(hr) (hr 2) (hr) (hr 2) (hr) (hr) (%)

0.893 0.888 0.789 0.793 0.319 4.9516 5.0059 -1.09

0.755 0.888 0.789 0.670 0.227 1.5575 1.5522 0.34

0.566 0.888 0.789 0.503 0.128 0.4926 0.4935 -0.19

0.330 0.888 0.789 0.293 0.044 0.1082 0.1087 -0.46

0.047 0.888 0.789 0.042 0.001 0.00155 0.00156 -0.64

"*1 TA: interarrival times

*2 Ts: service times

*3 War: average waiting times from simulation

*4 W,: average waiting times from queuing theory

*5 Devia.: deviation which is defined as (W,-Wt)/Wt*100%

The simulation results were then compared with the observed

data at Locks 22, 24, 25, 26, and 27 on the Mississippi River.

These locks were selected based on their criticality and available

data. The five locks were simulated as an interacting series.

Some of the validation results are summarized in Table 2. Each

result is averaged from 80 independent simulation runs. Table 2

225
shows that the difference between the simulated and observed

average waiting times for each lock are within the 95% confidence

interval based on the t test, except at Lock 25. The observed data

also show that tows sometimes were kept waiting at Lock 25 even

when the chamber was idle. Therefore, no direct comparison of

average waiting times at Lock 25 is appropriate.

Table 2. Comparison of Simulated and Observed Average Waiting

Times

Lock Wmi*I W.* Difference 95% Confidence

(min) (min) (min) Interval

22 4.09 3.73 0.36 3.49

24 6.12 6.36 0.24 6.72

25 4.49 10.94 6.45 -0

26 119.40 130.99 11.59 60.73

27 36.49 34.43 2.06 23.92

"*1 Wi,: simulated average waiting times

*2 Wb,: observed average waiting times

*3 The comparison is not appropriate.

3.2 Metaxodel Approzimations to Simulation

Each simulation run takes from a few seconds to a few minutes

on a personal computer depending on the values of various problem

parameters. Despite this high level of efficiency, simulation time

becomes expensive for evaluating large combinatorial problems such

as investment planning. Furthermore, the project combinations may

have to be evaluated over several time periods. A metamodeling

approach which statistically estimates unknown parameters of

equations from simulation results and then uses these equations as

226
substitutes has been developed to overcome the computational

requirements of simulation. The main difficulty with this approach

is to find structural forms for the approximating functions which

fit the simulation results as well as possible, This was

accomplished by queuing theory insofar as possible for these

functions.

In this study, a numerical method has been developed for

estimating delays through a series of queues. The method

decomposes systems of queues into individual queuing stations. The

analysis of each queuing station is further decomposed into three

modules, namely arrival processes, departure processes, and delay

functions. Arrival processes at a particular lock depend on the

departure distributions from the upstream and downstream locks and

the intervening speed distributions. The departure processes

depend on the interaction among the arrival distributions and

service time distributions at one lock. The delay functions relate

the waiting times to the arrival and service time distributions.

The basic concept of this method is to identify the parameters of

the interarrival and interdeparture time distributions for each

lock, and then estimate the implied waiting times.

To estimate delays in a queuing system, we need to know the

means and variances of the interarrival, interdeparture and service

time distributions. For series of G/G/I queues and bidirectional

servers, a difficulty arises in identifying the variances of

227
interarrival and interdeparture times. Because the interarrival

times at each lock depend on departures from both upstream and

downstream locks, and the variances of interarrival times cannot be

determined from one-directional scans along a series of queues. To

overcome such complex interdependence, an iterative scanning

procedure is proposed. The core concept is to decompose the system

into individual locks and then sequentially analyze each of those

locks. At each lock, the two arrivals from both directions are

first combined into an overall arrival distribution and then split

into two-directional departure distributions.

The algorithm is initiated by scanning along waterways from

either direction, sequentially estimating the interarrival and

interdeparture time distributions for each lock. Initially assumed

values for the variances of interdeparture times from the opposite

direction must be provided for the first scan. Then, the scanning

direction is reversed and the process is repeated, using the

interdeparture time distributions for the opposite direction

estimated in the previous scan. Alternating directions, the

scanning process continues until the relative difference in the

preselected convergence criteria stays within preset thresholds

through successive iterations. Waiting times at locks can be

computed in every iteration (and then used as convergence criteria)

or just once after all iterations are completed.

228
Arrival Processes

The mean and standard deviation of interarrival times are

estimated in two steps. First, the means and standard deviations

of directional interarrival times at a particular lock are

estimated from the interdeparture time distributions of the

adjacent locks. If flows are conserved between locks and if the

V/C ratio is less than 1.0, such relations are represented in Eq.

3:

{k=i-l,if j=1 (3) Caji = tdk k=i~l, if j=2

where

Taii : the average interarrival time for Direction j and Lock

i

Edjk : the average interdeparture time for Direction j and

Lock k

j : direction index (1 - downstream, 2 = upstream)

Because speed variations change headway distributions between

locks, Eq. 4 was developed to estimate the standard deviation of

directional interarrival times at one lock.

Gaj1 :djk+0. 0 2 5 11n(l+DD1 ) {(k-i-l-if j=1

I'vik k=i+l,ifji=2(4 (0.002)

R2 = 0.999954 n = 107 se = 0.0586 AY = 5.1685

229
where

standard deviation of interarrival times for Direction

j and Lock i

adk : standard deviation of interdeparture times for

Direction j and Lock k

Di& : distance between Locks i and k

PvL : average tow speed between Locks i and k

avk : standard deviation of tow speeds between Locks i and k

j : direction index (1 - downstream, 2 - upstream)

so : standard error of dependent variable

AY : mean of dependent variable

This suggests that, theoretically, the standard deviation of

directional interarrival times should be equal to the standard

deviation of directional interdeparture times plus an adjustment

factor depending on the speed distribution and distance.

Second, the overall mean and coefficient of variation of

interarrival times for this lock are estimated based on the

coefficient of variation of directional interarrival times.

= Tal* 1 .-- (5)

tli + taEl

A, = 0.179+0.41(Ca,, 2 + Cai) (6)

(0.027) (0.014)

R= 0.9188 n = 79 so = 0. 0059 ;- 0.988

230
where

the average interarrival time at Lock i

C12 : squared coefficient of variation of interarrival times

at Lock i

Caji 2 squared coefficient of variation of directional

interarrival times for Direction j and Lock i

In Eq. 6, the coefficients of variation of upstream and downstream

interarrival times carry the same weight in estimating the overall

variance of interarrival times, since the mean directional trip

rates are equal.

Departure Process

The departure module estimates the mean and coefficient of

variation of interdeparture times. Based on the flow conservation

law, if capacity is not exceeded, the average directional

interdeparture equals the corresponding interarrival time:

The coefficient of variation of interdeparture time is

estimated in two steps. First, the coefficient is estimated for

combined two-directional departures. Departure processes with

generally distributed arrivals and service times are analyzed using

Laplace transforms. The following metamodel was eventually

231
developed to bypass the difficulties of determining the variances

of the lock-idle times:

C02 = 0.207+0.795(l-p+p)+(p2 -p 2 ) = 0.207+0.795 = 1.002 - 1.0 (6)

Next the coefficient of variation of directional

interdeparture times is estimated. The following metamodel was

developed for this purpose:

Cdj 2 - 0.518+0.491C,& i (9)

(0.0056) (0.0068)

R2 = 0.9710 n - 158 so = 0.013 - 0.9164

where

COI: squared coefficient of variation of directional

interdeparture times for Direction j and Lock i

Cap: squared coefficient of variation of directional

interarrival times for Direction j and Lock i

C•2: squared coefficient of variation of interdeparture

times for Direction j and Lock i

Delay lunotion

The delay function is intended to estimate the average waiting

time at a lock. By applying Marshall's formula for the variance of

interdeparture times an exact solution for the average waiting time

W, was obtained as follows:

232
wz OA2*2032-°D (10) 2C (1-p)

where

W: the average waiting time

OA2: variance of interarrival times

as : variance of service times

D 2: variance of interdeparture times

tA: average interarrival time

p: volume to capacity ratio

In this delay function, the average waiting time increases as the

variance of interarrival and service times increase and decreases

as the variance ..f interdeparture times increases. The average

waiting time approaches infinity as the V/C ratio approaches 1.0.

Comparison of Simulated and Nuaerical Results

To validate the numerical method, its results were compared to

the results of the previously validated simulation model. Various

system configurations were compared, including a relatively large

20-lock system.

The parameter values for this test system (e.g. means and

standard deviations of input distributions and distances between

locks) were obtained from random number generators, except for

traffic volumes, which were assumed to be 10 tows/day in each

233
direction throughout the system. Table 3 shows the input

parameters and a comparison of waiting times, which are the output

variables of greatest practical interest. It can be seen that the

numerical model estimates aggregate waiting times within 8% of

those simulated. At individual locks, the percent errors are

slightly greater but within 10%. In its current form, the modeling

approach does not consider possible diversion to other modes on the

basis of excessive delay. However, the model might be applied

iteratively with a demand reestimation model.

