NBS

An Examination of the State of

the Art in Inland Waterways

System Lock Research

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Applied Mathematics

Washington, DC 20234

February 1 982

Final Report

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Prepared for; Navigation Division

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' NBSIR 81-241

AN EXAMINATION OF THE STATE OF

THE ART IN INLAND WATERWAYS

SYSTEM LOCK RESEARCH

Martin H. Pearl

MATIONAL BURJIAl' or aTANOAEUS

UBBABT

MAY 1 4 1982

U.S. DEPARTMENT OF COMMERCE

I National Bureau of Standards

National Engineering Laboratory

I Center for Applied Mathematics

Washington, DC 20234

February 1 982

Final Report

Prepared for: Navigation Division

Institute for Water Resources

Kingman Building

Ft. Belvoir, VA 22060

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Anrhler, Director
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Table of Contents

Table of Contents 1

List of Tables 11

Abstract Ill

Chapter 1. Purpose 1

Chapter 2. Navigable Waterways of the United States .3

Chapter 3. Maintaining the Inland Waterways System 7

Chapter 4. The Operation at a Lock 10

Chapter 5. Barge Traffic on the Inland Waterways System 16

Chapter 6. Attacking the Problem 19

Chapter 7. Simulations (The Early Years) 23

Chapter 8. Simulations (Part 2) 41

Chapter 9. Queueing Theory 48

Chapter 10. Queueing Theory Applied to a Single Chamber Lock 53

Chapter 11, Queueing Theory Applied to Multiple Chamber Locks 66

Chapter 12, Queueing Theory In the Foreign Literature 75

Chapter 13. Input and Performance Data 78

Chapter 14, Error Sensitivity 83

Chapter 15. Other Modes of Transportation 87

Chapter 16. Conclusions and Recommendations 93

References 98
ii

List of Tables

Table 1. Average Delay Per Tow (in Minutes) 12

Table 2. Typical Locking Times (in Minutes) 14

Table 3. Towboats and Barges of the United States (1971-1975) 17

Table 4. Towboats and Barges on the Inland Waterways System (1975) 18

Table 5. Predicted and Observed Total Waiting Time in Queue 54

Table 6. Comparison of Carroll's Simulation results with Wilson's M/G/1

Result and Results Using Lave and De Salvo's M/M/1 model 60

Table 7. Comparison of Observed and Calculated Averate Waiting Times for

Each Chamber at Lock and Dam No. 26 67

Table 8. Average Delays per Tow with Alternative Queueing Models

(in Minutes) .72

Table 9. Actual Data Compared with M/M/1 and M/G/1 ....74
iii

Abstract

The dams encountered on the locking rivers of the inland waterways system are

needed in order to maintain depths (usually nine feet) sufficient for tow

traffic. At the same time, the locks which pass waterborne traffic through

these dams act as bottlenecks and Interfere with the free flow of traffic. As

the level of traffic seeking to pass through a lock approaches the practical

capacity of the lock, congestion and delays develop. The simplest and

cheapest way to decrease the long delays commonly found at the busier locks in

the United States is to modify the operation of the lock in order to increase

its efficiency. However, the problem of determining the maximum capacity of a

lock and the operating procedures which achieve this efficiency is a difficult

one. Two complementary mathematical tools have been used in attempts to solve

this problem. They are

• computer simulations of traffic flow,

• the mathematical theory of queues.

The earliest computer simulation of the operation of passing a tow (consisting

of a towboat and a flotilla of barges) through a lock was performed by J, L.

Carroll in 1965 [9]

Since then, many (groups of) researchers working independently in various parts of the world have produced numerous computer simulations of barge traffic on locking waterways. In 1971, a large-scale,

multi-purpose simulation package was developed at the Pennsylvania Transportation and Traffic Safety Center (PTTSC), Pennsylvania State University,

University Park, Pa. by a team working under Carroll's direction. The portion

of this package devoted to traffic on a locking river consisted of three

parts
iv

• TOWGEN - the generation of a time stream of tows and an origindestination matrix,

• WATSIM - the movement of the tows of the system from origin to

destination,

• LOKSIM - the passage of each tow through each lock between its origin

and its destination.

Since 1971, the PTTSC package has been modified, reprogrammed, expanded and

generally refined (primarily under sponsership of the U.S. the Army Corps of

Engineers) and is today the accepted simulation package for modeling river

traffic in the United States. It appears to be far superior for general

purpose use than any competitor produced elsewhere in the world.

The second tool which has been used extensively for studying the capacity of a

lock is mathematical queueing theory. The first attempt to apply queueing

theory to the operation of a lock was made by Lave and DeSalve in 1968 [31,

56]. Shortly after these papers appeared, the question arose (and was debated

by Hayward [44], Wilson [82], and others) as to whether or not the basic

assumptions needed to make the queueing scenario mathematically tractable

(e.g. the assumption that the service times required by different tows are

independent) reflect the actual performance of a waterway. Subsequently,

several authors (e.g. Rochberg [72], Glassey and Ross [40], etc.) attempted to

follow up the work of Lave and DeSalvo in order to make the results predicted

by the model agree more closely with data obtained both from actual waterway

operations and from computer simulations. In general, these attempts have not

been satisfactory and the task of constructing an adequate queueing theory

model remains open.
V

Each mode of transportation has generated its own reservoir of research.

Queueing theory has been used in studying aircraft landings and take-offs at

an airport, automobile traffic at an intersection, automobile traffic on a

single-lane bridge or roadway, etc. One major purpose of this report was to

examine this model literature and to analyze the research developed therein

with an eye on applying portions of it to the lock and dam scenario. This

examination demonstrated that the state-of-the-art in lock and dam oriented

research is at least on a level with that applied to other model transport

problems and the possibility of finding helpful new ideas in this literature

is not sanguine.

1

1. Purpose

Since the end of World War II, all inodes of transportation, both in the United

States and in the rest of the world, have been experiencing ever-increasing

demands on their facilities. During this period, both passenger traffic and

freight tonnage has grown constantly. At the present time, the level of

demand on many of the transportation networks has approached the limits of

available capacity, resulting in serious delays, congestion and, ultimately,

the diversion of traffic. Faced with this ever-growing demand, the agencies

charged with the responsibility for maintaining a smooth, orderly flow of

traffic, have responded in two complementary ways:

• improving the efficiency of the existing facilities in order to

increase their capacities (usually by changing the rules under which

the facilities operate),

• expanding the facilities in order to increase the capacity of the system (usually by additional construction or new equipment).

Increasing the efficiency of an existing system presents a simpler, easier and

certainly cheaper way to accommodate additional traffic than is afforded by

construction.

Many of the problems arising in each of the various modes of transportation

are commmon to all of them and it has frequently been found that the same

basic problem has been studied in several different contexts. However, the

research efforts that have been engendered by these problems are invariably

directed at (as well as supported by, tested within, etc.) a single mode of

transportation. It is reasonable to expect that studies dealing with capacity

problems which have been developed under the auspices of one mode of transportation can contribute to the solution of problems facing other modes
2

of transportation . It is the two-fold purpose of this paper to examine the

state of the art in passing barge traffic along a locking waterway, and, to

evaluate research dealing with other modes of transportation with the aim of

applying this research to the lock and dam scenario.
3

2. Navigable Waterways of the United States

There are 25,543 miles of navigable waterways in the United States (including

those which have been improved by the Federal Government and other agencies

and those which have not been improved) which are capable of supporting commercial traffic V. Of this, 2,371 miles consist of the protected waters of

the Gulf of Mexico and The Atlantic Intracoastal Waterways, 490 miles of the

Great Lakes and the remaining 22,682 miles are composed of the various river

systems (the Inland Waterways System) of the continental United States and

Alaska. The Inland Waterways System comprises the Mississippi River, its tributaries and connecting waterways (8,954 miles), the Atlantic Coast Waterways

including the New York State Barge Canal and the Hudson River System (5,570

miles), the Gulf Coast Waterways, consisting of a network of rivers which

empty into the Gulf of Mexico (4,292 miles), and the Pacific Coast Waterways

(3,575 miles)

A waterway is called "locking" if the movement of traffic on it is made possible by the use of a system of locks and dams. Waterways which have no need

for locks and dams in order to provide sufficient draft to move traffic are

called "open". Where necessary, channel depth in open waterways in usually

maintained by dredging. The lower Mississippi River (south of St. Louis), the

Hudson and Missouri Rivers and the eastern portion of the Gulf Intracoastal

waterway are examples of open waterways. The upper Mississippi River (north

of St. Louis), the Ohio and Illinois Rivers and the western portion of the

Gulf Intracoastal Waterway are examples of locking waterways. This paper

deals with locking waterways exclusively.

/The Inland Waterways System serves both pleasure craft and commercial vessels. The commercial traffic is by far the greater of the two. In most analyses of waterway traffic, pleasure craft are ignored.
4

Since the second world war, there has been a very rapid growth in the use of

the Inland Waterways System. From 1961 to 1976, there was a 57% increase in

the tonnage carried on the Inland Waterways System and a 117% increase in ton2 miles /. During the same period, the increases in the Coastal Waterways

traffic have been 10% and 2% respectively. The growth in the use of the

Inland Waterways System has far exceeded all forecasts and has placed a considerable burden on the existing facilities. In 1975, over two trillion tonmiles of intercity freight were shipped by all modes of transportation in the

United States. Of this, 12% traveled by rivers and canals, and an additional

5% used the Great Lakes. The remaining 83% of freight transported between

cities traveled by rail (37%), highway (21%) and pipeline (25%).

Waterborne commercial traffic in the United States consists mainly of barge

transport of industrial commodities (coal, petroleum products, iron ore,

etc.), agricultural commodities (grains, soybeans, lumber, etc.) and lesser

amounts of manufactured products (paper, machinery, chemicals, etc.). In

1975, there were 946 million tons of waterborne commodities shipped domestically and 748 million tons of waterborne commodities in foreign shipments.

About 65% of domestic marine traffic travels by barge and the largest portion

of this (77%) is on the Inland Waterways System.

The Western Rivers System is the portion of the Navigable Waterways System

consisting of the Mississippi River, its tributaries and connecting waterways,

the Gulf Intracoastal Waterway and the navigable river system emptying into

the Gulf of Mexico. The largest part of the inland freight which is transported in shallow-draft vessels uses the Western Rivers System. Of the

/The large difference between these rates of increase will be discussed

later
5

roughly 25,000 miles of navigable channels in the United States which are

capable of carrying commercial traffic, about 14,000 miles are contained in

the Western Rivers System and about 8,700 miles of these have a limiting depth

^

of at least nine feet. It is in the Western River System that the most

rapid growth of waterborne traffic has taken place in recent years. The rapid

rise in ton-miles of freight (117%) compared to the rise in tonnage (57%)

between 1961 and 1976 reflects the increased use of the Western Rivers System,

where barge journeys are usually significantly longer than elsewhere in the

Inland Waterways System. Although this rapid growth in traffic has placed a

severe strain on all navigational facilities, it is at the locks and dams

which are needed to maintain navigable waters in the Western Rivers System

that the greatest impact has taken place. In times of heavy traffic (which is

virtually all of the time in some places) , locks and dams are not able to pass

vessels as quickly as they arrive and bottlenecks are created which slow river

traffic substantially. Vessels which cannot be served at a lock as soon as

they appear for service must wait (often for several hours) their turn in a

queue. The response to this problem of delay at a lock is the major focus of

this report.

When a direct water route between two points is available, the cost of transporting freight from one point to the other on the Inland Waterways System is

generally less than when using any other mode of transportation. In spite of

the long queues at locks which have become commonplace in recent years, many

shippers still feel that the low cost of waterborne freight movement overrides

other considerations. In addition, as the price of petroleum fuel (for trucks

3 /The limiting depth of a channel is the minimum depth that is maintained

along the entire length of the channel at all times.
6

and railroads as well as for river tows) goes up, the cost advantage of ship4 . ping via the Inland Waterways System can be expected to increase /, Thus,

the rapid growth in demand for services that the Inland Waterways System has

been experiencing since the second world war is likely to continue. Clearly,

unless some substantial changes are made either in the physical system itself

or in the way that the current system is used, the delays presently taking

place on the Inland Waterways System (especially on the Western Rivers System)

will become progessively longer. Eventually, the long delays will motivate

some shippers to seek faster, though more expensive, modes of transportation.

This added cost will be passed on, resulting in higher prices for the consumer

(for goods destined for the domestic market). For goods shipped in international commerce, the added cost will tend to make American exports less

attractive

/ The competitive price advantage of waterborne traffic may change if the

proposals for tolls at locks should be Implemented.
7

3. Maintaining the Inland Waterways System

In 1824, Congress appropriated $75,000 to improve navigation on the Ohio and

Mississippi Rivers. The task of making these improvements fell to the Army

Corps of Engineers. This marked the beginning of the Corps’ long involvement

with the national waterways. Since then, the maintenance and improvement of

the existing inland waterways and the planning, design and construction of new

waterways has remained their responsibility. For fiscal 1977, Congress appropriated 2.5 billion dollars to the Corps for civil works. Of this amount,

almost 660 million dollars were earmarked for projects dealing with navigation.

A substantial portion of the Corps’ current work deals with modernizing and

standardizing inland navigation -widening, straightening and deepening channels, building higher dams and larger locks and striving for uniform dimensions of barges, tows and locks. In recent years, the Corps has devoted a

great deal of effort to an attempt to alleviate the bottlenecks which now

exist at dams and locks, especially in the more heavily traveled portions of

the Western Rivers System. Two distinct approaches to this problem have been

used. The first requires the construction of new, larger and better facilities. This not only includes the construction of locks and dams but also of

larger barges, towboats with greater horsepower, etc. The second approach to

the problem of congestion is concerned primarily with making more efficient

use of existing (and, future) facilities. This includes possible changes in

the way services are distributed at a lock, in the make-up of tows, etc. The

Corps has commissioned numerous studies to determine how best to implement

each of these approaches to the problem.
8

The Corps has made significant strides in speeding the flow of traffic by

improving waterway facilities. Among these are: double locks at dams, the

construction of fewer but larger locks and dams on a channel, the standardization of sizes of both locks and barges, the greater use of large size barges

and high horsepower towboats, the use of helper boats, etc. However, building

new and better facilities is a very slow and expensive operation. Often the

increase in the volume of traffic that takes place between the beginning and

the completion of a large construction project more than overshadows the

improvements in capacity that have been made. Thus, the problem (long delays)

which led to the construction has not been solved. The queues at the locks of

the Inland Waterways System have been growing longer and as a result the

waiting time has been increasing.

There is another difficulty with relying heavily on more and more construction

to solve the problem of congestion. Every time that a new construction project is completed, it puts a severe limitation on the changes that can be made

in the future. Once a construction is in place, in the event that future

technology provides a means for obtaining substantial improvements in traffic

capacity by means of radically different designs or construction methods then

the nation has been forced into an obsolete, even though new, facility until a

better system can be built reflecting this new technology. For example, if,

after the locks on a channel have been constructed, it is found that substantial improvements in traffic capacity could have been achieved by changing

the locations of several of the locks, then the nation is faced with the

choice of replacing still usable facilities or living with a relatively

inefficient system. Thus, new construction is a last resort. It is important

that before construction takes place, all feasible non-construction approaches

to the optimal operation of a waterway be investigated.
9

Beginning in the early 1960’s, there was just such a shift in the Corps’

emphasis in its approach to the task of alleviating the congestion of the

nation's waterways. Increased attention was given to the problem of making

better use of existing navigational facilities, especially the locks and dams.

Basically, this problem was attacked using two separate (but often complementary) tools - computer simulation and theoretical mathematics. A description

of each of these tools and of the relation between them will be explored in

detail in the remainder of this report.
10

4. The Operation at a Lock

Most traffic along the Inland Waterways System consists of a flotilla of 6 to

15 barges lashed together and pushed by a towboat. The standard barge is 195

feet long, 35 feet wide and has a capacity of 1500 tons with a 9 foot draft.

They are designed to carry solid cargo and are usually uncovered. There are

also barges which carry liquid cargo. These tank barges have a capacity of up

to 3000 tons. Towboats range in length from 100 to 200 feet and have a horsepower rating of anywhere between 2000 and 10,000 hp. The trend in recent

years has been toward the more powerful towboats.

When a tow approaches a lock, it notifies the lockmaster (usually by radio)

and then takes its place in a queue waiting to pass through the lock. At

those locks at which there are two chambers, two queues are formed. If the

chambers are of different sizes, it is common for the queue waiting at the

smaller chamber to be smaller than the the queue waiting at the larger chamber. The tow captain can get in either queue to wait his turn. Until the tow

passes the lock, it remains under the direction of the lockmaster who tells

the tow captain where and when to move.

There will usually be another queue on the other side of the lock - vessel

traffic headed in the other direction. The most common queue discipline is to

alternately pass up-stream and down-stream vessels through the lock. Thus a

vessel headed (say) down-stream enters the chamber from the up-stream pool.

Water is pumped out of the lock until the water level in the chamber is the

same as that of the down-stream pool. The vessel then leaves the lock and an

up-stream vessel enters. Water is pumped into the chamber until the water

level is the same as in the up-stream pool, and the vessel then leaves the

lock. This procedure (one-up, one-down) is then repeated. It is customary to
11

choose vessels from within a queue on a first-come, first-served basis and to

Serve Opposing Queues Alternately. This is referred to in the literature as

SOQA queue discipline. If vessels are not served according to the SOOA queue

discipline then they are served in the order that they appeared at the dam.

This is known as the First-In, First-Out (FIFO) queue discipline. In a great

portion of the literature it is not clear which queue discipline is being

used. An entry of a tow into a lock begins when the tow captain is given permission to proceed from his waiting position into the lock. When there is

traffic in both directions on a river this waiting position must be sufficiently far away from the lock to permit vessels coming from the other direction to pass safely. This is called a long entry and can be as much as half a

mile. When there is only one queue at a lock then the waiting position for

tows in that queue can be just before the lock entrance. This is called a

short entry.

As previously stated, traffic on each side of a lock is usually passed on a

one-up, one-down basis. However, the lockmaster may expedite traffic by passing several vessels in one direction and then several vessels in the other

direction. This is called an n-up, n-down queue discipline. I^Hien successive

vessels travel in the same direction, the water level in the lock must be

changed (raised or lowered according to the direction of travel) while it contains no vessel. This is called a swingaround or empty turnaround. In practice, the time lost during an empty turnaround is less than the time lost during a long entry. If there are to be n passages through a lock in the same

direction without passages in the opposite direction, then all but the first

of these can use a short entry. Thus, more vessels can be passed through a

lock when the n-up, n-down discipline is used than can be passed through in
12

the same time when the one-up, one-down discipline prevails /, Although the

n-up, n-down queue discipline leads to a shorter average waiting time per vessel than does the one-up, one-down queue discipline, nevertheless, some few

vessels may be subject to unacceptably long delays.

Carroll, Rao and Wilson [16] state that the most effective queue discipline

varies from lock to lock, depending upon the level of traffic, the tow mix,

weather, etc. However, they present a table (Table 1 below) of average delay

times for various combinations of queue discipline and number of tow arrivals

per day, derived from a simulation analysis of Lock and Dam No. 27 on the

upper Mississippi River.

Number of Tows Per Day

Queue Discipline 11 19 28 36

First-come, first-served (FIFO) 5 22 63 199

Serve Opposing Queues Alternatively

( SOQA)

7 18 44 100

Three-up, Three-down 5 23 46 120

Table 1, Average Delay Per Tow (in Minutes)

/The same phenomenon is observed at an intersection of two roadways con- trolled by a traffic signal. If only one vehicle is permitted to cross the

intersection in any direction before the light changes, then the average waiting time per vehicle is very large. The minimum average waiting time per

vehicle is achieved by letting traffic continue in any one direction until

there is no queue waiting in that direction. Only then is traffic permitted

to move in the other direction -and it will continue to move in that direction

until that queue is empty.
13

Entry time for a tow begins when the tow captain is given permission by the

lockmaster to move from his position in the queue into the lock and ends when

the bow of the tow actually enters the lock chamber. Chamber time begins when

the bow of the tow enters the lock, continues while the tow is actually in the

chamber and ends when the stern of the towboat passes the other side of the

lock. Exit time begins when the stern of the towboat leaves the lock and ends

when the tow has cleared the lock area (so that a tow traveling in the opposite direction can begin its entry).

If an entire tow can be accommodated into a lock chamber without any changes

in the configuration of the flotilla then the passage is called a single lockage , Sometimes an entire tow can fit into a lock chamber, but only after the

configuration of the flotilla has been changed. Usually the towboat must be

moved from its normal position behind the barges to a position alongside them.

The towboat retakes its position behind the flotilla of barges on the far side

of the lock and the flotilla then continues its journey. Such a passage is

called a setover lockage

Clearly a setover lockage requires more time than a

single lockage. When a tow is too large for either a single lockage or a setover lockage then the passage of the tow must be made in several stages. The

flotilla of barges is broken up into two (or more) parts and each part is

passed through the lock separately. When all of the barges have passed to the

other side of the lock, the tow is reconstituted and proceeds on its way.

This is called a double (or more) lockage and requires substantially more time

than either a single or a setover lockage. Many factors beside tow size

affect the speed with which a tow can be passed through a lock. Foremost

among these are: the physical attributes and the condition of the lock itself

and of the nearby waterway, the tightness of the fit of the flotilla in the

chamber, the horsepower of the towboat and the weight of the load in the
14

barges, the currents in the river, etc. DeSalvo and Lave [31] and Kooman and

DeBruijn [53] have discussed a long list of other factors involved in tow

delays

Carroll, Rao and Wilson [16] have presented a table (Table 2 below) of typical

locking times (broken into its three components) for the various lockage types

at a 600 x 100 foot lock.

Entry

Lockage Type

Total (Long) Chamber Exit

Single 35-45 8-12 23-27 4-6

Double 105-135 10-22 90-106 5-7

Setover 50-70 8-16 37-47 5-7

Table 2. Typical Locking Times (in Minutes)

In 1970, in an attempt to reduce the locking time for double lockages, the

Corps placed an auxiliary towboat at Lock and Dam No. 26 on the upper

Mississippi River (Rao [69]). This dam, just above St. Louis, is one of the

most time consuming bottlenecks in the Inland Waterways System. The boat

(called a switchboat) was used to pull the first part of a flotilla of barges

of a double lockage out of the chamber and to tow it to a mooring where it

could wait for the remainder of the tow without Interfering with an entry into

the chamber from the opposite direction. The towboat and the second part of

the flotilla of barges then passed through the lock and the tow was reconstituted. This procedure reduced the chamber time for double lockages from an

average of about 93 minutes to about 69 minutes.
15

When a tow is barely able to fit into a lock chamber (e.g., nine 195 foot

long, 35 foot wide barges lashed together into a flotilla measuring 585 feet

by 105 feet, fitted into a standard 600 foot long, 110 foot wide chamber)

great care must be taken in order not to damage either the tow or the lock

facility. Clearly this need for caution tends to increase the time of the

lockage. This is a problem which will become increasingly more serious as the

size of both barges and towboats continues to grow.
16

5. Barge Traffic on the Inland Waterways System

The total length of the commercially navigable waterways of the United States

is currently 25,543 miles. Of this total, a draft of at least 9 feet is maintained on 15,675 miles. Sufficient draft for navigation (usually 9 feet) is

obtained by a system of 280 locks and dams containing 325 chambers on 44

6 waterways /. These locks provide lifts of up to 105 feet. The main chamber

at a lock varies from a minimum of 120 x 30 feet to a maximum of 1200 x 110

feet [79].

The trend in recent years has been toward the construction of larger barges

and more powerful tugboats. Immediately after World War II, the standard

solid-cargo barge was 175 feet long, 27 feet wide and 10 feet deep. The jumbo

dry-cargo barges of today are 195 feet long, 35 feet wide and 11 or 12 feet

deep. Tank barges (for liquid cargo) range from 200 to 300 feet long and are

about 50 feet wide. On locking rivers tows usually contain between 6 and 15

barges. On open waterways, tows having 40 barges are common. The increasing

size and power of the Inland waterways fleet is illustrated in Table 3.

/These figures include the Cross Florida Barge Canal which is not operational .
17

Type of Vessels

Self-Propelled

Towboats and Tugs

Number of Vessels

Horsepower

Non-Self-Propelled

Dry-Cargo Barges and Scows

Number of Vessels

Cargo Capacity (net tons)

Tank Barges

Number of Vessels

Cargo Capacity (net tons)

1971

4,059

4,217,064

17,527

19,710,605

3,420

7,486,718

1975

4,240

5,585,891

23,164

27,135,336

3,623

8,510,016

Percent

Increase

4.46

32.46

32.16

37.67

5.94

13.67

Table 3. Towboats and Barges of the United States (1971 -1975)^/

The largest portion of the nation’s tows operate in the Mississippi River System and the Gulf Intracoastal Waterway. Table 4 below presents the distribution of both towboats and barges in the Inland Waterways System.

/Table 3 is a portion of a table provided by the American Waterways Operators, Inc.
18

Mississippi River

Type of Vessels System and the Gulf Atlantic, Gulf Great Lakes

Intracoastal Waterway and Pacific Coasts System

Towboats and Tugs

Number of Vessels 2,541 1,554 145

Horsepower

Dry-Cargo Barges and

3,574,850

Scows

1,860,026 151,015

Number 18,049 4,917 198

Cargo Capacity

(net tons)

22,255,050 4,565,839 314,447

Tank Barges

Number 2,979 577 67

Cargo Capacity 6,295,236 2,056,813 157,967

(net tons)

Table 4. Towboats and Barges on the Inland Waterways System (1975)

Thus 59.9% of the nation's towboats, 77.9% of the nation's dry-cargo barges

and 82.2% of the nation's tank barges operate on the Mississippi River System

and the Gulf Introcoastal Waterway.

In conjunction with the recent trend to build larger barges and more powerful

towboats the Corps has been replacing locks of small chambers (600 x 110 feet)

with larger chambers (1200 x 110 feet). In addition, newer locks have a

higher lift than the older ones. Thus fewer locks are necessary on a waterway. For example, the Corps' modernization program for the Ohio River provided for the replacement of 46 existing locks with 19 larger locks.
19

6. Attacking the Problem

The Army Corps of Engineers has been striving since early in the nineteenth

century to maintain and improve the nation’s navigable waterways. Until the

period just after World War II, the major thrust of the Corps’ efforts was in

channel improvement and the building of new and larger facilities. With the

growing congestion on the inland waterways caused by the rapid expansion of

waterborne freight traffic, the Corps was forced to seek additional ways in

which to improve service. Previously there had been little effort to analyze

the locking procedures and use these analyses to speed the flow of traffic.

In the 1960’s, theoretical studies of lock operations were initiated. Various

local Corps offices had conducted systems analyses concerning their particular

operations since 1968. Shortly thereafter it became apparent that there was

need for a more integrated planning and development program. It was necessary

that this program be undertaken within the context of the total United States

transportation system and economy. In 1970, a task force was set up by the

Office of the Chief of Engineers to determine what role the Corps should take

in the general problem of navigation systems analysis. The report of this

task force in 1973 led to the institution of the Inland Navigations Systems

Analysis (INSA) program.

INSA is a planning capability. It is an integrated system of models, data,

and planning procedures developed to explain, predict and plan for U.S. inland

waterways transportation. It is designed to help the Corps achieve two

goals
20

• Operate the current inland waterways system as efficiently as possible ,

• Select the best size, location and timing of inland waterways improvements.