3.3 Project Sequencing and Scheduling

Either the simulation model or the metamodels may serve as a

project evaluation tool. That is, both are able to provide delay

estimates for a system of locks for different combinations of

proposed lock improvements (i.e. any measure that physically or

effectively increases the capacity of a lock). This is the basis

for estimating the benefits associated with such improvements. The

choice should be based on a tradeoff between precision for complete

lock operations (favoring simulation) and computational efficiency

(favoring the metamodels). Thus, the metamodels may be used for

preliminary screening and the simulation for the final detailed

evaluation.

The next step in the investment planning methodology is a

technique whereby the permutations of investment sequences may be

efficiently searched and a corresponding optimal schedule found.

234
The proposed approach for searching the solution space of possible

project permutations represents the solution space in two

dimensions and applies a heuristic search algorithm in selecting

the preferred sequence. Given a system cost evaluation function

for interdependent projects g(X,Y), the selection and sequencing

problem may be represented in two dimensional space. The function

g(X,Y) incorporates both benefit and cost factors into a

generalized cost while accounting for project interdependencies

where X is a vector of delay variables and Y represents a

particular combination of projects.

Assuming that each set of projects may be viewed as a system

generating a common time-dependent output, then a two dimensional

representation is quite feasible. For the lock rehabilitation

problem, the costs associated with a given combination of projects

in a given time period t, may be written as

(SC) • = Cy + g (X(X(t)),Y)(OC) (11)

where Cy is the total capital cost of construction for the set of

projects Y. The term g(X(X(t)),Y) represents the delay, and

corresponds to the function(s) obtained from some interdependent

evaluation, e.g. from a simulation model, while OC is the

opportunity cost of delay (which may be either a constant or a

function of time). Evaluating SC at different levels of output for

a combination of projects Y, defines a curve with annual system

costs SCi on the vertical axis and output level, X, on the

235
horizontal axis. Repeating for different values of Y (i.e.

different project combinations) produces a family of curves. By

always choosing the lowest cost curve for any given output level X,

i.e. by choosing the "lowest envelope" of the curves in Figure 2,

a sequencing and scheduling decision path is defined. Because the

output is assumed to be time dependent, the horizontal axis may

also represent time periods, e.g. years. Output and time may be

linked through a demand function, X(t).

Consider an example with interdependent projects A, B, and C.

Figure 2 shows a family of system cost (SC) curves corresponding to

the possible combinations of these three projects. Note that in

general, combinations involving only one project are preferable

(lower SC) for low levels of volume (thus earlier in the horizon

stage), and become less preferable as volume increases. Under this

representation, one combination is preferred to another at a given

output level (or time period) if its corresponding curve lies above

the other.'

In the example depicted in Figure 2, the selection and

sequence of projects is dictated by the lowest "envelope" defined

by the curves. This lowest envelope corresponds to the

minimization of the time integral of the system cost for feasible

expansion paths. Here, all three projects would be accepted if the

'Although the convex and monotonically increasing properties of the curves in Figure 1 are likely to occur for costs with a

delay component, they are not a prerequisite for the methodology.

236
volume level is expected to eventually exceed Q2 . We see also that

the sequence of projects should be A, B, C; this is because Curve

A lies below B and C, and AB lies below AC in the relevant regions.

Project A is preferred up to volume level Q, at the same time

Project B should be implemented since Curve AB falls below Curve A.

At volume level Q2 , Project C should be added to A and B, thus

implementing Combination ABC.

Unfortunately not all such families of curves can be

interpreted as easily as Cases 1 and 2. Consider a second case

shown in Figure 3 where Curves A and AB are unchanged but the

others are different. Here, Curves AB and AC intersect each other

before intersecting Curve ABC. It cannot be stated a priori

whether Combination AB or AC should be selected on the expansion

path between A and ABC. One would expect that if Area 1 is greater

than Area 2, then Combination AB is preferred to AC and Project B

should precede Project C on the expansion path. Areas 1 and 2

correspond to the difference savings when integrating over Paths AAB-ABC and A-AC-ABC, respectively.

Scheduling

Under the assumption that the benefits associated with a given

combination of projects in some period vary only with the output of

the system in that period, the start dates of the projects do not

affect the system costs. Thus the SC curves for a project

combination depend only on the presence, rather than start times,

237
of particular projects in that combination. The implications in

the context of waterways are that the capital cost of construction,

operating and maintenance costs, and benefits from reduced delays

are not affected by the age of the locks at any given time (i.e. by

project start dates) but only by the volume of traffic using the

locks. This assumption is very reasonable for the capital costs,

but somewhat simplifies the operating and maintenance costs. The

assumption is also reasonable for delay benefits although it

neglects the effect of long term economic changes induced by the

presence and performance of waterway investments.

AC ABAC

B

A

0

E

+J

U)

0

01 02 Volume

TI T2 TI me

Figure 2 Plot of System Cost for 3 Interdependent Projects (Casel)

Incorporating a Budget Constraint

238
In structuring the budget constraint, it will be assumed that

funds not spent in a given period will be available in subsequent

periods. Under this assumption, budget limitations have the effect

of delaying the earliest feasible start date of a given project

combination, just as they limit the earliest start of an individual

project. Consider the small example of two projects A and B. In

constructing the Curves A, B, and AB, the infeasible portion must

not be included. Figure 4 illustrates that Combination A is not

financially feasible until time T, corresponding to output Qn.

Combination AB is not feasible until time T2 . The three possible

expansion paths are then as follows:

A AS A AC ABC

(-

0

U

W Area 2 w

m Area 1

0

Volume

Time

Figure 3 Plot of System Cost for 3 Interdependent Projects (Case2)

239
1. start A at tine T, and B when Curves A and AB intersect

2. start B immediately and A when Curves B and AB intersect.

A S AB

0

LAB

E

4)

T1 T2

Time

Figure 4 Incorporating a Budget Constraint

In the validation systems of four and six locks were used to

compare the solution from the algorithm with that obtained through

exhaustive enumeration2 . In these four and six lock experiments

the optimal answer was found by the algorithm in 93.3 and 95

percent of the cases. In the suboptimal cases, the cumulative

costs were within 1% of those of the optimal sequence.

2

Conducting such tests on larger systems is not possible

because the optimal solution cannot be determined for comparison

with the solution obtained from the sequencing methodology.

240
3.4 Cash Flow Analysis

The output of sequencing and scheduling algorithm is the order

in which the projects are to implemented and the project start

times (i.e. the time construction is complete and the facility is

returned to full operation). Unfortunately, the implementation of

construction schedules are not without uncertainties. Often,

projects may be delayed due to funding interruptions, technical

complications, cost overruns or other unforseen conditions. Such

delays and overruns can be binding on the Inland Waterway Trust

Fund (IWTF). For example, if soil and geological surveys

incorrectly assess the type of foundatic-al rock, a project might

be interrupted to permit further engineering and design. For this

reason, it is helpful to have a methodology for evaluating the

financial sensitivities to changes in project costs and schedules.

Such a methodology was developed and programmed for the Corps

of Engineers to conduct sensitivity analysis of the IWTF with

respect to numerous scheduling and budgeting parameters. The

primary computational objective behind the methodology is to reveal

the resulting Trust Fund balance profile over a specified planning

horizon. The methodology allows for the inclusion of the numerous

factors in obtaining the cash flow profile of the IWTF, for

example:

1. project sequence and start dates,

241
2. distribution of project costs over the construction period,

3. duration of the construction period,

4. length of any project interruptions,

5. interest rate accrued on unspent sums over the planning horizon,

6. fuel consumption rates over the planning horizon, and

7. fuel tax rates over the planning horizon.

The computer program that implements the cash flow analysis

consists of four modules and a comprehensive user interface. The

scheduling module provides utilities for controlling project

specific parameters such as project start time, constru -ion

duration, and interruptions. The expenditure module considers four

basic Trust Fund parameters: distribution, federal matching share,

inflation, and base year for discounting. This morcale provides for

three types of expenditure distributions (normal, uniform, and user

defined. The revenue module incorporates the fuel tax, fuel

consumption and account interest rates to determine the total

revenues available in each time period. The output module provides

a summary table of the Trust Fund balances and a host of graphic

utilities and summary statistics. The computer program has been

successfully applied to analyze the sensitivity of the Trust Fund

balance to many of the possible uncertainties.

4. Additional Applications of the Methodology

Various applications can be envisioned for this entire

methodology or for some of its components. These included the

242
following:

1. Estimation of lock delays under various conditions such as

congestion levels, stall patterns, traffic mix, operational

improvements, major capacity improvements, and closures for

maintenance.

2. Computer evaluation of various lock operating options such as

chamber assignment selection for tows, grouping of vessels in

chambers, use of helper boats, priorities among vessels, and

platooning (m-up-n-down).

3. Investment planning and programming including selection and

timing of new projects and smaller scale improvements under

financial constraints.