To achieve these goals, INSA uses its models, data and planning procedures to

• Forecast commodity flow,

• Predict modal share of traffic,

• Simulate and monitor inland waterway transportation,

• Predict economic impact of inland waterway improvements.

The INSA simulation models will be discussed in detail later in this report.

Interest in analyzing lock procedures and in using these analyses to speed

traffic flow coincided in time with the emergence of large-scale electronic

computers. The availability of these computers made it possible for the

first time to perform elaborate simulations of a waterway. Although simulations include the constructing of physical models, usually in miniature, and

the actual manual manipulation of them [4], the vast majority of simulations

are done abstractly on a computer. For mathematical simulation of any system,

a computer program must be written which describes the essential features of

that system. In addition, demand and capacity data are required, as well as

some start-up information describing the condition of the system before the

simulation begins. All of this must be entered into the computer in a form

compatible with the simulation program. The computer then operates on all of
21

the stored information, using the program provided, and produces the performance statistics of the operation. Hopefully, these results are similar to

the actual performance of the system being modeled.

The main advantages of computer simulations are speed and the relatively low

cost of testing alternative scenarios. The disadvantages of computer simulations are

• Difficulty in defining and writing a program which adequately captures

the essential features of the system being modeled,

• Necessity for introducing an element of randomness into a problem so

that a simulation often cannot be exactly duplicated,

• Necessity to search for optimal solutions and the difficulty of

recognizing whether the simulated operational strategies could be

further improved by investigating yet another set of parameter values,

• Inflexibility of a particular simulation program, that is, making a

minor change in the scenario or introducing additional parameters to a

computer program which was not designed to accomodate them would

require extensive changes in the computer program.

At about the same time that computer simulations were being introduced as a

method for modeling traffic on a waterway, the first attempts were being made

to use queueing theory as a tool with which to model the operation of a lock.

Queueing theory is a branch of applied mathematics developed early in the

twentieth century, primarily to study congestion in telephone circuits. As

in, the simulation technique, the essential features of the system must be

identified and incorporated into the model. The task of the mathematician is

to express these features in the form of equations, or some other mathematical

expression, and then to provide a solution. Once the mathematical solutions

are found, it is usually very easy to obtain directions for the optimal use of
22

the system being modeled /. The main advantages of a theoretical mathematical solution are:

• the parameters of the system can easily be varied over a broad range,

• sensitivity to changes in the parameters can be studied,

• the optimal solution is usually identifiable and readily available,

• results can readily be duplicated.

The major disadvantages are:

• the difficulty in expressing the essential features of the system

mathematically,

• the difficulty in solving the mathematical formulation once it has

been obtained.

It is generally agreed that when a viable theoretical approach can be found,

it is superior to simulation as a tool.

/When the features of the system are probabilistic, rather than deterministic, it usually happens that the solution obtained yields the optimal on- the-average use of the system, rather than the optimal use obtainable from a hypothetical policy that could somehow predict and allow for all random

fluctuations. Clearly the lock and dam scenar o falls into this category.
23

7. Simulations (The Early Years)

While the present study is concerned with the state-of-the art in estimating

lockage delays and lock capacities, it is not feasible to ignore the body of

work which deals with peripheral areas of waterways traffic but which, nevertheless, acted to stimulate research devoted specifically to locks and their

operations. The first attempts to simulate traffic on a waterway were made at

a time when the computer was just beginning to be a serious tool in scientific

research. The computer was expensive to use, had only a small fraction of the

capability it was later to have and computer tine was not easily available.

Early theoretical studies of lock and waterway capacity were generally small

scale, using a minimal amount of computation and requiring limited amounts of

data.

In 1954, the Melpar Corporation was commissioned by a private company which

operated tows on the Ohio and Mississippi Rivers to improve the company's profits, In a paper which was the outgrowth of this projects, O'Brien and Crane

[66] devised a Monte Carlo simulation of the fleet movements of the company's

tugs with the aim of improving performance by keeping towboats and barges

"busy". The simulation was to be performed without the use of electronic computers. The data required for the simulation were the number of available

barges left at each port served by the company and the number of days barges

were held before they were returnable as "empties", (All freight traffic was

in one direction.) A table of random numbers was used to obtain distributions for input into the simulation. However, in order to simplify calculations it was soon decided to replace these distributions with their average

values. This resulted in only minor changes in the output of the simulation.

Although the company's traffic passed through a number of locks, the work by
24

O’Brien and Crane did not consider either the operation of the lock and dam

scenario or the movement of traffic in a waterway. However, their paper did

seem to stimulate a long list of more theoretical and broadly based studies.

It is interesting to note that O’Brien and Crane’s paper had no references.

It has long been recognized that in order to determine the efficiency of operations at a lock, some theoretical maximal capacity must be postulated. Then,

the actual capacity achieved at that lock can be compared with the theoretical

capacity to obtain a measure of efficiency. In 1966, Bottoms [6] defined a

maximal tonnage capacity for a lock in terms of the lock size and the lockage

times. He also defined the practical tonnage capacity and showed that the

efficiency (i.e., the ratio of practical tonnage capacity to maximal tonnage

capacity) was about the same for all of locks he investigated. Since the

efficienty was consistently in the 25% to 32% range, he concluded that much

more tonnage could be gotten through a lock using the present facilities. In

1967, Schultz [76] continued the work of Bottoms but showed that the capacity

of a lock depended on a number of factors which Bottoms had ignored. For

example, the stretch of waterway leading to the lock, the depth and the

breadth of the floor of the channel, the hydraulics of the lock, etc. all

influence the true maximal tonnage capacity. Schultz introduced a graphical

method, using scale model paper cutouts, to analyze the capacity of a waterway.

Since these pioneering efforts, there has been a steady stream of increasingly sophisticated papers treating the theoretical maximal capacity (in tonnage, in cost-benefit, etc.) of a lock and attempting to improve lock efficiency by non-construction methods. In its bare-bones form, the capacity of a

lock (in vessels per time^ period) is determined as follows:
25

(i) Find the mix of vessel-types nj, n2» ...

(ii) Find the locking time for each type of vessel in the mix, tj^, t2>...

(iii) Find the time between vessels, tg,

(iv) Calculate the total time needed to pass a mix of vessels through the

lock, t = (tj + tQ)n2 + (t2 + tQ>n2 + ...

(v) Calculate the average time required per vessel.,

t = t/ (nj + n 2 + . . .

(vi) Calculate the number of vessels, N, which can pass the lock in a time period T,

N = T/t

A simple change in step i is needed if capacity is to be expressed in tonnage

per time period. The input data for this bare-bones capacity model is

• the vessel mix,

• the locking time for each type of vessel,

• the time between vessels.

When capacity of an existing lock is being determined, the needed data can be

obtained by observations made at the lock. This is not possible when the lock

under consideration has yet to be built, or when new procedures are contemplated at an existing lock then. To obtain input for the model in these cases

a study is necessary based both on the data available at similar existing

locks (if any) and on estimated date derived from the contemplated features of

the specific lock under consideration.

Once the bare-bones model has been established, improvements in the model can

be made in several ways;

• distinguish between situations which have been lumped together in the

bare-bones model and treat them separately, e.g., up-stream and

down-stream vessel mix, locking time, etc.; lockage in winter

and in summer, in darkness and in light, etc.
26

• incorporate into the model distinguishing features of the lock being

considered, e.g, the depth of the chambers, the separation of the

chambers (when there are more than one), the speed, currents and

geometry of the river, the capabilities of the hydraulics system, etc.

• consider the effect of treating the traffic sequentially rather than

lumped by the proportion in vessel mix.

In 1975, in the Hague, Kooman and DeBruijn [53] presented a long, detailed

paper giving an algorithm for finding the capacity of a lock. It was a

refined version of the bare-bones model described above. Although this model

was developed for study of the inland navigational system of the Netherlands,

it can be applied in the United States as well. Among the particular aspects

of the lock facility and traffic mix considered in this paper are:

• the wet cross sections of both the upper and lower gates

• the distance for a long entry,

• the dead weight of each vessel,

• the vessel length,

• the vessel utilization ratio,

• laden and unladen vessels.

Using their model for lock capacity, Kooman and De Bruljn estimated the delay

time at a lock in terms of the ratio of volume to capacity. In general they

used average values of the data in their simulation. According to Kooman and

and De Bruijn,

For a given lock the duration of successive lockages varies in practice

under the influence of the number of vessels to be locked and the actual

exit and entry times per vessel. These quantities are not, however,

constant but are in turn a function of such different variables as the

volume of traffic, sizes of vessels, vessel utilization, type of vessel,

the way vessels are navigated, etc. A simulation of the locking process

on the basis of the variable lockage time , taking into account all the

important influencing factors, is extremely complicated. No computer

program is so far available for an expanded simulation of this kind.
27

Apparently, the earliest application of c aputer simulation to the operation

of a lock was made by Joseph Carroll [9] in 1965. Working under the auspices

of the Tennessee Valley Authority, Carroll constructed a Monte Carlo model of

the Pickwick Lock on the Tennessee River, The Pickwick Lock is a single chamber lock and so represents the simplest locking situation. At the initial

stage of the simulation, no attempt was made to integrate the operation of the

single lock into the entire waterway system of which it is a part, (A simulation in which the whole of the system is included in the model, in the full

spirit of a "systems analysis", is considerably more difficult to construct.)

Carroll’s earliest model contained provisions for single, double, triple and

setover lockages. In addition, the program considered entry times (both long

and short), exit times and times for empty lockages. Tow arrival patterns

were part of the input to the model and both random and regular arrivals were

used. The output of the model was the lockage delay and idle times of each

tow as well as the number and types of tows processed. The output of the

model was compared with data obtained from the Pickwick Lock. This paper was

the first in a long series which culminated in the INSA program ten years

later

At about the same time that Carroll was constructing his model of the

operation of a lock, Howe [48], at Resources for the Future, Inc. (RFF) modeled the speed of tow as a function of channel depth and barge-flotilla configuration. As an outgrowth of these two models, the first systems model was

developed jointly by RFF and the Pennsylvania State University [10]. This

model simulated the movement of shallow-draft tows through a single straight

waterway. The model could accommodate up to 10 locks, each with one or two

chambers, 20 ports and 10 delay points 'channel restrictions). The input for

this simulation consisted of
28

• Tow characteristics (horsepower, tonnage, average draft and flotilla

configuration)

• tow itineraries (sequences of points visited in traveling through the

system)

• physical attributes of locks, channels and delay points,

• operating characteristics of locks (in the form of frequency distributions for up to 8 distinct locking time components for each chamber in

each direction).

The arrival times of the tows at the origin points (which, in this model, were

restricted to the two end points of the channel) were generated randomly by

means of a Poisson process. The movement of each tow in the channel was based

on the speed attainable by the tow (obtained from the RFF model) and the passage time (entry, lockage and delay times) of the tow through the lock

(obtained from Carroll's Pickwick Lock model). The output of the model was a

description of the movement of each of the tows until it reached its preassigned destination (or until the model run was terminated). From this, a

large variety of statistics describing what took place in the channel could be

compiled. These include:

0 the number of tows and barges processed at each lock,

• delay times,

• tonnages and tonnage-miles,

• the number of single and multiple lockages,

• average queue lengths.

Finally, the adequacy of the model was judged by comparing its output with

actual data collected by the Corps of Engineers on a portion of the Ohio

River.
29

The RFF-Penn State model described above was refined and generalized at the

Pennsylvania State University by Carroll and Bronzlni in 1971 [12], It was

intended that the new model be applied to a Y-shaped channel (rather than sim

ply a straight channel) based on a portion of the Upper Mississippi River con

taining 3 locks and a portion of the Illinois River containing 7 locks. In

addition, limits on the number of different tow characteristics and on the

travel Itineraries were expanded. Also, it was required that the number of

towboats and barges which enter a port be substantially equal to the number

which leave. While the model was being prepared, it became evident that the

construction of the input data was in Itself a difficult problem. For this

reason, the model was broken into two parts and a separate model was used for

the generation of traffic. This first part, called TOWGEN, consisted of a

program to generate a time-stream of tows from the various origins (no longer

restricted to be the end points of the channel) to their destinations. The

input to TOWGEN consisted of :

• An origin-destination (0-D) tonnage matrix for each commodity,

• a table of barge data, showing commodities carried, average loading

and commodities backhauled by each type of barge,

• an 0-D mileage table,

• a frequency distribution of flotilla size and towboat horsepower for

each commodity

• cost functions for each type of barge.

Tows were generated stochastically within the computer. The number and place

ment of empty barges required to support the assumed commodity flow were computed and made a part of the waterway traffic.

In the second part of the model, the tow generated by TOWGEN re put through

waterways simulation program (WATSIM) taking them from their point of origin
30

to their destination. In addition to the output of TOWGEN, the input for

WATSIM consisted of:

• parameters specifying the particular program options to be used,

including the attainable speeds in the channel, taken from Howe’s RFF

model

• frequency distributions for the components of locking time,

• a description of the waterway including ports, locks, delay points and

channel junctions.

The output of WATSIM consists of 12 tables describing the traffic that the

model has created. Typical outputs available from the program are:

• The number of barges and tows of each type originated and terminated

at each port,

• The number of tows and barges and total tonnage passing through each

lock,

• Total and average delay and process time at each lock,

• Maximum delays and queue lengths at each lock,

• Number of single and multiple lockages and locking times for each

chamber at each lock,

• Frequency distributions of tow delays and headway spacings

• Instantaneous and cumulative inventories of towboat and barge

utilization

• Summary of system delays.

The WATSIM model accommodated up to 30 ports, 20 delay points and 75 different

lock chambers distributed among 30 different locks. The waterway modeled by

WATSIM was permitted to have up to 5 branches. Ten types each of barges, tow

horsepowers and flotilla sizes were allowed. The programs were written in

FORTRAN IV(G) and run on the IBM 360/67 at Pennsylvania State University.
31

One critical parameter for the WATSIM model is the amount of time required

for a tow of a given type (number, size, load and configuration of barges,

size and horsepower of towboats, etc.) to pass through a given type of lock

(size of chamber, single or multiple chambers, switchboat, etc.) by means of a

given type of lockage (single, double, setover). As part of the Pennsylvania

State University simulation, a separate model (LOKSIM) was developed by

Hayward [42] in 1971, under the direction of Carroll, in order to better

estimate this parameter and, at the same time, provide additional flexibility

in the model. LOKSIM permits up to 3 chambers at a lock, various chamber

assignment and queue disciplines, and a mix of pleasure craft and commercial

vessels. After its development, LOKSIM was integrated into the T0WGEN-V7ATSIM

package

In August 1971, the Pennsylvania Transportation and Traffic Safety Center

(PTTSC) of the Pennsylvania State University submitted a 6 volume report about

the TOWGEN-LOKSIM-WATSIM package (to be referred to as the PSU-COE model) to

the North Central Division, Army Corps of Engineers. Volumes 2 [39], 3 [8]

and 4 [42] are devoted to in-depth detailed descriptions of the WATSIM, TOWGEN

and LOKSIM models respectively. Each volume lists the data requirements and

other inputs, the operation and the outputs of the model and includes a

discussion of the testing that the model had undergone. Volume 1 is a summary

report of the entire project and has a chapter devoted to two applications of

the program made by PTTSC to study alternative designs for lock and chamber

configurations. The first of these was to a portion of the Ohio River

9 (described in greater detail in Volume 6 of the report /). The second

9 /Volume V, [70] of the PTTSC report is concerned with navigation on a deep

draft waterway and will be discussed later.
32

application was to the 10 lock subsystem of the Illinois and upper Mississippi

Rivers which was the original impetus for the projects. In addition, the

North Central Division, Army corps of Engineers used the TOWGEN-LOKSIM package

in a separate study of the upper Mississippi River. These 3 applications provided a verification of the versatility and usefulness of the package.

The waterway system simulations described above formed a series in that each

paper built on the previous research. However, at about the same time, there

were a number of isolated simulations of locked river being developed in various parts of the world. In 1969, in France, Rousselin [74] studied the capacity of the upper Seine River. He presented a theoretical method for determining the maximal capacity of a navigable waterway. The optimal economic

capacity was obtained by multiplying various factors of the maximal theoretical capacity by reduction coefficients. However, Rousselin determined that a

computer simulation of the operation of the waterway would give a more precise

estimate of the optimal economic capacity than does the "classical method"

previously described. In 1970, Dunan and Coutheillas [33] continued the work

of Rousselin and described the simulation model in detail. The model is small

scale, having a maximum of 4 locks (though they may be of different sizes) and

not permitting branches in the channel. In addition to a flow diagram of the

model, Dunan and Coutheillas provided

• a list of inputs needed for the model (location and dimensions of the

locks, times required to fill and empty the chamber, number and types

of tows, etc.)

• a list of outputs (the number and categories of the vessels passed,

average time of passage, total number lockages, etc.).

No mention is made of the model ever being tested by comparing its output with

real data. These papers do not present any advances over the PSU-COE model.
33

In 1967, in West Germany, Leutzbach and Lenz [58] developed a model of a

single chamber lock which combined a small amount of queueing theory with a

computer simulation. They showed that for a waterway with Poisson arrivals of

tows, the average waiting time predicted by the model was the same as that

given by queueing theory. An interesting feature of this model is that the

lock can be held open to await an arriving tow for a specified period of

time.

All of the models that we have discussed thus far are for vessels traveling in

a shallow-water channel. There are a number of other studies dealing with

deep-draft navigation and these will be discussed shortly. Undoubtedly other

shallow-draft simulation models have been constructed in various parts of the

world. However, the PSU-COE simulation models appear to be more detailed,

more versitile and more useful than those developed elsewhere. According to

the INSA report. Volume 5 [80]

In general, foreign models were developed to satisfy the needs of particular project studies, and there has been little work on generalized

models. These models have not been relied on in U.S. efforts, probably

because of lack of domestic exposure, language barriers, computer differences, and differences in navigation conditions and operating practices.

Also, models directly applicable to U.S. waterways were developed at

about the same time as the foreign models.

In the previous paragraphs I have endeavored to show the path along which simulation of shallow-draft waterways developed through the early 1970's. During

this period, parallel research projects concerning deep-draft waterways were

taking place. The problems encountered in studying deep-draft waterways are

quite different from those met in modeling shallow-draft waterways. For

example, traffic in deep-draft waterways consists largely of ocean-going

ships. Thus the many-facted problems engendered by towboats and barge

flotillas do not apply to deep-draft waterways. On the other hand, deep-draft
34

waterways present a number of problems / of their own (e.g. the existence of

significant tides, the use of tugs, etc.). As stated in the INSA report. Volume 5

the substantive differences in the salient characteristics and operating

procedures of the two systems account for the lack of direct model transferability.

A number of models were developed in the period 1968-70 to study interoceanic

ship traffic through the Panama Canal. In 1969, A.T. Kearney & Company [1]

developed a sea-level canal simulation, written in the SIMSCRIPT programming

language in order to evaluate the effect of nonstructural improvements on the

canal’s capacity. This model contained several novel features. Inputs

included ship size, arrival rates, weather conditions operating restrictions,

lock staffing and equipment, and scheduled down-time for maintenance. Outputs

included lock utilization, tug requirements and traffic congestion.

A simulation based on a proposed sea-level canal in Panama was developed by

Harrison [41] in 1969 for the Waterways Experiment Station (WES) of the Army

Corps of Engineers. This program was written in FORTRAN and moved ships

through the canal by computing their maximum safe speeds in terms of canal

geometry and each ship’s individual characteristics. The effect of tides was

incorporated into the model and various tidal ranges and canal geometries were

used

The Welland Canal connects Lake Ontario with Lake Erie. The elevation of Lake

Ontario is 326 feet higher than that of Lake Erie. This 326 foot difference

is made up in a 7 mile stretch of the 27 mile Welland Canal. This portion of

the canal contains 7 locks with an average lift of 46 feet. In fact, locks

/ Generally, treatments of these problems have been of a less mathematical

nature and so are of less interest in this study.
35

4, 5 and 6 are constructed so that a ship sails directly from one lock right

into the next. There is an eighth lock which compensates for the fluctuating

levels of lake Erie. The Welland Canal runs parallel to the Niagara River and

so permits ocean-going vessels to bypass Niagara Falls.

In 1965 Luce and Sandor [59] studied the Welland Canal with a view to increasing its effective capacity. In examining the operation of the canal they concluded that the major causes of congestion were

• irregular arrivals leading to an effective capacity of less than 75%

of the maximum capacity,

• interruptions in service due to weather conditions and accidents,

etc.

• one particular lock (Lock 7) was a severe bottleneck,

• the entries to the lock were too long,

• ship captains unnecessarily took too much time for each lockage.

In order to increase the efficiency of the waterway, they recommended that

• traffic traveling toward Lock 7 be given priority over traffic

traveling away from the lock,

• fast vessels be given priority over slow vessels,

• ship captains be made aware of the importance of the short time

period (about 5 minutes) that they waste during a lockage,

• lock and dam priorities be arranged so that when bad weather is

expected, those operations which can continue may do so (e.g. heavy

vessels are less susceptible to bad weather than are light vessels.

• special rules to be established to apply when traffic is essentially

in one direction,
36

• there be greater communication between locks, leading to dynamic

graphic display models of all traffic on the canal being maintained at

each lock.

The implementation of these recommendations was able to reduce the lock cycle

from 100 minutes to 80 minutes. Although this paper is devoted solely to the

Welland Canal , it contains a number of features which should be of general

interest. The two unusual queue disciplines (fast one first, and heavy one

first) which were successfully implemented on the Welland Canal, do not seem

to have been considered elsewhere. In addition, the authors emphasize the

sensitivity of the number of lockages per day and the queue length to the time

required for an individual lockage. This emphasis appears to be a very useful

outlook on waterway research and should be more widely employed than it is.

In conjuctlon with the study of the Welland Canal by Luce and Sandor, the

Kates, Peat, Marwick & Company developed a simulation program, written in

ALGOL. The St. Lawrence Seaway Authority (SLSA) rewrote this program in FORTRAN in 1968 [77] and made a number of improvements which permitted the model

to be applied more widely. In this model, locking times were based on the

specific ship, crew and lock crew characteristics.

As previously stated, the fifth volume of the PTTSC report was concerned with

a deep-draft waterway. Rea and Nowadig [70] developed a simulation model for

a multiple channel deep-draft navigational system. The impetus for this work

was a proposed new canal paralleling the Welland Canal, but on the U.S. side

of the Niagara River. The earlier SLSA model was incorporated into this one

and several additional features were added. Since there would be two channels

connecting Lake Ontario with Lake Erie, a means of assigning vessels to one or

the other would be necessary and this became an important part of the model.
37

Another feature of this model was a provision for a lock to remain open for a

fixed period of time to await the arrival of an expected vessel. (This capability also appeared in the model developed by Leutzbach and Lenz [58] in 1967

in West Germany.)

In 1973, Carroll, Rao and Wilson [15] introduced a new deep-draft simulation

model called NETSIM. It was an expanded version of the one Rea and Nowadig

had developed two years earlier for the proposed canal parallel to the Welland

Canal. The aim of NETSIM was to model the entire Great Lakes seaway system.

The following features were added to the 1971 package:

• the passage of a vessel through the entire Great Lakes seaway

system could be simulated,

• an 0-D matrix for each commodity carried along the system,

• a dynamic, real-time algorithm for scheduling vessels in accordance

with customer demand for transport.

The program language was changed from IBM General Purpose Simulation System

(GPSS) used by Rea and Nowadig to SIMSCRIPT.

Several foreign studies of navigation on deep-draft waterways have been made.

In 1969, Nishida, Watanabe and Eguchi [65] constructed a simulation of traffic

on the Uraga Channel, connecting Tokyo Bay with the Pacific Ocean. The Uraga

Channel is an open waterway. However, vessels entering the area are given

very specific instructions as to which route to take. The Ministry of Transportation proposed Increasing the capacity of the Urage Channel by enlarging

and straightening the current routes and by changing port regulations. This

paper provides a simulation of traffic to examine the route improvement plan.

A detailed flow chart is provided, together with a list of inputs and outputs.
38

In 1972, Deazeley and Wilson [28] described a simulation they performed to

model the Bristol Channel as a closed waterway. An interesting feature of

this model is the assumption that when a tow appears which is not large enough

to fill the chamber a lockage will not take place until a second tow appears

to fill the chamber. Inputs for the model include;

• the density of traffic,

• tidal conditions,

• ship-size characteristic patterns,

• the discipline at the lock,

• the size of a lock or set of locks,

• the number of locks

• the service time at a lock.

The output of the simulation is a curve of the mean waiting time per ship as a

function to the traffic density. It is pointed out that the mean waiting time

per ship will not occur when traffic is lightest because, in such a situation,

ships may have to wait for a lockage until the chamber contains enough traffic. The program was written in ALGOL but no details of the flow chart are

given.

In 1971, Matlin, Gokhshteyn and Kunakhovich [60] presented a comprehensive

discussion of an integrated deep water navigational system in the European

part of the USSR. This report was designed to serve several purposes. One

was that the work would serve as a text for students at Soviet universities.

According to the authors

The development of the river network and improvement of the navigation

conditions on the rivers of the USSR are taking place primarily on the

basis of all around utilization of the water resources: the construction

of hydro-engineering complexes on the navigable rivers to generate

electric power and simultaneously increasing the navigation dimension;

construction of canals used for irrigation or water supply with provision
39

of navigable conditions on them; the creation of reservoirs to regulate

the runoff and at the same time maintained for navigation purposes, and

so on. In addition, in our country there are no canals or supporting

structures and none planned which do not also serve for the purposes of

land improvement, water supply and the generation of electric power.

[sic]

In this paper, there are many descriptions of the waterways of the Soviet

Union and much discussion of the specific commodities which will have to be

shipped along the country’s waterways. However, no mention is made of simulations, or, models of any kind, being used to obtain indications of optimal

use of the waterways. Very little concern is given to problems of delay time

or of different lock disciplines and operating procedures. All of the references in this paper are to Soviet publications and appear to be of the same

type as the paper itself.

In addition to those research projects which have been described above, there

were undoubtedly many other papers written both in the United States and

abroad during this period (until about 1973) describing analyses of navigation

on inland waterways. The typical model appearing in the literature consisted

of a bare-bones flow diagram of a computer program, coupled with observed data

used to provide numerical values for the model’s inputs. The majority of

these models were aimed at one particular lock or channel which the authors

were interested in. One striking feature of this early literature is the

infrequency with which one author (or team of authors) cites the work of other

authors. It seems plausible to assume that very few authors were familiar

with previously-made models. It then follows that very similar models are

repeatedly independently discovered. Sometimes very broad, general remarks

appear by an author claiming that his model is better than (unnamed) others.

This claim is seldom if ever substantiated. For example, in 1972, Deazeley and

Wilson [28] stated that
40

so far as the writers are aware this is the first time a simulation model

of such versatility has been devised for the shipping traffic.

The only reference in the paper was to an earlier paper by Deazeley. Each

author lacked the sense that his work is a single step along a long path.

This can be seen by how few people described the workings of their model in

terms subsequent authors could build on.

This section would not be complete without mentioning a simulation of a very

different kind which was conducted by Armstrong [4] in 1970 as part of the

project considering replacement of Lock and Dam No. 26 on the upper Mississippi River. In order to test the desirability of each of several possible

major construction projects, a physical model was constructed on a scale of 1

to 120 to reproduce a 6.7 mile portion of the upper Mississippi River. Several alternate lock and dam configurations were constructed which could be

fitted into the scale model in various combinations. The effects of each of

these on the flows and currents within the miniature waterway were measured.