4. Improved management decisions for tow operators, e.g. optimizing

fleet schedules and operating speeds under various levels of lock

congestion and unreliability.

5. Improved management decisions for shippers, e.g. inventory

policies, mode choice and facility location decisions.

6. Improved demand forecasting, based on an improved estimate of

future service levels. Beyond such waterway applications, it

appears that the approximation methods for queuing networks may be

applied in other types of systems such as road networks,

243
communication networks, manufacturing plants, and parallel computer

processors. The algorithm for scheduling interdependent projects

should have even wider applicability.

5. Conclusions and Extensions

A fairly comprehensive methodology has been developed for

evaluating and scheduling waterway system improvements. Some of

the elements may be separately used in several other important

applications. Some relatively complex aspects of the waterway

system, such as the interactions among delays at adjacent locks,

the effects of relatively rare lock failures on delays, and the

effects of reliability and congestion on tow operating decisions

and shipper inventory policies can be analyzed with this

methodology.

Further research would be desirable in several areas,

including the following:

1. improved microsimulation components to analyze, in greater

detail, various lock operating options,

2. improved metamodels for the approximation of operating

characteristics at multiple chamber locks,

3. hybrid model switching automatically between simulation and

metamodels depending on required model sensitivity

4. new variants of the scheduling algorithm, which trade

244
computation time for improved solutions,

5. connections to a model that predicts equilibrium demand over

time in a multimodal network.

245
6. References

1. Albin S.L. (1984) Approximating a Point Processes by a Renewal

Process, II: Superposition Arrival Processes to Queues.

Operations Research. 32. 1133-1162.

2. Au, T. and Au, T., Engineering Economics for Capital

Investment Analysis, Allyn & Bacon, Inc., Newton, MA, 1983

3. Carroll J.L., Bronzini M.S. and Hayward J.C. (1972) Analysis

of Waterway Systems: Summary Report. Pennsylvania

Transportation and Traffic Safety Center, University Park,

Pennsylvania.

4. Carroll J.L. and Bronzini M.S (1973) Waterway Transportation

Simulation Models: Development and Application. Waterway

Resources Research. 9. No 1, 51-63.

5. Carroll J.L. and Wilson H.G. A Zero-Sum User Charge System for

Rationing the Use of Inland Waterway Locks. Proc. the Int.

Conf. on Transportation Research.

6. DeSalvo J.S. and Lave L.B. (1968) An Analysis of Towboat

Delays. J. Transpn Econ. Policy. 2, 232-241.

7. Howe C.W., et al. (1969) Inland Waterway Transportation

246
Studies in Public and Private Management and Investment

Decisions. Johns Hopkins Press, Baltimore.

8. Janson, B., Method of Programming Regional Highway Projects,

J. of Transpn, Engg., ASCE, 114, no. 5, 584-605.

9. Kleinrock L. (1975) Queuing Systems Volume I: Theory. John

Wiley & Sons, Inc., New York.

10. Martinelli, D., and Schonfeld, P., Approximating Delays at

Interdependent Locks, J. Waterway, Port, and Oceanagraphic

Engg., ASCE (to be published).

11. Newell G.F. Applications of Queuing Theory.

12. Newell G.F. (1979) Approximate Behavior of Tandem Queues.

Springer-Verlag

13. Whitt W. (1983) The Queuing Network Analyzer. The Bell System

Technical Journal. 62. 2779-2815.

14. Whitt W. (1984) Approximations for Departure Processes and

Queues in Series. Naval Research Logistics Quarterly. 31. 499-

521.

15. Wilson H.G. (1978) On the Applicability of Queuing Theory to

Lock Capacity Analysis. Transpn Res. 12. 175-180.

247
248
METAMODELS FOR ESTIMATING WATERWAYS

DELAYS THROUGH A SERIES OF QUEUES

by

Melody D. M. Dai and Paul Schonfeld

249
250
METAMODELS FOR ESTIMATING WATERWAY DELAYS

THROUGH SERIES OF QUEUES

by

Melody D. M. Dai

Department of Transportation and Management Science

National Cheng-Kung University

Tainan, Taiwan

and

Paul Schonfeld

Department of Civil Engineering

University of Maryland

College Park, MD 20742

Revised in January, 1993

Forthcoming in

Transportation Research

251
ABSTRACT

A numerical method has been developed for esdmating delays on congested waterways.

Analytic and numerical results are presented for series of G/G/1 queues, i.e., with generally

distributed arrivals and service times and single chambers at each lock. One or two-way traffic

operations are modelled. A metamodelling approach which develops simple formulas to

approximate the results of simulation models is presented. The structure of the metamodels is

developed from queueing theory while their coefficients are statistically estimated from

simulation results.

The numerical method consists of three modules: (1) delays, (2) arrivals and (3) departures.

The first estimates the average waiting time for each lock when the arrival and service time

distributions at this lock are known. The second identifies the relations between the arrival

distributions at one lock and the departure distributions from the upstream and downstream locks.

The third estimates the mean and variance of interdeparture times when the interarrival and

service time distributions are known.

The method can be applied to systems with two-way traffic through common bi-directional

servers as well as to one-way traffic systems. Algorithms for both cases are presented. This

numerical method is shown to produce results that are close to the simulation results.

The metamodels developed for estimating delays and variances of interdeparture times may

be applied to waterways and other series of G/G/1 queues. These metamoiels for G/G/I queues

may provide key components of algorithms for analyzing networks of queues.

INTRODUCTION

Inland waterway transportation is quite important in the U.S. and other regions, especially

for heavy or bulky commodities, since it is inexpensive, energy efficient ard safe. Most U.S.

waterways consist of stepped navigable pools formed by dams across natural rivers. The lock

structures used to raise or lower vessels between adjacent pools constitute the major bottlenecks

in the waterway network [19] and generate extensive queues. Some locks have only one

chamber, -while others may have two parallel chambers whose characteristics may differ. The

most common chamber sizes are 110*1200 (i.e., 110 ft wide, 1200 ft long) and 110*600. Each

chamber size can accommodate a limited number of barges at one time. For example, a 110 ft

252
* 1200 ft chamber can accommodate at most 17 standard barges plus a towboat while a 110 *

600 chamber can accommodate at most 8 standard barges plus a towboat. If a tow has more

barges than the chamber can accommodate, it must be disassembled into several pieces (called

"cuts") to move through the chamber and must later be reassembled. Therefore, the service time

distributions depend on chamber size and tow size distributions. Sometimes, chambers will be

out of service (i.e., "stalled") due to various causes such as freezing, accidents, and mechanical

failures.

A reliable and efficient method for estimating lock queueing delays is essential for evaluating

and scheduling waterway investments. Unlike pure queueing theory, simulation methods can be

used to model the complexities of waterway operations. However, when many interdependent

lock investment proposals are considered, their selection and scheduling becomes a large

combinatorial problem [181 and simulation may require too much computer time for practical

applications. Hence a numerical approximation method which combines queueing theory and

simulation results is proposed here for estimating delays through series of waterway queues.

Figure 1 shows a simple diagram of a lock queueing system. Locks are the servers and tows

are customers waiting to be served by locks. In the lock queueing system tows from both

directions, upstream and downstream, share the same lock servers, while in most other queueing

systems the servers are exclusively one-directional. In this paper, the term "two-way traffic

operations" characterizes the lock queueing system while "one-way traffic operations" describes

a more general queueing system.

Arrival and service time distributions at locks are fairly complex. Carroll [2] and Desai [81

found that service times are not exponentially distributed, and arrivals are not Poisson distributed.

Other standard distributions have been tested for the present study, without consistent success.

Thus, empirical distributions (specified for 50 ,itervals) are used here for simulation while

general distributions, described only by their means and variances are used for queueing models.

Although isolated locks with a single chamber may be modeled as G/G/1 queueing systems, locks

with two parallel chambers may not be treated simply as G/G/2 queueing systems unless these

parallel chambers are identical.

The lock service time distributions are affected by the chamber assignment discipline at locks

with two dissimilar chambers. There the "main" chamber is larger than the "auxiliary" chamber

253
O Lock 0 Node • Inflow i Outflow

Figure 1. Lock Queueing System

254
and can accommodate without disassembly large tows which might require several cuts and far

larger service times to move through the auxiliary chamber. However, if the same number of

cuts is required through either chamber, the auxiliary chamber may provide faster service.

Generally, an auxiliary chamber has the same width as a main chamber (110 ft), but a shorter

length (360 or 600 rather than 600 or 1200 ft). When a tow requires the same number of cuts

in either chamber, it typically takes less time to move through the shorter auxiliary chamber.

Therefore, lock service time distributions are dynamic and depend on the chamber assignment

discipline.

Considerable interdependence may exist among locks in a series. The departure distributions

differ from the arrival distributions since the service time distributions change the tow headways.