Two model towboats with remote control were added to the system to determine

the ease of maneuvering for each configuration. The actual maneuvering of the

model towboats was done by a group of 17 towboat pilots who had spent many

years on that portion of the Mississippi River and were familiar with the

river and with the operation of tows and who were invited to participate in

the project. After maneuvering the two towboats through the various configurations, the opinions of the pilots were solicited and then tabulated.
41

8. Simulations (Part 2)

Up to 1973 a large number of simulation models of varying sizes, and scopes

were developed in the United States and abroad. After 1973, very few new

models were constructed. Instead, navigation research turned toward the unification, refinement and expansion of existing models. Also, the emphasis

changes from models geared to specific waterways (or portions of specific

waterways) to models which could be used to simulate widely differing navigational systems. After the PSU-COE package was accepted by the Waterways

Experiment Station (WES) of the Corps of Engineers, Daggett, Ankeny and Eiland

[22] in 1973 expanded WATSIM into a more sophisticated program (called WATSIM

IV) by adding a number of features which reflected traffic conditions on the

Gulf Intracoastal Waterway. This constituted a new area of application. The

PSU-COE package had been aimed primarily at the Western Rivers System and the

Great Lakes System. The major innovations of WATSIM IV were:

• the provision for multiple lockages,

• the provision for open lockages (that is, lockages in which both

of the locks are open at the same time).

The long period of development through which the PSU-COE model passed culminated in 1976 in the Inland Navigation Systems Analysis (INSA) [79]. The

INSA program is an integrated system of models, data and planning procedures.

It was put together by the Corps of Engineers in order to provide a comprehensive model with which to examine the effects on waterway capacity and on tow

delays of changes in construction, regulation, operations, etc. Although the

INSA program was a direct descendent of Carroll's and Howe's trailblazing

models of 1965, INSA also contained a systhesis of a great volume of research
42

on waterway navigation from all parts of the world. Its aim was to provide a

general-purpose, state-of-the-art model, together with directions for obtaining the data necessary to drive the model and a study of the sensitivity of

the model to the data. Among the major improvements incorporated into the

INSA program from the earlier PSU-COE model and the WATSIM IV revision are:

• the separate program (TOWGEN) converting commodity flows into vessel

generation was replaced by commodity flows being fed directly into the

simulator; then a stored representation of port operations (barge

loading and unloading, flotilla assembly, dynamic tow dispatching,

etc.) was used to generate vessel traffic,

• the ability of the model to simulate navigational systems of arbitrarily large size,

• greater flexibility in barge mix, type of lockage, flotilla size, etc.

• the availability of various lock queueing disciplines,

• the assignment of traffic among parallel routes when they are available ,

• the Inclusion of pleasure craft into the model,

• the monitoring of a greater number of output variables.

In addition, computer languages especially designed for use in traffic simulation were employed. The computer language used by INSA was SIMSCRIPT II. 5.

Since they became available, the WES TOWGEN-WATSIM IV model and the INSA program have been used for numerous studies on various waterway systems. One

important application of the \-/ES model was made in 1975 to the long-standing

problem of the proposed replacement of Lock and Dam No. 26 on the upper
43

Mississippi River near St. Louis, Missouri [79]. Other studies were made on

the Illinois River, the proposed Red River waterway and the Black Warrior

River System in Alabama. By 1976 over 500 computer runs had been made using

the WES simulation model.

In 1976, Daggett and McCarley [23] begin a comprehensive study of traffic

through the Winfield Locks on the Kanawha River in West Virginia. This project included a simulation based on the TOS TOWGEN-WATSIM IV package. The

main purpose of this study was to determine the capacity of the existing system (two 360 ft. by 56 ft. chambers with a normal lift of 28 feet) using current policies and to estimate the capacity using several alternative operating

policies. In addition, it was planned to determine the effect that several

proposed structural changes in the existing waterway (Including the replacement of the existing locks) would have on traffic capacity and delay times.

Finally, a system-wide analysis was planned for the entire waterway. This

study had to be suspended several months later with only a portion of the

anticipated work completed in order to reanalyze the proposed replacement of

the Gallipolis Lock. Nevertheless, a substantial amount of work was done.

The capacity relative to the existing facilities and to several alternative

operating policies was determined. The abreviated project provided an indepth test of the TOWGEN-WATSIM IV package at a single lock. Detailed comparisons were made between observed delay times and those predicted by the simulation. Four alternative operating policies were simulated:

• First In-First Out (FIFO) Unrestricted,

• First In-First Out with the assumption of a 10% reduction in lockage

component times

• FIFO Ready-to-Serve

• 3 Up-3 Down Unrestricted.
44

The ready-to-serve restriction prohibits breaking or remaking of tows sufficiently close to the lock to cause traffic delay. In unrestricted traffic,

tows need not adhere to this regulation. The ready-to-serve operating policy

would require several switchboats at the lock to facilitate the passage of

vessels and and so may not be practical. However, this policy resulted in the

most efficient use of the lock chambers. The 3-up, 3-down policy proved to be

worse than the FIFO policy because the short entry was not sufficiently

shorter than the long entry and so did not compensate for the empty turnaround

of the chamber.

The report put out by Daggett and McCarley [23] provides an unusually detailed

picture of the way in which a simulation package is used in a particular

study. First, data collected by the Performance Monitoring System (PMS) program for July 1975 was obtained to be the primary inputs for the simulation. These consisted of over 300 tow lockages. Also, lockage data for May,

September and December were used in formulating average statistics for comparison with the July data and for determining season effects on lockage and

traffic characteristics. Before the model could be used, it had to be calibrated. That is, the parameters inherent in the model must be set at levels

reflecting the operation of the Winfield Lock. Using the December 1975 data,

the following adjustments were made:

• the horsepower vs number of barges per tow frequency distribution was

revised to permit the model to more closely reproduce the actual mix

of the several types of lockages.

/The Performance Monitoring System will be discussed in greater detail later

in this report.
45

,1 • dedicated equipment percentages for several of the barge types were

adjusted in order to reproduce the correct ratio of empty to loaded

I

' barges

I • chamber penalty times for selected lockage types were varied in

several runs so that the use of the two chambers by various size tows

f. could be more closely represented.

Explanations are given for the deviations in the case of the Winfield Lock as

compared to the general situation incorporated into the WES model. The output

I of the final calibration run matched the prototype data exceptionally well.

i

\ For example, the total number of tows passing through the lock during the sam-

*

j pling period was reproduced within 4% by the model. Using the calibration

I input data (lockage component times, flotilla makeup, tonnages, etc.) and the

July 1975 0“D matrix, the tonnage passing through the lock as obtained by the

model was within 1% of the actual tonnage as obtained from the available data.

Other outputs reproduced in this way were unusually accurate. With one exception, all output figures and the July 1975 data were within 10% of each other.

The one exception was the number of barges using the second chamber. The devI iation in this case was 12% and was caused in part by the heavy use of the

I second chamber by pleasure craft.

I After the completion of the calibration, the effect of seasonal variations was

investigated using data available from the six year period 1970 -1975. As a

result of this investigation it was decided that variations due to the season

were negligible and could be ignored.

The input to the TOWGEN program is a listing of commodity flows. These data

were available for 1975 and were projected for eight different years in the

future out to 2017. Two primary approaches were used to analyze the capacity
46

of the existing lock facilities: increases in delay times and increases in

lock utilization, both as functions of increased commodity tonnages. Delay

12

times reflect the economic / costs to shippers as well as the physical capacity of the lock. For each of the four queueing disciplines described above,

the analysis involved plotting the values obtained from the simulation, determining the range of experimental sample values, fitting functions, and plotting curves to these data points. These curves (monthly delay vs monthly tonnage, annual tonnage vs utilization, and annual delay vs utilization) are presented in the report. It was found that, especially during periods of high

utilization of the lock, delay time was quite sensitive to changes in the

make-up of the traffic assumed for the waterway. Even when assuming constant

traffic levels and examining several time periods, it was found that delay

times varied greatly. It was believed that a major part of this variation was

due to the wide range in the number of cuts required by some tows in order to

pass the lock (tows requiring as many as 8 lockages occur at the Winfield

Lock) and to the wide range of lockage times required to service various

tows

The Winfield Lock study, though incomplete, has been discussed here in more

detail than other studies for two reasons;

• it is quite recent (appearing in February 1977) and so reflects the

state-of-the-art in waterway simulation,

• such step-by-step details as are provided by the authors are found in

very few other papers.

/Since delay time reflects economic costs to shippers, it is a more relevant

factor than lock utilization in a multi-modal study where excessive delay

using one mode of transportation can bring about a change by the shipper to

another mode of transportation.
47

In fairness, it should be pointed out that the detail presented in a printed

research paper is often not determined by the author but, rather, by the editor of the journal in which the paper appears. Since editors of technical

journals are universally anxious to save space, few authors are permitted such

full exposition as is found in the paper by Daggett and McCarley. For example, neither of the two printed papers [13, 16] describing the PSU-COE model

contains more than a very small fraction of the detail contained in the six

volume PSU report.
48

9. Queueing Theory

It is generally agreed that queueing theory was founded by A.K, Erlang in a

series of papers which appeared during the period 1909 -1920 [35]. The context in which Erlang worked (at the Copenhagen Telephone company) was the problem of determining what number of automatic telephone exchanges was sufficient in order to provide a satisfactory level of service to telephone customers. After 1920 interest in queueing theory grew rapidly and in 1957 Doig

[32] published a bibliography devoted solely to queueing theory that contained

over 600 items. Since 1957 research in queueing theory continued to expand

and, in addition, has diversified steadily so that today it is a widely pursued discipline within applied mathematics. A major impetus for this expansion has been the successful application of queueing theory to widely divergent fields. For example, Pritsker and Young [68] have pointed out that

queueing theory has been applied to production lines, job-shop scheduling, all

sorts of traffic flows, fire department operations, courts of justice, etc.

In addition to the queueing theory studies devoted to solving applied problems, a large body of general abstract queueing theory was developed at about

the same time. Abstract queueing theory refers to that portion of research

which is not application specific. The first text books devoted to queueing

theory appeared in the early 1960’s. They made accessible to the mathematical

community a powerful tool which could be applied to appropriate problem

areas

Considering the wide variety of application of queueing theory mentioned

above, it is not surprising that the same basic queueing problems have been

repeatedly attacked with different applications in mind. Usually such

research is found in the literature devoted to the area of application and it
49

is unlikely to be seen by researchers in a completely different field. Such

duplication of effort will undoubtedly continue until a convenient and efficient means can be found to search the entire queueing theory literature for a

particular queueing scenario. However, the problem is compounded by the fact

that each area of interest maintains its own vocabulary and it becomes difficult to "translate" a paper from the one application area to another.

In general, a queueing problem is specified by describing three basic components :

• the arrival of customers to be served,

• the servicing of these customers by the server(s),

• the servers.

Thus, for example, it is necessary to know not only how many customers are

expected in a given time period, but also whether they come singly or in

batches, whether they come in a random order or in a preassigned sequence,

etc. There are similar questions dealing with the service provided by the

servers. Although it is necessary to specify the rule describing the order in

which customers are to be served, in the vast majority of queueing theory

research it is assumed that service is provided on a first-come, first-served

(FIFO) basis. When there is more than one server, it is necessary to know how

customers are assigned to each server. Finally, any special features of the

particular queueing problem must be described, e.g., are customers likely to

1 3 give up and leave the queue after a prolonged wait, / or, when there is more

/Clearly this situation is not applicable to a queue of tows waiting at a lock. However, a shipper faced with long delays in the past may choose

another mode of transportation and so "leave" the queue.
50

than one server does service of one customer interfere with the service of

14 , another customer? /

Although it might be reasonable to consider queues in which the arrival time

of each customer and the length of time required to serve each customer are

known precisely, such situations are seldom studied since they rarely occur in

practice and they provide little fodder for mathematical attack. Typically,

both the arrival of the customers and the service time for each customer are

assumed to be governed by a probability distribution function. The standard

notation in the mathematical literature for a queue in which

• the arrival times of customers are governed by probability

distribution X,

• service times are governed by probability distribution Y and

• there are s servers

is X/Y/s. The average number of customers that arrive in a fixed time period

is called the arrival rate and is usually denoted by X . The average number of

customers that can be served in the same time period (assuming that customers

are always available to service) is called the service rate and is usually

denoted by \i. The ratio of X to p , that is, p = X/y is called the traffic

intensity of the queue and is a critical factor in determining the length of

time that a customer will have to wait for service. There is one queueing

model which is used as a starting point in every treatment of the subject.

Fortunately, this model fits the lock and dam scenario fairly well. In this

model it is assumed that customers arrive at random, that is, if time is

thought of as being broken up into a large number of "small” non-over-lapping

intervals of the same size, then

14

/This question is certainly applicable to a queue at a multiple chamber

lock.
51

• it is not possible for two or more arrivals to take place in a single

interval

• the probability of one arrival is the same in all of the small

intervals (and is proportional to the length of those Intervals),

• the probability of an arrival in any one interval does not depend on

whether or not arrivals have occured in any previous Interval

This is known as Poisson input and its level is given in terms of the mean

arrival rate X. The probability distribution for the service time is assumed

to be of the same type as for arrivals, that is,

• there is no possibility that two or more customers will finish service

in the same small interval of time (even if there is more than one

server)

• the probability that a customer will finish service in a small

interval is the same for all intervals and for all servers (and, for

each server is proportional to the length of the interval)

• the probability that a customer finishes service in any one interval

does not depend on whether or not customers have finished service in

any previous interval.