The departures from one lock usually affect the arrivals at the next lock. Therefore, it is

improper to assume such locks are independent. That drastically impairs the applicability of

stochastic queueing theory. The interdependence among locks increases the difficulty in

estimating delays for the lock queueing system since it is necessary at each lock to identify the

interarrival time distributions of flows from adjacent locks.

Two-way traffic operation through common servers complicates the interdependence of lock

delays and precludes the use of some otherwise interesting queueing models. Delays are

determined by the arrival distributions and service time distributions. It is much more difficult

to identify the arrival distributions for two-way traffic systems than for one-way traffic systems.

The arrival distribution at one lock is affected by departures from both upstream and downstream

locks, while departures from this lock also affect the arrivals at upstream and downstream locks.

For example, in Fig. 1 the arrivals at Lock 2 would be affected by the departures from Locks 1

and 3. The departures from Locks 1 and 3 toward Lock 2 are highly correlated with the arrivals

at 1 and 3 from 2. At least some of the arrivals at I and 3 represent departures from 2. Hence,

the arrival distributions of these three locks are interdependent. Thus, two-way traffic operation

greatly complicates the estimation of the two arrival distributions at each lock.

The arrival distributions cannot be determined directly from the departure distributions of the

adjacent locks since tow speed variations between locks alter the arrivals. Thus the arrivaldistributions at one lock are affected not only by the departure distributions from adjacent locks,

but also by the distances and speed distributions between these locks.

255
Random failures, which in inland waterways are called stalls, contribute significantly to the

difficulties in estimating delays. Stalls, which interrupt lock operations and thereby increase

delays, are relatively rare compared to other events. Their occurrence is very difficult to predict.

Thus, Kelejian's efforts to model stalls and stall durations have not yet yielded strong results

despite the rigorous statistical methods employed [11].

The purpose of this research is to estimate delays for a realistic lock queueing system,

assuming the interarrival time distributions at the system boundaries and the lock service time

distributions are known. The difficulties in estimating delays for such a lock queueing system

are summarized as follows:

1. Arrival and service time distributions are generally distributed.

2. Parallel chambers are not identical.

3. Service time distributions would be affected by the chamber assignment discipline.

4. Considerable interdependence exists among a series of locks.

5. Two-way traffic operation through bi-directional chambers complicates the analysis.

6. The arrival distributions depend on not only the departures from previous locks but also

on the distances and speed distributions between locks.

7. Stalls increase the means and variances of delays.

The available analytic solutions for estimating delays in G/G/i queues are quite inadequate.

Kleinrock [13] suggested an approximation solution for a G/G/1 queue with heavy traffic. In

fact, this approximation solution is an upper bound for average waiting times in G/G/i queues.

It works well when the volume to capacity (V/C) ratio approaches 1.0 but generates significant

errors in estimated delays for low to medium traffic levels.

Bertsimas [1] derived an exact delay solution with mixed generalized Erlang distributed

arrivals and service times. This result could be applied to more realistic situations than Poisson

arrivals and exponential service times. However, without a departure function, this result is

difficult to extend to a series of locks.

DeSalvo [7] and Wilson [23] tried to estimate lock delays by treating locks as M/M/1 and

M/G/1 queues, respectively. These two models did not account for interdependence among locks

and their assumptions significantly limited their applicability.

System simulation models to analyze lock delays and tow travel times have been developed

256
by Howe [10] and by Carroll and Bronzini [3]. These modelled the waterway systems in

considerable detail and required considerable input data and computer time. Both models

assumed Poisson distributions for tow trip generation and did not account for stalls. However,

simulation models should, at least in principle, be able to represent the complexities of traffic on

waterway networks much better than analytic queueing models. Thus, a more efficient simulation

model able to accom te stall effects was desirable for this work.

Such a simulation model was developed in the early stages of this study [6]. This model can

accommodate generally distributed trip generations and service times. It can also evaluate the

stall effects. Although this simulation model requires only a few seconds to a few minutes on

PS/2 computer for each run, that is still hardly affordable for direct application in large

combinatorial network investment problems.

To avoid the computational expense of simulation when evaluating numerous combinations

and schedules of network improvements, a metamodelling approach [16, pp. 679-689] was

developed to approximate the results of the simulation model. The complete methodology used

in this study consists of (1) developing and validating a simulation model to represent waterway

networks with queues at locks, (2) formulating functions developed from queueing theory for

delays through series of locks, (3) statistically estimating the parameters of these functions using

simulation results, and (4) employing an iterative sequential scanning procedure to estimate

interarrival and interdeparture time distributions lock by lock until results converge at each lock.

Thus, relatively simple equations may serve as a proxy for the simulation model.

Basically, the numerical method is a decomposition model. It decomposes systems of queues

into separate queueing stations. The analysis of each queueing station is decomposed into three

steps, namely arrivals, departures and delays. Such decomposition techniques have been widely

applied for analyzing networks of queues [4,14,15,20,21,24]. These previous studies show that

it is usually sufficient to approximate all the flows in networks of queues by renewal processes

characterized by two parameters (the mean and the variance).

The proposed numerical method may be applied not only to the lock queueing system, but

also to some other series of queues. Th'e departure processes module and delay function may be

applied to networks of G/G/l queues. Further research in estimating interarrival time

distributions with multiple unequal inflows would be necessary for extending the numerical

257
method to general networks of queues which may have inflows and outflows at any node.

I. Simulation Model

A detailed discussion of the simulation model developed for this work is included in Dai and

Schonfeld [6]. A brief description of the simulation model and its validation is provided below.

1.1 Data Bs

The simulation model was developed on the basis of PMS (lock Performance Monitoring

System) data collected since 1975. This data base includes very detailed information on traffic

through the locks as well as physical aspects of lockages [9]. It is very useful for understanding

and quantifying waterway characteristics, such as lock operations, arrival distributions, service

times distributions, tow size distributions and stalls.

1.2 Model Features

The simulation model is programmed in Fortran-77 which provides great flexibility in

modelling. Basically, it is a stochastic, microscopic and event-scanning simulation model which

can handle any distributions for trip generation, travel speeds, lock service times and tow sizes.

These distributions can be specified for each interval in tables or by standard statistical

distributions. Currently, normal distributions are used for travel speeds while general

distributions based on empirical observations are used for other input variables. Tows are

allowed to overtake other tows. A FIFO (First-In-First-Out) service discipline is currently

employed. This model simulates two-way traffic through common servers and accounts for stalls.

1.3 Validation

To check the logic of this simulation model, its results are first compared to theoretical (but

very well established) results from queueing theory. This also checks the model's ability to

represent general series of queues. The model's results are then compared with observed data

to demonstrate how closely the model represents real systems and verify its ability to simulate

the special features of waterways.

The model's predicted waiting times at a single lock are compared with those obtained from

258
queueing theory when arrivals are Poisson distributed and service times are generally distributed.

The validation is conducted for a variety of volume/capacity (V/C) ratios ranging from .0471 to

.8934. To reduce the simulation variance each result is obtained by averaging the output from

30 independent simulation runs. To insure results are compared for a steady state, each

simulation run discards the first 10,000 tow waiting times and collects the next 12,000 values for

computing the average waiting time. The results are shown in Table 1. They confirm that the

simulated and theoretical average waiting times are extremely close. Such results verify that the

overall mechanism of the simulation model is correct. They also show that generally distributed

service times are generated satisfactorily in the simulation model. That is reassuring since the

same logic is also used to generate generally distributed interarrival times for G/G/1 queues and,

ultimately, to develop metamodels for series of G/G/i queues.

Table 1 Comparison of Theoretical and Simulated Results for a Single Lock Queue (M/G/1)

Interarrival Tines Service Times V/C W,•. Wt2 Deviation

Avg Var Avg Var

(hr) (hi) (hr) (h9) (hr) (hr) (M)

.8880 .7886 .7933 .3188 .8934 4.9516 5.0059 -1.09

.8880 .7886 .6701 .2274 .7546 1.5575 1.5522 0.34

.8880 .7886 .5025 .1280 .5659 0.4926 0.4935 -0.19

.8880 .7886 .2930 .0435 .3300 0.1082 0.1087 -0.46

.8880 .7886 .0418 .0009 .0471 0.00155 0.00156 -0.64

1 W,.: average waiting times from simulation

2 W,: average waiting times from queueing theory

The simulation results are then compared with the observed data (from January 1987) at

Locks 22, 24, 25, 26 and 27 on the Mississippi River. At that time, Locks 22, 24 and 25 had

single 600 ft long chambers. Locks 26 and 27 had two chambers each (600 ft and 360 ft long

at Lock 26, 1200 ft and 600 ft at Lock 27). The validation results are summarized in Tables 2,

3 and 4. Each result is averaged from 80 independent simulation runs. The initial condition for

simulation is assumed to be an empty system, which is consistent with the observed condition

for this system in winter.

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Table 2 shows that the simulated average waiting times for each lock and for the whole

series of locks are close to those observed except at Lock 25. The observed data also show that

tows sometimes were kept waiting at Lock 25 even when the chamber was idle. Such operation

is somewhat unusual. Therefore, no direct comparison of average waiting times at Lock 25 is

appropriate. Tables 3 and 4 also show that the simulation model represents the real system quite

wenl.

Table 2 Comparison of Simulated and Observed Average Waiting Times

Lock W5h1 Ww, 2 Difference oa.i oa,' 95% Confidence Interval

(min) (min) (min) (min) (min) (min)

22 4.09 3.73 0.36 2.87 15.19 3.49

24 6.12 6.36 0.24 4.14 29.74 6.72

25 4.49 10.94 6.45 4.50 19.19 -5

26 119.40 130.99 11.59 28.91 271.11 60.73

27 36.49 34.43 2.06 30.37 106.42 23.92

1 W,,.: simulated average waiting times

2 W,: observed average waidaig times

3 c..: standard deviation of simulated waiting times

4 a,,: standard deviation of observed waiting times

5 The comparison is not appropriate.

Table 3 Comparison of Chamber Volumes

Lock Chamber Vol,.' Volý 2 Difference 95% Confidence Interval

(tows/month) (tows/month) (tows/month) (tows/month)

22 1 44.61 45 0.39 1.88

24 1 56.00 56 0.00 2.44

25 1 51.40 52 0.60 2.13

26 1 275.20 265 10.20 2.83

26 2 155.96 167 11.04 3.81

27 1 390.95 389 1.95 10.25

27 2 306.73 306 0.73 10.62

1 VoI,.: simulated volumes

2 Vole: observed volumes

260
Table 4 Comparison of Cut Volumes

Lock Chamber Cuts' Voi, 2 Vol.W3 Difference 95% Confidence Interval

(tows/month) (tows/month) (tows/month) (tows/month)

22 1 1 30.35 31 0.65 1.47

22 1 Ž2 14.26 14 0.26

24 1 1 39.76 40 0.24 1.76

24 1 22 16.24 16 0.24

25 1 1 35.88 37 1.12 1.65

25 1 n2 15.52 15 0.52 0.97

26 1 1 75.46 74 1.46 2.04

26 1 >2 199.74 191 8.74 3.25

26 4 1 137.25 147 9.75 2.87

26 4 n2 18.71 20 1.29

27 1 1 390.95 389 1.95 10.25

27 4 1 269.05 265 4.05 5.59

1 "Cuts" are subsets of barges into which tows are subdivided for passage through lock chambers

2 Vol,,.: simulated volumes

3 Vol,: observed volumes

Each simulation run takes a few seconds to a few minutes on a personal computer, depending

on traffic volumes, duration of simulation periods, network size, and other factors. Despite that,

simulation time becomes expensive for evaluating large combinatorial investment scheduling

problems. For example, when there are n=20 possible investment projects, it is necessary to

simulate 2?* combinations to make the best decision. 30*22 separate simulation runs are then

required if each performance measure is based on the average over 30 independent replications.

Furthermore, the project combinations may have to be evaluated over several time periods.

Therefore, as n increases, direct evaluation by simulation becomes very expensive.

A metamodelling approach is proposed to overcome the computational requirements of

simulation. A simulation model is then treated as a function with unknown explicit form which

261
turns input parameters into output performance measures. The mctaxdelling approach provides

a method to develop simple formulas to approximate this function. However, the structural forms

of these formulas are quite important and not intuitively obvious. In queueing systems

applications, the structural forms should be based as closely as possible on queueing theory.

2. Numerik Methd

2.1 Methodology

In this study, a numerical method has been developed for estimating delays through a series

of queues. This method was originally developed for systems with bi-directional servers. With

a few simplifications, this method can be adapted for the more generally encountered systems

with one-directional servers. The numerical method decomposes the entire networks of queues

into each individual queueing stations. The method consists of three major modules: arrival

processes, departure processes, and delay functions. Arrival processes at a particular lock depend

on 6' c eparture distributions from the upstream and downstream locks. The departure processes

depend on the interaction among the arrival distributions and service time distributions at one

lock. The delay functions define the relations among waiting times, arrival distributions and

service time distributions. The basic concept of this method is to identify the parameters of the

interarrival and interdeparture time distributions for each lock, and then estimate the implied

waiting times. To be more concerned about the dependence among successive arrivals, the

authors also test the correlation of successive arrivals in the waterway systems. The results show

that the correlation among successive arrival intervals is very small. Most are close to 0 and no

more than 0.2 or less than -0.2. Currently, the following assumptions are used in the numerical

262
method:

1. Arrivals and service times are generally distributed.

2. A single service time distribution applies to both directions of passage through a chamber.

3. All traffic units are identical and have service times at a lock chamber governed by a single

probability distribution.

4. Each lock has one chamber.

5. Inflows and outflows occur only at the two end nodes of a series of locks.

6. The average upstream volumes are equal to the downstream volumes.

7. The long-term volume to capacity ratio (V/C) is less than 1.0 at every lock.

8. Idle times and service times are independent.

It should be noted that Assumptions 2, 3, 4, 5, 6 and 7 are only applicable to the numerical

method. The simulation model is not limited by those assumptions. The numerical method can

provide a quick and inexpensive approach for the analysis of lock delays. However, Assumptions

4, 5, 6 and 7 limit fairly significantly the applicability of the currently developed numerical

method and necessitate the substitution of the simulation model when significant deviations from

those assumptions must be considered. With some extensions to the numerical method expected

in the near future, Assumptions 4 and 5 may be eliminated. Assumption 6 could be relaxed even

though it is usually realistic for waterways.

The general service time distribution (Assumption 1) may be adjusted to incorporate the

frequency and duration of random failures provided such stalls depend mostly on traffic volumes

rather than the passage of time. Although the simulation model underlying the metamodels has

only been tested with a First-In-First-Out (FIFO) service discipline, the average delay estimated

263
by a metamodel should not depend on the discipline.

Each module of the numerical method consists of one or more metamodels. The procedures

used in developing each metamodel are summarized as follows:

1. Use queueing theory to identify the input (independent) variables which will affect the

output measures (dependent variables) and to propose functional relations between the input

variables and the output measures with appropriate structure form.

2. Plot the relevant output measures (dependent variables) versus the input (independent)

variables. This step helps confirm the relations between the output measures and the input

parameters.

3. Compute Pearson correlation tables. This step confirms correlations between selected

dependent variables and the independent variables and help avoid multicollinearity problems

among the independent variables.

4. Estimate parameters for the proposed functional relations (or "metamodels") and select the

preferred metamodel. The coefficient of determination (R2) is used to compare the

explanatory power of the alternatives. In general, the metamodel with R' closest to 1 is

preferred since it best accounts for the variation of dependent variables. It is also important

to test if the independent variable is significant in explaining the dependent variable's

variation.

5. Perform residual analysis to check whether this metamodel violates certain regression

assumptions, such as normality and homoscedasticity, and to detect outliers. The residual

analysis should include the property-analysis and the graphical analysis of residuals. The

basic residual properties to be examined include the mean, variance, skewness, and kurtosis.

264
The graphical analysis is the most direct and revealing way to examine a set of residuals.

The residuals can be displayed in one or two dimensions. The useful one-dimensional plot

of residuals includes histograms (or stem-and-leaf plots), schematic plots, and normal

probability plots [12]. The two-dimensional plot examines the relationships of the residuals

to either dependent or independent variables and is useful for identifying violation of

regression assumptions such as independence of residuals [12].

The variables and their corresponding ranges which were used in developing the metamodels

are listed in Table 5.

Table 5 Ranges of Variables Used to Develop Metamodels

Variable Minimum Maximum Description

CA 0.91 1.01 Coefficient of variation of interarrival times

CD 0.63 1.01 Coefficient of variation of interdeparture times

Cs 0.58 0.90 Coefficient of variation of service times

D 4.00 156.00 Distance between locks (mi)

W 0.03 18.50 Average waiting time (hr/tow)

X 0.04 1.67 Average trip rate (tows/br)

P, 75.60 295.00 Average tow speed (mi/day)

p 0.04 0.97 V/C ratio

oa2 0.35 608.46 Variance of interarrival times (b9/tow2

) o,2 1.38 2457.43 Variance of directional interarrival times (hr2/tow2)

GD2 0.17 606.44 Variance of interdeparture times (hr/tow2)

G2 1.05 2457.51 Variance of directional interdeparture times (hr/tow2

)

a32 0.04 4.34 Variance of service times (hr2/tow2)

'a 2.22 136.30 Standard deviation of tow speeds (mi/day)

2.2 Arrival Proesses

The following two steps are used for estimating the mean and variance of interarrival times:

Step 1. Estimate the means and variances of directional interarrival times at a particular lock

from the departure distributions of the adjacent upstream and downstream locks.