This is known as exponential service time. The mean service time is given in

terms of y. Finally, the number of servers is denoted by s. The simplest

case is when there is just one server. This queueing model is denoted by

M/M/1. Fuller mathematical discussions of M/M/ 1 can be found in any standard

text book on queueing theory, e.g., [71] or [78]. It is not possible to

discuss here the many generalizations and variations of this model which are

found in the mathematical literature. Instead we will restrict out attention

to those queueing systems that arise in the modeling of locks and dams.
52 \

1 Thus far, all of the information about a queueing system which has been discussed above is concerned with how the system behaves once it is started. In

addition to this ongoing information, in order to completely specify a queueing situation, it is necessary to provide start-up information. That is, a

beginning time (t = 0) is selected and the description of the system at this

time is given. On the basis of the ongoing queueing assumptions and the

start-up information, it is possible (at least in theory) to find the probability that the system will be in any given state at any later time, (Of

course, the state of the system can only be specified in terms of probabilities since the arrival and service times are provided in probabilistic

terms ,

It can be shown mathematically that all (well-behaved) queueing systems settle

down to a steady-state condition. That is, although the probabilities for the

various states of the system at a time t depend on the value of t, unless t is

quite small these respective probabilities are nearly the same for all values

of t. In fact, the larger t is, the less is the variation of the system in

terms of t. In the long range, the system settles down to a (probabilistically specified) condition from which it does not change. This condition is

called the steady-state of the queue, and it is usually much easier to find

than the description of the system in terms of t. Moreover, as t increases,

the effect on the system of the start-up arrangements dies out. Fortunately,

for most practical purposes, the steady-state condition of a queueing system

provides most of the useful information about its long-term behavior.
I

I 53

^ 10. Queueing Theory Applied to a Single Chamber Lock

I The use of queueing theory to analyze delay and waiting times at a lock

appears to have been introduced by Lave and De Salvo in two related papers

! both published in 1968 [31,56]. The impetus for these studies was an investi15

i gation of the percentage of the active time of a towboat accounted for by

each of 10 categories of delay. Data for these 10 categories of delay were

i obtained for a sample of 19 towboats during the period 1960-1964 on the Lower

1 Mississippi, Upper Mississippi, Ohio and Illinois Rivers. Total delay

I

i amounted to about 25% of active time. Waiting for and being serviced by a

j lock was found to be the greatest single cause of delay, accounting for 30% of

!

j total delay time and 7.29% of active time. In fact, since the Lower

j Mississippi River contains no locks and, during the high water season on the

Ohio River most locks are "collapsed" to permit open river operations, delay

caused by waiting for and being served by a lock represented a much higher

percentage of total delay time and of active time on a locking waterway.

; ; In order to estimate the expected waiting time in a queue and the expected

’ service time for a lock. Lave and De Salvo assumed that the arrival rate X was

Poisson and the service rate p was exponentially distributed, that is, they

I

assumed that the operations of a lock could be modeled by an M/M/ 1 queueing

system. In order to test their model, they tabulated the delays experienced

by one towboat operating on the Illinois River during the first six months of

1967. This represented about 35 transits of each of the locks being studied.

To predict waiting time, it was necessary to obtain the number of tows (represented by K) for each lock for a period of one year. The arrival rate X (in

/The active time of a towboat exclt ies only the periods when it undergoes

major repairs or is otherwise unready for service.
54

tows per hour) for each lock is equ_l to K divided by 8760. The service rates

y (in tows per hour) were obtained from a study of these locks for 1949 and

1950. Table 5 presents a comparison of the predicted and observed total

waiting time in queue T^ (in hours)

d

for each lock.

Traffic

Predicted Observed Relative Intensity

Lock K X y Tl Error P q q

Starved Rock 6243 .71 1.13 1.50 1.38 .09 .63

Marseilles 5953 .68 1.01 2.04 2.02 .01 .67

Dresden Island 6245 .71 1.39 .75 1.09 .31 .51

Brandon Road 6491 .74 1.19 1.38 1.39 .01 .62

Lockport 6424 .73 1.62 .50 .55 .09 .45

Table 5. Predicted and Observed Total Waiting Time in Queue

The predicted T^ is given by the formula

q

(1) Tl = ^

. q u (m-x)

The relative error is the difference between the predicted and the observed

waiting times divided by the observed waiting time. On the basis of these

relative errors, Lave and De Salvo concluded that the M/M/ 1 model seemed to

fit the situation at the locks quite well. No attempt was made to explain the

significantly larger relative error at Dresden Island.

Of the 10 categories of towboat delay listed by Lave and De Salvo, 6 are not

related to the waterway itself (e.g., waiting for supplies, cargo or crew,

making minor repairs, etc.). Of the 4 categories related to the waterway, 2

account for significant percentages of active time ^^/. The first of these,

locking, has already been discussed. The second is delay caused by the making

/The remaining two categories, delay at bridges spanning the waterway, and

running aground account for .12% and .24% of active time respectively.
) 55

!

: and breaking of tows, that is, by the assembly of the tow at the beginning of

a trip and the disassembly at the end. Also included are the times needed to

pick up and drop off barges during an intermediate stop. Making and breaking

tows consumes approximately 3.16% of towboat delay. Although the time

required for either making or breaking a tow reflects the availability of auxiliary equipment (e.g., a switchboat waiting at docks to handle dropped

barges), it has been observed that a major Influence on the delay was the

number of barges involved. However, the specific relation between the delay

time and the number of barges was not known. Lave and De Salvo set up and

tested linear regression models for the time spent in making a tow, T^, and

for the time spent in breaking a tow, T^, in terms of the number N of barges

added or dropped

(2) = a + b N,

(3) Tb = c + d N.

They also set up and tested log-linear models

(4) log = a + b log N,

(5) log Tb = c + d log N.

Data were collected from the log books of two towboats. There were 242

observations on the time required to make a tow and 247 observations on the

time required to break a tow. On average, 1.7 hours were needed to make a tow

and 1 hour was needed to break a tow. The number of barges added ranged from

1 to 24 with an average of 3.4. The number of barges dropped ranged from 1 to

16 and averaged 3.1. It was found that the linear regression model provided a

better fit to the data than did the log-linear model both for making and

breaking a tow. For the linear regression model, 78% of the variation in time

taken to make a tow and 63% of the variation in time taken to break a tow is

accounted for. The corresponding figures for the log-linear regression model
56

are 66% and 49% respectively. Moreover, the significance levels of the coefficients in the linear regression model are all below .06. Lave and De Salvo

conclude that the statistical results for the linear regression model are

quite good.

In 1972, Hayward [44] Investigated the assumptions which underlay Lave and De

Salvo's decision to use the classical queueing model M/M/1 to study the operation of a single chamber lock. For his study, Hayward selected the Pickwick

Lock on the Tennessee River. The Pickwick Lock is a single chamber lock that

is in many ways typical of locks on the inland waterways system. The chamber

is 600 feet long by 110 feet wide - a quite common size. Moreover, Carroll

[10] had previously performed an analysis of the operations of the Pickwick

Lock while developing the original TOWGEN-LOKSIM simulation package and so

considerable simulation data already existed for comparison with data obtained

from various queueing models. Nowhere does Hayward use real-world data .

In 1968, Carroll [10] determined that arrivals at the Pickwick Lock were

random in nature and so fitted the assumption of Poisson arrivals. In 1972,

Desai [30] confirmed Carroll's finding and showed that arrivals at most of the

locks on the Illinois and Ohio Rivers systems could be considered random.

Therefore, in all the models considered by Ha3^ward , he assumed that arrivals

were random. He compared the outputs of 5 models (Lave and De Salvo's queueing theory model and 4 computer simulations) for traffic passing through the

Pickwick Lock. The 5 models Hayward considered were

• the classical M/M/1 queueing theory model of Lave and De Salvo,

• Carroll's computer simulation (the ancestor of LOKSIM) which used SOQA

queup discipline and service time expressed as the sum of entry time,

lock ng time, exit time and recycling time.
57

[

I

(

(

; • LOKSIM simulation modeling the M/M/ 1 queue as closely as possible

using a FIFO queue discipline,

• LOKSIM simulation modeling the M/M/ 1 queue but using SOQA queue

discipline

• LOKSIM simulation with SOQA queue discipline and the same type of service time as in Carroll's 1968 model.

In each case, it was assumed that traffic consisted of 70% single lockages and

30% double lockages with an average service time of 82.7 minutes per tow.

These figures were considered typical by Carroll and were used in his 1968

study. For each of the 5 models described above, a graph was constructed to

show how average delay time depends on the arrival rate. For the queueing

theory model, the formula for delay time obtained by mathematical analysis was

- graphed. For each of the 4 computer simulation models, 4 to 6 different

arrival rates were selected as input. For each arrival rate and for each

ij model, a simulation was run until 1000 tows had been passed through the lock.

j|

The average delay time per tow was obtained as the output of these simulations

yielding a single data point per tow (arrival rate, average delay time). The

i! data points were connected to form a graph.

The average service time of 82.7 minutes per tow assumed by Ha}n^ard is equivalent to a service rate of y = .73 tows per hour. The arrival rate, X, was

varied from .03 to .58 tows per hour yielding corresponding traffic intensities, p, varying from .11 to .79. Although all five curves drawn by Hayward

have the same general shape, they fell into two distinct groupings. The

queueing theory and simulated queueing theory curves consistently predicted

longer average delay times than did the other 3 models. The models based on

! 1 queueing theory and simulated queueing theory were the only ones that used the
58

FIFO queue discipline. It is difficult to determine from Hayward's work

whether the longer average delay times predicted by these models were due to

• queueing theory vs lockage simulation

• FIFO vs SOQA

• chance

• none of the above.

However, Hayward ascribed the longer predicted delay times to the queue disciplines .

Predicted average delay times did not appear to be influenced by the choice

between Poisson service times and service times based on the various

components of a lockage. When p reached .75, Lave and De Salvo’s model, as

applied to the Pickwick Lock, predicted an average delay per tow more than 100

minutes longer than the average tow delay predicted by the LOKSIM simulations

of traffic through a lock. Since information about delay time is needed

mostly for larger values of the traffic intensity, Ha3rward claimed that Lave

and De Salvo’s model performed inadequately. It should be noted that in Lave

and De Salvo’s study, traffic intensity, p, ranged only from .45 to .67.

In 1977 Wilson [82] continued Hayward’s analysis of the adequacy of Lave and

De Salvo’s M/M/ 1 queueing model for a single chamber lock. Wilson noted that,

according to Carroll’s observations at the Pickwick Lock, the empty chamber

cycle time was approximately equal to the difference between the times for a

long and a short entry. Thus, in the case of the Pickwick Lock, the choice of

queueing disciplines, (FIFO vs SOQA) should not affect the average delay time

per tow although it might have a considerable effect on the delay time of a
59

17 , specific vessel /, Wilson pointed out that exponential service times are

unrealistic since the cycle time was approximately 15 minutes and so complete

service times of shorter duration are not possible. For this reason, he

reverted to Carroll's service time which consisted of the sum of entry time,

1

lockage time, exit time and recycling time / . The model with Poisson

arrivals Carroll's service times, one server and unspecified queue discipline

was referred to by Wilson as M/G/1. The assumption of a 70% single lockage

and a 30% double lockage mix used by Carroll and Hayward was retained by

Wilson, The average lockage times for single and double lockages were taken

as 59,1 and 133,0 minutes respectively. These result in an average lockage

time of 81,27 minutes (Hayward had used 82,7 minutes). In addition to the

average lockage times , Wilson also made use of the variances of the lockage

times. His formula for the average delay time (given without derivation or

19

reference) was /

( 6 )

X W

M

^ a ^ + 1

2(1 - P)

where is the variance of the lockage time. Using Carroll's version of

service time, the variance of the lockage time is 1347,7, Using exponential

service times, the variance of the lockage time is 6604,8, Both of these

figures are based on an average locking time of 81,27 minutes. According to

Wilson, it is the large difference in the variances which was responsible for

the consistently higher expected average delay times obtained using Lave and

/ Note that Hayward came to the opposite conclusion, claiming that the difference between the two queueing disciplines, FIFO and SOQA, was the major

cause of the failure of Lave and De Salvo's model,

18 ,/ According to Hayward, changing from exponential service times to Carroll's

service times did not affect the average delay time per tow.

19

/ This formula differs from the one given by Hilller and Lieberman [46] and

« used by Rao [69],
60

De Salvo’s M/M/1 model than were obtained from either the M/G/1 model or from

computer simulations of lockages. (See Table 6.) Wilson expressed surprise

that Lave and De Salvo's relative errors were as small as they were considering both the much greater variance of lockage time In their model and the

relatively poor results obtained by him on the Pickwick Lock using their M/M/

model

Arrival Rate

(tows/day) p

Mean Delay

(minutes)

simulated

Mean Delay

(minutes)

M/G/1

relative

error

Mean Delay

(minutes

M/M/1

relative

error

2 .11 6.4 6.2 .03 10.3 . 66

4 .23 13.7 14.3 .04 23.7 . 66

6 .34 24.8 25.1 .01 41.6 .66

8 .45 40.0 40.3 .01 66.9 .66

10 .56 66.0 63.4 .04 105.3 .66

12 .68 111.5 102.7 .09 170.5 .66

14 .79 209.7 184.2 .14 306.0 . 66

16 .90 477.4 455.5 .05 756.6 . 66

Table 6. Comparison of Carroll' s Simulation results with Wilson's M/G/1

results and results using Lave and De Salvo's M/M/ 1 model.

In 1976, CACI, Inc. [20] prepared a study on waterway and rail capacity for

the Department of Transportation. An Integral part of this study was the Lock

Capacity Function Generator (LCFG) , a model of traffic on a waterway using

mathematical queueing theory. The LCFG assumed Poisson arrivals, a general

probability distribution with known mean and variance for service times and

one server. Thus the CACI model, like Wilson's, Is In the M/G/1 category.

Service time was expressed as the sum of approach time, entry time, chamber

time and exit time, together with a provision for recycling time when needed.
61

I Unliked the other queueing models, up-stream and down-stream traffic were

j treated separately. Input data on lockage times, barge loadings and tow size

distributions were obtained from the Performance Monitoring System (PMS) of

I the COE. The output of LCFG was compared with the output of the Inland Navigational System Model which is a part of the INSA program. The LCFG output

jn was also compared with real-world data supplied by PMS. When average locking

I times predicted by the LCFG model were compared with data obtained from PMS,

[iti l;i

[i relative errors ranged from .01 to .54.