265
If flows are conserved between locks and if the V/C ratio is less than 1, the average

directional arrival rates at one lock should be equal to the average directional departure rates

from adjacent upstream and downstream locks. Therefore, the average directional interarrival

r, at that lock should also be equal to the average directional interdeparture times from

adjacent upstream and downstream locks. Such relations are represented in Eq. 1:

- = - {k-i-1, ifij1 (1) t.#, •j• k-i+l,if J-2

where

tam : the average interarrival time for Direction j and Lock i

t do :the average interdeparture time otr Direction j and Lock k

j : direction index (I = downstream, 2 = upstream)

If each tow moves at the same speed, the directional arrival distributions at one lock will be

the same as the directional departure distributions at the preceding lock. However, speed

variations change headway distributions between locks. Using the approach outlined in Section

2.1, Eq. 2 is developed to estimate the variance of directional interarrival times at one lock.

2=_ -. 2 a 2*D -i_#J=

a -354+.998°.+ 1271n D +. 1621nl (2)

(.17s) (.003) (.046) OW46)

R2 = 0.9997 n = 72 s. = 0.1263 i = 12.3459

where

o•2 : variance of interarrival times for Direction j and Lock i

o : variance of interdeparture times for Direction j and Lock k

DIA : distance between Locks i and k

O, : average tow speed between Locks i and k

aVIA : standard deviation of tow speeds between Locks i and k

j : direction index (1 = downstream, 2 = upstream)

s. :standard error of dependent variable

266
p . mean of dependent variable

Standard errors are shown parentheses under the estimated parameters of Eq. 2.

Currently, there is less theoretical basis for Eq. 2 than for the other metamodels developed

in this study. This metamodel was developed largely by empirical analysis. The dependent

variable was plotted versus possible influential factors, including the variance of directional

interdeparture time distributions, the distance between two locks, the average tow speed, and the

standard deviation of tow speeds. These plots help confirm the structural form of this

metamodel.

It is noteworthy that the coefficient for the variance of directional interdeparture times from

the preceding lock is 0.998, which is approximately equal to 1. This suggests that, theoretically,

the variance of directional interarrival times should be equal to the variance of directional

interdearture times plus an adjustment factor depending on the speed distribution and distance.

The lugh R2 value and the small standard error (only about 1% of the mean value of the

dependent variable) are also noteworthy.

Step 2. Estimate the overall mean and variance of interarrival times for this lock based on the

variances of directional interarrival times which are obtained from the step above.

- Iasi * (3)

tA

'A2 = o.o12+.173(6 2 + 2) (4)

(0.00379) (0.000083)

R2 = 0.99999 n = 108 Se = 0.00109 p = 3.09753

whereatA : the average interarrival time at Lock i

OA02 : variance of interarrival times at Lock i

The meaning of Eq. 3 would be clearer if viewed in terms of the average trip rates rather

,ban the average interarrival times. Eq. 3 implies thrt the overall arrival rate at certain lock is

267
the sum of the average directional arrival rates from upstream and downstream. In Eq. 4,

variances of upstream and downstream interarrival times carry the same weight in estimating the

overall variance of interarrival times, since directional trip rates are equal here (Assumption 6).

Eq. 4 should be reestimated when applied to a directionally imbalanced general network of

queues. It may be noted that the arrivals metamodel may estimate the variances of arrival

distributions not only at locks but at any distance from a departure point with a known departure

distribution (i.e. known mean and variance) such as a junction or port.

2.3 Departure Processes

The departure processes module estimates the mean and variance of interdeparture times.

Based on the flow conservation law, if the V/C ratio is less than 1, the outflow rate should be

equal to the inflow rate. Therefore, the average directional interdeparture time can be determined

from the corresponding interarrival time:

The variance of interdeparture times is estimated in three steps:

Step 1. Estimate the coefficient of variation of interdeparture times. Departure processes with

generally distributed arrivals and service times are analyzed by using Laplace transforms. The

use of Laplace transforms for derivations in queueing theory (which is quite frequent) is

presented in texts such as Kleinrock [13]. Some analytic relations obtained in this study are

shown below using the following notation:

Let:

AOA : mean and variance of interarrival times

tss o2 : mean and variance of lock service times

-DO 2 : mean and variance of interdeparture times

, o2 : mean and variance of lock idle times

p : V/C ratio

CA, Cs, CD : coefficients of variation for interarrival times, service times, and interdeparture

times

268
fA(), f3(t), fD(g), ft) : probability density functions (pdf) for interarrival times, lock service

times, intetdeparture times, and lock idle times, respectively

F;(z), F3(z), F;(z), F*(z) :Laplace transforms for fAt), fW(t), fD(t), fW(t)

For example, for interarrival times, the Laplace transform is expressed as

F;(Z = f fW(t)e - 0

The departure process in a queueing station may be analyzed for two different conditions:

with and without a queue. The interdeparture time distribution would be equal to the service

time distribution while there are queues waiting for service. However, the interdeparture time

would be equal to the sum of the idle time and the service time while there is no queue.

Therefore, the Laplace transforms for the interdeparture time distributions can be represented as

follows:

FKZ) 1,4, ' = FS{z) (6)

F; O= (- (7)

The probability of having a queue is given by the volume/capacity ratio p [13]. Then the

probability of not having a queue is (l-p). Therefore, the Laplace transform for the interarrival

time distribution can be represented by Eq. 8.

F;(z)=(1-p)F;(z) I... M ,+pF;(z) I.m ,'SW=(1-p)Fx*(Z)F(Z)+PF (Z) (8)

The mean of a distribution can be represented by the negative value of the first derivative

of its Laplace transform when z equals 0. Therefore, the average interarrival time, service time,

interdeparture time, and idle time can be represented by Eqs. 9a to 9d.

'A - z 12.0

269
-o = m•t.(9I€) t3 & C0( b

at '(90

= - o (9d)

The variance can be expressed as the difference between the second derivative of the Laplace

transform and the mean. Eqs. lOa-lOd express such relations for interarrival time, service time,

interdeparture time and idle time distributions.

a2= &2 P(z)la.o - '2 (10a)

A z

2

2 = O )Z (1l0) OD & 2

"

G 2= ___) Z. - i2(1d

a? = & 2(l )

When z equals 0, the Laplace transform is equal to 1, producing the following relations for

interanrival time, service time, interdeparture time and idle time distributions:

F;(O)=I (Ila)

270
F;(O)=I (1ib)

F;(O)- 1 (110)

F*1 (O)= I (lid)

Combining Eqs. 8, 9b, 9c, 9d, lIb, and lId yields

=D -_ a;• -0 = -((1-pX-t,-t,)+p(-ts)) (12)

Due to flow conservation, if the V/C ratio is less than 1, then the average interdeparture time

would be equal to the average interarrival time:

tD = 'A (13)

Therefore, Eqs. 12 and 13 can be combined:

'D = (1-p)(t,+ts)+Pts = tA (14)

Since p = ts/tA, Eq. 14 yields

t= tA (IS)

Combining Eqs. 8, 9b, 9d, 10b,10c, 10d, llb, lIc, lId, and 15 yields

2 =Ijz)°-.OtD2 (1-P)GI 2+U 2+(tA-tI)tI (16)

Dividing Eq. 16 by t 2 , we can obtain the following relation for the coefficient of variation CD:

271
02

C2 = (I-p)--- +p+C 2p2 -p2 (17)

In the special case where the arrival process is Poisson distributed and the service times are

exponentially distributed, then due to the memoryless property of the Poisson distribution, the

variance of idle times would be equal to the variance of interarrival times. Since the interarrival

times for a Poisson process are exponentially distributed and since the mean and standard

deviation of an exponential distribution are equal, we can state the following:

2o 2 (18)

of CA t A

oa2 = S

2 (19)

Therefore, in this special case, Eq. 16 can be simplified to

OD0 2= (20)

which is consistent with Burke's theorem [5]. In that theorem Burke proved that when the

arrivals are Poisson distributed and the service times are exponentially distributed, then the

departures must be Poisson distributed with the same mean and variance as the arrivals.

The main difficulty in estimating the variance of interdeparture times (Eq. 16) when arrivals

and service times are generally distributed is determining the variance of the lock idle times.

These depend on the way in which the previous busy period terminated. This problem may be

bypassed by developing a metamodel for directly estimating the variance of the interdeparture

times. Following the approach outlined in Section 2.1, the following metamodel was developed:

CD 2 = o.396+0-.6• CA 2(-p)+p)+j1.Ooo2(C 3

2p2-P) (21)

(0.052) (0.053) (0.0046)

R2 = 0.9977 n = 120 s. = 0.000056 p = 0.8267

Eq. 21 was originally developed by using four separate variables: (1) CA2

(1-p), (2) p, (3) CS2

p2,

and (4) p2. However very high correlations were observed between Variables 1 and 2 and

272
V3riables 3 and 4. Moreover, the coefficients were almost equal for Variables 1 and 2 and for

Variables 3 and 4. To avoid multicollinearity problems in developing Eq. 21, Variables 1 and

2 were combined into a single variable while Variables 3 and 4 were combined into a second

variable. It is noteworthy that the dependent variable has a standard error of 0.000056, which

is only 0.0000677 of its mean.

Since the mean and standard deviation of an exponential distribution must be equal, its

coefficient of variation must be 1.0. Thus, for the special case of an M/M/I queue:

CA2 = 1 (22)

Cs2 = 1 (23)

Substituting Eqs. 22 and 23 into Eq. 21, the latter may be simplified as follows:

CD2 = 0.396+0.606(1-p+p)+(p 2 -p 2) = 0.396+0.606 = 1.002 - 1.0 (24)

This result is also consistent with Burke's Theorem [5].

The parameters and structural form of this metamodel are similar to those of Eq. 17 which

was analytically derived for GIG/I queues. In addition, its standard error is extremely tight and

it satisfies Burke's Theorem very closely when applied to .he special M/M/I case. Since the

general distributions are not specific to waterway applications and are specified only by their

mean and variance, this metamodel seems quite reliable and useful for predicting interdeparture

time distributions from G/G/i queues embedded in larger systems, such as series and networks.

Step 2. Estimate the variance of interdeparture times. By the definition of the coefficient of

variation, the variance of interdeparture times can be represented as follows:

OD 2 =CD2t-D 2 (25)

Step 3. Estimate the variance of directional interdeparture times for upstream and downstream

wraffic. For this purpose the following metamodel was developed:

273
o~ 2 *@

o2=0.28.62 A(26)

J-1

(0.08) (0.03)

R2 = 0.9971 n = 216 so = 1.0697 p-11.9695

The above metamodel was developed empirically, without a structural form well grounded

on queueing theory. Although its R2 is still quite high, this metamodel accounts for most of the

error produced by the complete numerical method. Hence, improvements in this step would

clearly be desirable.

2.4 Delay Function

The delay function is intended to estimate the average waiting time at a lock for which the

arrival and service time distributions are known. Marchal [171 has proposed the following upper

bound for waiting times in a G/G/l queue:

w0. (4A2+3 2)(1 +Cl()

(1 -p)( 1 +2 ) (27)

p2

This upper bound is exact for M/M/1 and M/G/1 queues.

To develop the delay metamodel, the structures of several upper bounds, including Marchal's,

wenr considered. Since the metamodel with the structure of Marchal's upper bound has the oest

coefficient of determination (R2), the smallest standard error, and no multicollinearity problem,

it (Eq. 28) was selected to approximate the delay:

(1 -p .1 2 +CSU2) (28)

(0.013) (0.001)

R2 =O0.9990 n = 120 s. = 0.0151 p = 2.1602

In this delay function, the average waiting time is almost proportional to the sum of the variances

274
of interarrival times and service times. The average waiting time approaches infinity as the

valmrne/capacity ratio p approaches 1.0.

Comparing Eqs. 27 and 28, it is seen that they are very similar. Theoretically, in Eq. 28, the

parameter for second term should be equal to 0.5. Therefore, the first term should be negative

since the second term is an upper bound for the average waiting time. Such comparison supports

the reasonableness of the metamodel obtained in Eq. 28.

2.5 Algorithm For Two-Way Traffic Systems

The algorithm to estimate delays for a series of locks is developed based on the metamodels

discussed in Section 2.1 to 2.4.

Delays depend on the arrival and service time distributions. Therefore, to estimate delays,

we need to know in advance the means and variances of the interarrival and service time

distributions. For two-way traffic systems with series of G/G/i queues and bi-directional servers,

a difficulty arises in identifying the variances of interarrival times. The variance of interarrival

times at a certain lock labeled k is affected by the departure distributions from adjacent upstream

and downstream locks. The departure distributions at adjacent upstream and downstream locks

depend on their arrival distributions, which are affected by the departure distributions from Lock

k. Hence, the variances of interarrival times at adjacent locks depend upon each other.

Therefore, the variances of interarrival times cannot be determined from a single one-directional

scan along a series of queues. For example, if we tried to estimate delays by scanning from

upstream toward downstream, we could determine at Lock k the variance of interarrival times

from upstream, but the variance of interarrival times from downstream would be unknown and

would be affected by the departure distribution from this Lock k. To overcome such complex

interdependence, an iterative algorithm is proposed. It starts scanning in one direction while

using some initialized assumed values for the variances of interdeparture times from the opposite

direction. It can thus sequentially estimate the interarrival and interdeparture time listributions

for each lock. Then, the scanning direction is reversed and the process is repeated, using the

interdeparture distributions for the opposite direction estimated in the previous scan. Alternating

directions, the scanning process continues until the variances of interdeparture times computed

in successive iterations converge. Then the algorithm stops reestimating the arrival distributions

275
and proceeds to estimate delays.

In the first scan the initial values for variances of interdeparture times from the opposite

direction are suggested to be equal or close to the variance at the corresponding locks from the

scanning direction. Such assumed initial values hasten the converge but are not required. Brief

convergence and stability tests, conducted by the authors, show that poor initial values increase

the number of iterations for convergence but do not affect the final results. Computation speeds

for various problem sizes and convergence criteria are summarized in Wei, Dai and Schonfeld

[22].

The following algorithm is designed to apply the metamodels in estimating single-chamber

lock delays. The notation used in this algorithm is previously discussed as follows:

g• : the average interarrival time for Direction j and Lock i

t4A : the average interdeparture time for Direction j and Lock k

a sn2 : variance of interarrival times for Direction j, Lock i and Iteration n

2 : variance of interdeparture times for Direction j, Lock k and Iteration n 0,.

ow 2 : variance of interdeparture times from origin node

0 2 : variance of interdeparture times from destination node

DIA : distance between Locks i and k

IL,,, : average tow speed between Locks i and k

a,,VI : standard deviation of tow speeds between Locks i and xc

j : direction index (I = downstream, 2 = upstream)

tA : the average interarrival time at Lock i

GAIn2 : varianci of interarrival times at Lock i and Iteration n

Do : the average interdeparture time at Lock i

2 : variance of interdeparture times at Lock i and Iteration n

0 2 : variance of lock service times at Lock i

276
Pt : volume/capacity (=V/C) ratio at Lock i

it• : average inflow rate at Lock i

CS, : coefficient of variation of service time at Lock i

CA,,, CD,, : coefficients of variation for interarrival times and interdepamture times,

respectively, at Lock i and Iteration n

c, : constant (assumed value)

M : total number of Locks

W, : the average waiting time at Lock i

The required input of this algorithm includes the service time distributions, inflow

distributions, distances between locks, and speed distributions. This algorithm provides delays

as well as interarrival and interdeparture time distributions. The algorithm consists of the

following steps:

1. Compute the average directional interamrival times and the average directional interdeparture

times for each lock.

d d k=-l#Jfi=()

2. Compute the average interarrival time for each lock

tA (3a)

du+ am:

3. Estimate the variance of interarrival and interdeparture times for each lock.

3.1 Set n=l

3.2 Assume initial values for the variances of interdeparture times in Direction 2, at Locks 2

through M

277 |I |IM HH H Il bma i .. ..
o,12 = c. i=2, ... M

3.3 Starting from Lock 1, let i = 1

3.4 Compute the variance of directional interarrival times at Lock i using the metamodel

expressed in Eq. 2:

2=.354+.99804A -- 1271n1Lfj=2n 'i. (2a) °.•.•'-'354LV ÷'#,A. ' '• '• , k'lv -i÷l#J-2

3.5 Compute the variance of combined interarrival dimes at lock i

oA,2= 0.0128g+0.1243(u 2 + a (4a)

3.6 Compute the coefficient of variation for interarrival times at Lock i

At. -

'Ag

3.7 Estimate the coefficient of variation for interdeparture times at Lock i using the metamodel

expressed in Eq. 21:

CD 2 =0.396+O.606C 2(1-)p C 2 2 2 (21s) 'o,. o. oA-P A3.8 Compute the variance of interdeparture times at Lock i

"ODk2=C Di.2t-D

3.9 Compute the variance of directional interdeparture times at Lock i

2 = 0.52+8.16 V D (6

E a don2

3.10 Repeat Steps 3.5 - 3.9 for i = 2, ...M

3.11 Set n - n+1

278
3.12 Starting from Lock M, let i = M, and repeateps 3.5 - 3.10 for i = M, ...

3.13 Set n - n+1

3.14 Repeat Steps 3.4 - 3.13

3.15 If the following condition is satisfied, then go to Step 4. Otherwise, go to Step 3.14

S2~ -O2 I

-2 - - 0.001 1-1, ...M, J1.,2

4. Estimate the average waiting times using the meumoclel expressed in Eq. 28:

= 2 +o2 X1 + c, 2

) (C82)

(1 -P.X -2 +CS12)

2.6 Algorithm For One-Way Traffic Systems

Although the numerical method was originally developed for two-way traffic systems, with

a few simplifications, this method can be adapted for the more generally encountered systems

with one-directional servers. One-directional systems may be treated as a special case of twodirectional systems. The one-directional algorithm should perform better since the arrival

distributions will be affected by the departure distributions from upstream only and are not

subject to circular interdependence. Therefore, the arrival distributions may be determined in a

single one-directional scan without any iteration. The notation used in this algorithm for one-way

traffic is as follows:

Do : distance between Locks i and k

ILV, : average tow speed between Locks i and k

a 0 : standard deviation of tow speeds between Locks i and k

tt : the average interarrival time at Lock i

24 : variance of interarrival times at Lock i

ti) : average interdeparture time at Lock i

279
A 2 : variance of interdeparture times at Lock i

2 : variance of interdeparture times from origin node

oaS : variance of lock service tites at Lock i

PA : V/C ratio at Lock i

it : average inflow rate at Lock i

CA, CD? CS, : coefficients of variation for interarrival times, interdeparture times, and

service thm respectively, at Lock i.