!

1! The LCFG model was used to study 9 portions of the Mississippi River/Gulf

'i

S' Intracoastal Waterway System. The CACI study investigated locks of various

II sizes (from 360x56 to 1200x110) on several rivers of the Mississippi River

j system as well as two locks on the Gulf Intracoastal Waterway system. Among

j the outputs of the model were estimates of the delay time of a tow and the

J expected average queue length. The formula for the average delay time was the

S'

1 same as the one used by Wilson (equation 6).

'!3

II

I

The studies by Lave and De Salvo, Hayward, Wilson and CACI, Inc. described

above appear to represent the totality of research on the application of

queueing theory to a single chamber lock performed in the United States.

Furthermore it appears that the application of queueing theory to traffic

through a lock has not been treated in the foreign literature. There are several other studies performed in the United States devoted to multiple chamber

i': locks and these will be discussed in the next section. However, aspects of

the problems associated with multiple chamber locks often raise questions

f about the analysis of a single chamber lock. For example, although Glassey

:|t and Ross [40] were concerned exclusively with parallel chamber locks, their

ijS questioning of the prevailing assumption of Poisson arrivals at a lock
62

opens the door to reinvestigating this point for single chamber locks. The

question of Poisson arrivals was also studied by Desai [30] who concluded

that arrivals followed a Poisson pattern in 67 of 80 queues at 40 locks.

Glassey and Ross were concerned primarily with Lock and Dam No. 26 on the

Upper Mississippi River and their data are drawn from the traffic there.

According to their observations, waterway traffic is not homogeneous but is

• highly seasonal,

• heavier during daylight hous than at night,

• lighter on weekends than during the week.

The seasonal variation in traffic intensity is considerably greater than the

day/night variation. How significant these factors are is not studied in

detail, but nevertheless bring into question the assumption of Poisson

arrivals. In fact, the average waiting time per tow as calculated with the

assumption of Poisson arrivals is shown to be a lower bound for the true aver

age waiting time. Whether this lower bound is significantly lower than the

true average waiting time has not yet been determined.

In 1979, Rochberg [72] looked at the various methods that had been used to

analyze the capacity of a lock, the average waiting time of a tow, etc. and

presented a critique of each. In only a limited way did he present any alter

native suggestions for modeling which he felt would be an improvement over

existing methods. Rather he was concerned with reevaluating the models put

forward by other authors. For examples Rochberg pointed out that while average delay time per tow remains the standard variable to be solved for in both

queueing theory models and by computer simulations, other related variables

(e.g., average delay time per barge, per ton of cargo, or even, per ton of a

specific cargo) might be more relevant. In addition, a situation which

achieves a low average delay time per tow by having a few tows wait an
63

. inordinately long time while the majority of tows have virtually no waiting

time (say a 12-up, 12-down queue discipline) would not be acceptable. Thus,

minimizing the maximum expected waiting time that a tow might be faced with

could be a better criterion for satisfactory locking facilities than minimizing the average waiting time per tow.

It was pointed out by Rochberg that many real-world variables can not be satisfactorily introduced into a theoretical treatment (e.g., horsepower of towboats, weather conditions, etc.) and that no attempt had thus far been made to

estimate the Importance of such variables in obtaining meaningful results. In

order to use queueing theory properly, the validity of the assumptions and

inputs must be carefully evaluated. In general these are usually roughly correct but seldom completely accurate. Rochberg demonstrated just how sensitive

‘ the results obtained in queueing theory can be to parameters which either do

I not remain constant or are difficult to measure with precision (e.g., traffic

intensity, locking time, etc.). For example, using an M/M/ 1 queueing model

with an average service time of one hour and tows arriving at an average of

every 61 minutes, there would be, in the long run, a sixty hour wait. However, if tows arrive at an average of one very 70 minutes then the average

wait in the long run would be only 6 hours. If we change the assumptions and

20

assume that the service time is exactly one hour / (rather than an average

of one hour) then the average waiting time is cut to 3 hours.

According to Rochberg, the average waiting time per tow is given by

c Xy

(7) w =

y - X

20 ,/For example, this phenomenon occurs when traffic passes through several

locks in a short span of waterway.
64

where c is a quantity to be specified and depends on whether arrival and

service times are random or not. When the quantity y-X is small, w is

extremely sensitive to even small changes in X and y.

Although the queueing models M/M/ 1 and M/G/1 are capable of being solved

mathematically, changing the queueing model to reflect more accurately the

operations of a lock usually results in a mathematical system which is

untractable by currently-available techniques. There are schemes for

approximating the solutions of such a queueing system by numerical methods.

However, these methods themselves introduce an error of their own.

The problems involved in producing information about the operation of a

waterway by means of a computer simulation are quite different from those

which arise when queueing theory is used. Much more detailed data is needed

as input for a computer simulation model. For example, queueing theory

requires the average time for a lockage and, perhaps, the variance of the

lockage time. A computer simulation uses a spectrum of lockage times as well

as the probability distribution for the various lockage times. One of

Rochberg*s main criticisms of the way in which computer simulations are used

is that the lengths of computer runs are usually arbitrarily chosen (except

that they must be "long"). Of course, "long" is a very vague term. Rochberg

pointed out that even a computer run which is generally considered to be

sufficiently long might nevertheless not be long enough to show that a queue

at a lock would grow indefinitely or would reach unacceptable length, not

immediately, but in the foreseeable future.

With the current state of the art, simulations seem to provide more accurate

results than does queueing theory, even though, because of the randomness

introduced into a computer simulation, results cannot be exactly duplicated.
65

However, Rochberg suggested that simulations can be improved by making use of

information that a queueing theory analysis can provide. For example, queueing theory is often able to evaluate the variance of delay time (or of other

variables involved in a simulation). This information can be used to determine how long a computer simulation run should be in order to provide a specified confidence level for the output. For example, when the variance of a

variable is known, it is not difficult to determine how long a computer run is

necessary in order to have, say 95% confidence that the computed value of the

variable differs from the variable with a relative error of at most .01. The

variance of a variable such as delay time can be estimated by using a large

number of computer simulation runs but such a procedure is time consuming,

expensive and probably produces less accurate results than those obtained

using queueing theory methods.
66

11. Queueing Theory Applied to Multiple Chamber Locks

For locks with two or more chambers, there are a number of factors which can

add serious complications to the comparatively simple model of a single chamber lock. For example, when there are two chambers at a lock, separated as

much as possible by having one adjacent to each bank of the river, then the

interference of traffic through one chamber with traffic through the other is

obviously less than when the two chambers are adjacent to each other. If the

assumptions of independence for the arrivals, operations, service times,

exits, etc. at the two chambers can be justified (e.g, by observations at the

lock) then the mathematical analysis of the lock using queueing theory is substantially simplified. Another simplifying assumption (whether or not traffic

at one chamber behaves Independently of traffic at the other chamber) is that

the two chambers are identical - at least from a mathematical point of view.

A major impetus for the study of multiple chamber locks is Lock and Dam No.

26 which is located on the Mississippi River at Alton, Illinois (north of St,

Louis) below the confluence of the Illinois and Upper Mississippi Rivers, The

lock contains two chambers - a main chamber 600 feet in length and an

auxiliary chamber which is 360 feet in length. Both chambers are 110 feet

wide. Barge traffic on the Upper Mississippi and Illinois Rivers increased

rapidly after World War II and Lock 26 became a serious bottleneck. Numerous

studies were commissioned to determine whether Lock 26 should be expanded,

replaced, or left alone. An integral part of several of these studies was an

evaluation of the effect on average delay time of the size of a second chamber

at the lock. Several researchers were concerned with comparing traffic

through one chamber with traffic through two parallel chambers of the same

size, although studies which received impetus from Lock and Dam No. 26

considered two unequal parallel chambers.
67

Lave and De Salvo [56] considered the benefits that would result from adding a

second chamber to a single chamber lock. In the case they studied, the added

chamber was to be twice the size of the existing one and they computed the

average delay time per tow by assuming that the availability of a parallel

chamber would cut the average service time per tow in half. The average delay

time per tow was then calculated by treating the double chamber lock as a

single chamber lock with the reduced average service time per tow, Hayward

[44] disputed Lave and De Salvo’s claim that the average service time per tow

would be cut in half. He presented his own formula for the average service

time but provided neither a reference nor a derivation.

In 1976, Glassey and Ross [40] compared observed average waiting time per tow

for the two chambers of Lock and Dam No, 26 with calculated average waiting

time using the M/G/1 model coupled with the assumptions of independent chambers, Poisson arrivals and independent service times.

X (Tows /hr ,

E[S]

(hours)

E[s2] Wq

(hours)

Wq

(hours)

relative

% error

P

600’ chamber ,646 1,466 2,753 16,79 6,53 157 .947

360’ chamber ,4865 ,8523 1,320 ,549 ,814 32.5 .415

Table 7, Comparison of Observed and Calculated Average Waiting

Times for Each Chamber at Lock and Dam No, 26

E[S] is the average service time, E[S^] is the average of the square of the

service time, Wq is the calculated average waiting time and Wq is the observer

average waiting time. The observed data were taken from the lock operating

log for a 26 week period beginning March 1, 1974 and represent a total of 5171

observations
68

Glassey and Ross suggested that the poor performance of the M/G/1 model in

predicting the average waiting times at both chambers was due to the inaccuracy of the assumption of homogeneous random arrivals. However, they concluded, this cannot be both the cause of the estimate for the average waiting

time at the large chamber to be too large and the cause of the estimate at the

small chamber to be too small. Thus, another explanation was needed to

explain the poor performance of the M/G/1 model. To this end, they argued

that, since average waiting time at the small chamber was considerably less

than at the large chamber, more tow captains chose the small chamber than

would have been expected to under the assumption of independent chambers. In

order to Incorporate this idea into the model, they presented a modification

of the M/G/1 model by assuming that there was a number N such that at any time

when the queue at the larger chamber reached a length of N, then all traffic

2 1 would be diverted to the smaller chamber / . In order to select an appropriate value for N it was assumed that whenever the small chamber was found to

be empty by an approaching tow, then the tow would use the small chamber

instead of the large one. Thus, N was obtained in terms of the proportion of

time that the queue could be expected to be empty using the M/G/1 model. The

average waiting time at each chamber as calculated with the model incorporating this modification was considerably closer to the observed average waiting time than had been obtained from the model prior to the modification.

Glassey and Ross argued that tows containing a sufficiently small number of

barges would have less reason for preferring the large chamber than would tows

/The possibility of queues of length N at both chambers or at the smaller

chamber alone was not discussed.
69

with a large number of barges since some tows which could pass through the

large chamber in a single lockage would require a double lockage when using

I the small chamber. As a means of avoiding the difficulty involved in making a

i| specific assignment of tows to chambers, they developed a model in which barI ges form a single queue which feeds both chambers. The barges were assumed to

arrive in batches of random size. In this model the lock is a two server sysI tern. However, since queueing systems of the form M/G/2 are mathematically

complicated when service rates are unequal, Glassey and Ross developed an

! "equivalent" single server system (M/G/1) having approximately the same service characteristics. Such a model yielded the average waiting time per

barge, rather than the average waiting time per tow. Although detailed numerical calculations were not carried through, Glassey and Ross pointed out that

changing customers from tows to barges did not change the data requirements of

the model. In either case, these were

i • the tow size distribution,

• the service time distribution (depending on tow size),

• the arrival rate distribution for each tow size.

On important point not made by Glassey and Ross is that when p is so large

(p=.947 for the large chamber) the average waiting time is extremely sensitive

to the data. For example, if we change the values of p and E[S] for the large

chamber from .646 and 1.466 to .6 and 1.4 respectively then Wq is changed from

16.79 to 5.16.

»

I In 1977, Rao [69] presented a comprehensive study of the many aspects of evaluating navigation investments with the intention of optimizing the returns

I from expenditures whose aim is to reduce congestion. This study was the

7

I author’s dissertation for the Ph.D. degree in Business Adiministration at

I
70

Pennsylvania State University, and is considerably broader and deeper than any

of the previously-mentioned papers. Rao took the Corps of Engineers’ proposal

for the replacement of Lock and Dam No. 26 as the impetus for his work.

Although the data presented throughout the study refers only to Lock and Dam

No. 26, the discussion is consistently presented in terms of the general

problem of evaluating navigation investments. The study is comprehensive and

includes the political and environmental aspects of waterway navigation and

construction as well as discussion of various cost/benefit analyses, alternate

modes of freight transportation, appropriate times for investment, etc.

The problem of estimating the capacity of a lock was treated by Rao in detail

as an integral part of navigation investment analysis. Tow traffic was

divided into classes according to the type of lockage required. For each

lockage type, service time was expressed as the sum of entry time, chamber

time and exit time. The average service time per tow is the weighted average

of the service times for the various types of lockages required. It then

became a straight forward operation to calculate the maximum number of lockages which can be performed at the lock in one year. Provisions are included

for the average down-time of the lock and for portions of the year when the

lock cannot be used because of weather conditions. The maximum number of

lockages per year is one measure of capacity investigated by Rao. Another

measure of the capacity of a lock is the total tonnage which can pass through

the lock in one year. This is equal to the product of the average tonnage per

lockage and the number of lockages. The average tonnage per lockage was

expressed in terms of the number of barges of each type and the loading of

each barge. Other measures of capacity stulded by Rao were

• average tonnage per tow,

• average number of barges per tow,
71

o average number of loaded barges per tow.

These were examined in order to determine whether the yearly through-put measures of capacity he obtained reflected prevailing traffic characteristics or

represented an idealized situation in which lock capacity can theoretically

attain some maximum level. For each measure of capacity, actual 1972 data for

each chamber of Lock and Dam No, 26 were obtained and compared with the capacity as calculated using the model. The capacity of a lock was not Rao's

major concern. On the contrary, it was studied only because it was needed in

the calculation of cost/benefits, in finding the appropriate time for navigation investments and in determining the modal demand for waterway transportation.