M : total number of locks

Wj : the average waiting time at Lock i

This algorithm has the same input and output as the algorithm for two-way systems. It consists

of the following steps:

1. Compute the average interarrival time and interdeparture time for each lock

= tA " t,. i=I .. (Ib)

2. Estimate the variance of interarrival time and interdeparture time for each lock.

2.1 Starting from Lock 1, let i=l

2.2 Compute the variance of interarrival times at Lock i using the metamodel developed for twoway traffic (Eq. 2):

2 =-.3,54÷.998o 2+ .1271n DlI'*'-1 ÷.1621no (2b)

2.3 Compute the coefficient of variation for interarrival times at Lock i

C-- tA,

2.4 Estimate the coefficient of variation for interdeparture times at Lock i using the rnetamodel

developed for two-way traffic (Eq. 21):

280
2 2 (21b)

2.5 Compute the variance of interdeparture times at Lock i

oo=

2.6 Repeat steps 2.2 - 2.5 for i = 2, ...M

3. Estimate average waiting times using the metanmodl developed for two-way traffic (Eq. 28):

XA +0 2)(1 +CSj)

W, = -0.035.0. 1S (nb)

2.7 Comparion of Numerical and Simulated Results

An experiment was conducted to test how well the numerical method duplicates the results

of simulations. Three-lock, two-directional systems were tested. The controlled variables in this

experiment include the V/C ratio p, the variance of lock service times, inflow rate, distance

between locks, and tow speed. In this test the values of the controlled variables were uniformly

sampling within the range of 0.01 to 0.89 for the V/C ratio, 0.0007 to 0.3332 for the variance

of lock service times, 12.0 to 57.0 tows per day for the inflow rate, 5 to 60 miles for the distance

between locks, 108 to 325 miles per day for the average tow speed, and 33.84 to 101.52 miles

per day for the standard deviation of tow speeds. The ranges of the controlled variables were

chosen to cover the possible operating conditions of waterways. For this system the variances

of the interdeparture times converged within 0.1% for every lock and direction in no more than

5 iterations. The results are shown in Table 6.

The largest absolute deviation is 0.0944 hr when the average simulated waiting time is

2.3165 hr while the numerical method estimates 2.4109 hr. This represents a deviation of

approximately 4.08%. The relative deviations are larger when V/C ratios and waiting times are

close to zero, but very small in the more congested situations where investment evaluations have

practical value. In general, these results indicate that the numerical method may be used to

281
screen alternatives and gr'eady reduce the number of lock improvement combinations that have

to be evaluated by the more detailed microscopic simulation model.

Table 6 Comparison of Numerical and Simulated Results (Two-Way Traffic System)

V. V/C Dist Speed as1 W.do Wn.' Deviation

tows/day mi mi/day hr hr hr

12.0 0.01 5 270 0.0007 0.0003 0.0000 -0.0003

12.0 0.07 5 270 0.0360 0.0153 0.0000 -0.0153

12.0 0.17 5 270 0.1897 0.0989 0.0636 -0.0353

24.0 0.15 5 325 0.0309 0.0334 0.0000 -0.0334

24.0 0.34 5 325 0.1620 0.2316 0.2029 -0.0287

24.0 0.25 5 325 0.0915 0.1139 0.0813 -0.0326

36.0 0.22 5 108 0.0309 0.0542 0.0198 -0.0344

36.0 0.03 5 108 0.0006 0.0008 0.0000 -0.0008

36.0 0.50 5 108 0.1618 0.4621 0.4588 -0.0033

48.0 0.50 5 162 0.1883 0.4355 0.4125 -0.0230

48.0 0.29 5 162 0.0646 0.0962 0.0698 -0.0264

48.0 0.67 5 162 0.3330 1.2028 1.1981 -0.0047

54.0 0.75 10 108 0.2271 1.3926 1.4037 0.0111

54.0 0.57 10 108 0.1279 0.3901 0.3782 -0.0119

54.0 0.89 10 108 0.3167 4.9837 4.9400 -0.0437

54.0 0.75 20 216 0.1616 1.2203 1.2222 0.0019

54.0 0.57 20 216 0.0909 0.0329 0.3076 -0.0210

54.0 0.89 20 216 0.2259 4.4608 4.4634 -0.0073

57.0 0.60 5 325 0.1557 0.5430 0.5579 0.0149

57.0 0.05 5 325 0.0011 0.0012 0.0000 -0.0012

57.0 0.80 5 325 0.2738 2.0874 2.1815 0.0941

57.0 0.35 60 162 0.0645 0.1372 0.1039 -0.0333

57.0 0.60 60 162 0.1882 0.6381 0.6465 0.0084

57.0 0.80 60 162 0.3?72 2.3165 2.4109 0.0944

A.: two-way flow rate

2 osz: variance of service times

3 W,,: simulated waiting times

4 W.: waiting times estimated with numerical method

282
3. Conclusions and Recomnmendations

A numerical method has been developed for estimating delays through a series of queues

with inflows and outflows occurring only at end nodes. This method was originally developed

for systems with bi-directional servers. With a few simplifications, this method can be adapted

for the more generally encountered systems with one-directional servers. The two-direction

algorithm employs an iterative alternating direction scanning procedure to estimate the interarrival

and interdeparture time distributions lock by lock until the interdeparture time variances for

successive iterations converge. The performance of this two-direction algorithm is tested with

satisfactory results. The one-direction algorithm only scans the interarrival time and

interdeparture time distributions from the first to the last lock without any iteration and should,

theoretically, be less subject to interdependence errors.

Both the two-direction and one-direction algorithms rely on several metamodels estimated

from a previously developed simulation model. These metamodels provide the following

valuable results for series of G/G/i queues.

1. The delay function metamodel (Eq. 28) indicates how the V/C ratios, arrival distributions,

and service distributions affect the average waiting times for G/G/i queues. This delay

function is inspired by the structure of Marchal's upper bound for G/G/I queues. The

standard error of the estimated delay is 0.0151 hours which, compared to its mean of 2.1602

hours, is sufficiently tight for evaluation purposes.

2. The relations among the coefficients of variation of interdeparture times, interarrival times,

service times, and the V/C ratio are formulated in the departures metamodel. The structure

of the departure function (Eq. 21) is based on functions for the squared coefficients of

variation of interdeparture times. By applying Laplace transforms, these functions (i.e., Eqs.

16 and 17) are derived theoretically in this paper. Statistical estimation of the parameters

yields a very good fit. The function's standard error of 0.000056 is extremely tight

compared to its mean of 0.8267. The parameters also have very tight standard errors. In

addition, this departure function is consistent with Burke's Theorem. The results show that

the metamodelling approach combining queueing theory and statistical estimation based on

simulation outputs is quite successful. This departure function should be very useful for

analyzing networks of queues.

283
3. The arrivals module provides the relation between the variance of interarrival times and the

variance of interdeparture times from the adjacent queue stations when speed variations

change the headway distributions between successive queue stations.

The numerical method is useful in analyzing series of GIG/l queues. However, the following

extensions would be desirable to increase its applicability.

1. A function should be developed to estimate the variance of interarrival times when inflows

and outflows occur between queue stations. With such an arrival function the numerical

method might be applied to general networks of queues, including tree and grid networks.

The proposed approach would compute the variance of overall arrival rates as the sum of

the arrival rate variances from all inflows, assuming individual inflows are independent of

each other, and then develop the relation between the variance of interarrival times and the

variance of arrival rates.

2. The effects of random failures (i.e., stalls) might be incorporated by treating stalls as a

second class of users with its own arrival and service time distributions.

3. If possible, the numerical method should be extended to locks with two or more chambers.

This is rather difficult for chambers with different characteristics, because the chamber

assignment process affects lock capacity.

4. Queues with limited storage.

Additional statistical and computational tests are also desirable to further validate this method

and extend its applicability.

The final methodology for estimating waterway delays may combine simulation and

numerical methods. The simulation model may be used to accurately estimate delays for

relatively small systems of locks. The numerical method may be used in analyzing large

combinatorial problems. Guidelines should be developed for switching between the numerical

method and simulation in applications requiring intermediate accuracy.

Acknowledgement

The authors wish to thank Dr. George Antie and the Institute for Water Resources of the U.S.

Army Corps of Engineers for sponsoring this research. They also wish to thank the Associate

Editor, Professor Edward Moriok and two anonymous reviewers for their very helpful comments.

284
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