Unlike capacity which was an integral part of Rao’s study, average delay time

was relegated to an appendix. He discussed the classical general X/Y/m queueing model and then applied it separately to each of the I classes of lockage

requirements, calling the resulting model X/Y/m/I. The simplest example of

such a model is M/M/l/I consisting of I classes of lockage requirements, Poisson arrivals for each class with an arrival rate of for the i^^ class,

exponential service times for each class with a service rate of for the i^^

class, and one server, (This model had been treated by Ross but not in the

context of navigation.) Rao then expanded the model to incorporate bidirectional traffic. In both cases, algebraic relations were given from which the

average delay time per tow can be found. Unfortunately, the algebra quickly

became extremely complex and numerical solutions for a given input data

required extensive calculations. For specific numerical values of the

arrival rates, service rates and maximum queue lengths, the average waiting

tine per tow can be calculated but this requires the use of a large scale computer.
72

Rao next presented a more sophisticated queueing model, M/E^/l/I. Again the I

classes of customers (possibly tows or barges) referred to the types of lockages required. The service time distribution for the i^^ tow class is assumed

to be kj^-Erlang. The k-Erlang distribution is the distribution of the sum of

k independent exponential random variables. Thus a tow of the i*^" class is

conceived of as requiring k^ distinct steps of a lockage. Each of the steps

is presumed to have an exponential distribtuion. This model was solved by the

"method of stages" as given by Kleinrock [49] and was illustrated by Rao for a

specific simple case.

The average delay time per tow was computed for traffic at 7 locks on the

Illinois Waterway and at Lock and Dam No. 25 using the four models introduced

by Rao as well as the standard M/G/1 and M/E]^/l models. The output of these

six models were compared with the results of the simulation model developed by

Carroll and Bronzini in 1971.

Lock Name Simulation M/M/1 M/M/ 1/3

Bidirectional

M/M/ 1/3 M/G/1 M/Eg/l M/Ei^ /1/3

i

Lockport 62 74 87 87 40 43 52

Brandon Road 111 111 135 139 83 64 76

Dresden Island 21 45 53 53 26 26 29

Marseilles 29 52 61 62 29 30 33

Starved Rock 37 46 56 55 26 27 31

Peoria 38 58 66 67 40 34 35

LaCrange 38 49 54 55 28 28 29

Lock & Dam No. 25 25 30 33 34 19 17 18

Table 8. Average Delays per Tow with Alternative Queueing Models

(in minutes)

1

'
73

The values for in the last column were 2, 14 and 23 for single, setover and

double lockages respectively. At the Lockport and Brandon Road Locks, traffic

intensity ranged from 50 to 60 percent. At all other locks the traffic intensity ranged from 30 to 40 percent. Moreover, all locks were single chamber

locks.

Several findings can be observed from Table 8. All models which assumed

exponential service distributions performed poorly when compared to the simulation results. This agrees with Wilson's findings several years earlier and

was the reason he switched from the M/M/1 mode to the M/G/1 model. Also, the

assumption of bidirectional traffic (and SOQA queue discipline) has little

effect on the output of the model. Further, dividing traffic into three classes according to the type of lockage required resulted in poorer output when

the M/M/ 1 model was used and gave mixed results when the Erlang distribution

was used. Unlike several other authors, Rao is satisfied with the assumption

of Poisson arrivals.

There were 3 locks (with a total of 4 chambers) for which actual data were

available, providing the possibility of making performance comparisons of the

various queueing models. The lock with two chambers was Lock and Dam No. 26.

Rao chose to compare the available data only with the M/M/1 and M/G/1 models.
74

Traffic Average Delay per Tow (In minutes)

Lock Name Intensity Actual M/M/1 M/G/1

Brandon Road .72 117 194 126

Meldahl .17 13 9 6

Lock and Dam No. 26 .91 319 620 363

Main Chamber

Lock and Dam No. 26 .49 53 54 49

Small Chamber

Table 9. Actual data compared with M/M/1 and M/G/1

The actual data for the two chambers of Lock and Dam No. 26 were collected by

the Corps of Engineers in 1975 and Showed substantial agreement with the data

used by Wilson. (See Table 7.)

Rao was well aware that his treatment of the traffic at a two chamber lock was

in terms of single server models. He stated

The reader may wonder why multiple server models have not been mentioned

for use in modeling congestion at multiple chamber locks. There are three main difficulties with the use of multiple server queueing theory.

First, where multiple chambers now exist at lock sites on the Inland

waterways, they are generally of unequal size and have dissimilar characteristics.... Treating multiple servers of unequal size and characteristics via queueing theory becomes rapidly intractable. [Also] at

many multiple chamber locks the smaller or auxiliary chamber is not used

very frequently, its main purpose being to stand by as a chamber to be

used when the main chamber is being repaired or when traffic is extremely

heavy. Even in heavy traffic, tow operators generally prefer to use the

larger chamber for reasons of safety and ease of maneuverability. Where

additional locks have been proposed at existing sites, the design sometimes calls for the new chamber to be separated from the existing chamber

rather than adjacently located.... This type of situation can also be

appropriately handled by treating the site as consisting of two separate

facilities rather than a single multiple server facility.
75

12. Queueing Theory in the Foreign Literature

The first phase of this study consisted of locating research dealing with the

problem of analyzing traffic on a waterway which had been performed anywhere

in the world. To this end, the On-Line Bibliographic Retrieval System, available through the library at the National Bureau of Standards, was used. Three

data bases accessible through this retrieval system were selected for search:

• the Engineering Index,

• the Science Citation Index,

• the National Technological Information Service Index (NTIS).

The NTIS Index is concerned primarily with domestic research but also contains

a number of translations of foreign language papers and reports on research

which was performed abroad but was supported by the United States Government.

The Engineering Index and the Science Citation Index are very broadly based

and contain references to virtually all of the world's technical journals.

In order to use the retrieval system, a list of pertinent keywords was drawn

up to be used as input. These were: lock, queue, channel, traffic, delay,

waiting, tow, barge and simulation. When any pair of these words were fed

into the retrieval system a listing of all papers contained in the Engineering

Index whose titles or keywords contained that pair was produced. This listing

was supplemented by a number of central papers on lock traffic which had been

supplied by the Army Corps of Engineers. Using the Science Citation Index,

the retrieval system was then asked to find

• all other papers by the authors of the papers in the listing,

• all papers by any author which used any of the papers in the listing

as a reference
76

Finally, as every new paper was found, its reference list was examined in the

hope of finding additional relevant papers. In this way, a huge list of

potentially relevant papers was obtained. It was then necessary to examine

these papers individually in order to determine which of them would contribute

to this study. Although the possibility remains that there are important

papers in the literature which still have not been found, both the thoroughness of the indices and of the search procedures makes this unlikely. On the

other hand, some unpublished papers (such as [4A] which was presented as a

classroom exercise and was never published or submitted to a government

agency) are virtually unknown to the scientific community and are not easily

found. Another class of papers which would proved difficult to discover are

unpublished reports submitted by private consulting firms (either domestic or

foreign) to foreign governments. At present no evidence of the existence of

any such papers has been found.

One thing that was immediately apparent when examining the output listings of

these searches was the large preponderance of domestic research. Papers from

outside the United States accounted for a small fraction (estimated at less

than 15%) of the total. Whereas simulation models for analyzing traffic on a

waterway have been constructed and used in many parts of the world (See Section 7 of this report.) there appear to have been very few attempts outside

the United States to apply queueing theory to the problem. Moreover, the

level of sophistication in these foreign papers is considerably below that

found in the domestic literature. Koehler [51,52] and Leutzbach and Lenz

[58], writing in German, examined both vessel flow and cargo flow through a

lock of a waterway using queueing theory. However, neither of these treatments consider such aspects of the lock-queueing problem as multiple chambers,

different queue disciplines, multiple lockages, etc.
77

As several other authors have done, Koehler questioned the usual assumption of

Poisson arrivals. Other authors have tended generally to accept this assumption on the basis of a comparison of the available data and the functional

form of the Poisson distribution. Koehler used a standard statistical test,

called a goodness-of-fit test, which is based on the (read: chi-square)

distribution, to determine whether arrivals are random. Desai [30] used the

same approach to the same problem.

It is interesting to note that a substantial portion of the items on the list

of references in the German language papers (including the major reference to

mathematical queueing theory) were written in English and published in the

United States. On the other had, foreign language references of any kind in

domestic papers are virtually non-existent and, when they are found, are

usually included for completeness.
78

13. Input and Performance Data

There is an enormous amount of data continually being published in the United

States dealing with all aspects of the cargoes moved in commercial waterborne

traffic and of the vessels which carry them. For example, every year the

American Waterways Operators, Inc. prepares a report [2] containing extensive

listings of the commodities carried along each of the nation's waterways.

These listings give the tonnage carried (broken down into ocean-going and

inland portions) for each type of commodity on each waterway. In addition,

data are provided about the types and sizes of barges used, the number and

horsepower of towboats, etc. The prime source of data concerning commercial

waterborne traffic is the United States Government which periodically makes

available a vast array of statistics. There include detailed information

about the origin and destination of traffic, seasonal variations in flow

intensities, dollar-value of cargoes, net tonnage, ton-miles, etc.

In spite of the large quantity of data being published concerning all aspects

of traffic on a waterway, the specific data needs of some researchers may not

be met. However, in reading the overwhelming majority of literature dealing

with traffic through a lock, it is impossible to ascertain just what data the

author would have liked to have had, what data he actually had available to

him, and what portion of that he actually used. For example, it has not been

uncommon for an author to describe in great detail exactly how he obtained,

either by a computer simulation or by a mathematical study using queueing

theory, a distribution of arrival times or waiting times of tows at the particular lock in which he is interested. He might then conclude that his model

is satisfactory on the basis of comparisons he has made of his results with

the available data - without presenting those data or even giving the reader

any indication as to the exact data that he used and how he obtained them.
79

Even in those papers where the data are presented for comparison, there may be

no source provided and so It remains Impossible for the reader to verify the

author’s work.

This off-hand treatment of data often manifests Itself in another way. It is

quite rare for an author to complain about, or even comment about, the lack of

data available to him. When Hayward [43] modeled the Pickwick Lock using a

queueing theory model, he compared the results he obtained with the output of

a TOWGEN-LOKSIM computer simulation rather than with real-world data. He presented no explanation for this procedure - not even stating whether real-world

data existed or not. Because of his faith in the TOWGEN-LOKSIM package, Hayward did not seem to consider the lack of any comparison of his results with

real-world data to be a serious flaw in his study.

When Lave and De Salvo [56] used a queueing theory model to find the expected

waiting time per tow at a lock, their input and comparison data consisted of

the waiting times experienced by a single towboat during 35 transits of each

lock under consideration. They recognized that depending upon data from a

single towboat had a number of drawbacks.

The final input required for the queueing model is the service rate at

each lock. As argued above, the service rate observed by this towboat

probably constitutes a biased estimate of average service rate. The most

recent study on the service rate of these locks was for the two years

1949 and 1950,

Lave and De Salvo caution the reader that these figures were almost two

22

decades old at the time and so might not prove reliable. In 1978, / Wilson

[82] reassessed the models developed by Lave and De Salvo and by Hayward, In

his paper, he said

/Note that Wilson's paper appeared 4 years after the Performance Monitoring

System had been begun.
80

The scarcity of delay data prevented De Salvo and Lave from making

comparisons over a broader range of circumstances. Although empirical

data are not available, results of computer simulations of locks have

been published.

Lock and dam studies can be divided roughly into two categories - those which

deal with the general problem of (unspecified) traffic passing through an

(unspecified) lock(s) and those which are concerned with the operation of a

single specific lock. The extensive literature dealing with the replacement

of Lock and Dam No. 26 is a prime example of the latter. For those studies

centered on a specific lock, data are much more easily obtained. For more

general research projects, it is necessary to determine the variation in a

given category of data among the locks of the types of waterways under consideration. If this variation is large, then either

• the category can be treated as a parameter and varied according to

need , or

• the study must be restricted to a class of locks for which the variation in the parameter is small.

However, in either case, the appropriate data must be available for input,

calibration and comparison with output. Moreover, the data from various locks

must be comparable - that is, the same things must be measured and, in as much

23

the same way as is possible. As an example, the entry time / for a tow

might be defined (and measured) to begin when

• the water level in the chamber has reached the proper level (so that

the recycling time has ended) , or when

/Even greater difficulty is encountered when trying to measure the entry

time of a tow which, approaching a lock and finding no queue, can enter the

lock directly from the reach without making a stop beforehand.
81

the lock gates have been opened sufficiently so that a tow can enter

the chamber, or when

the lockmaster gives the tow captain permission to begin his approach

I to the lock, or when

I

• the waiting tow actually starts its approach to the lock.

Similarly, chamber time, exit time, delay time, etc. for a tow can be defined

in several different ways.

I In 1974 the Corps of Engineers introduced the Performance Monitoring System

(PMS), a program under which a wide variety of data would be collected by lock

personnel for each lockage, vessel and shift. The PMS was intended to measure

the service provided to the navigation industry on the inland waterways. The

Peat, Marwick, Mitchell Company prepared a data collection manual [67] for use

at all locks on the waterways of the United States. This manual contained the

forms and procedures to be followed by lock personnel to record and prepare

the data inputs for PMS, These data were used by the Corps of Engineers to

prepare extensive statistical information about the state and operations on

American waterways and to provide both input and comparison data for analyses

of traffic congestion, waiting times in a queue, lock capacity, etc. It is

evident, however, from reading the available domestic literature on waterway

traffic, that in the years since the PMS program was begun, researchers have

not made adequate use of the PMS data that are accessible to them.

One Important advantage of using a single highly detailed collection manual

throughout the entire inland waterways system was that researchers could

expect data obtained from different portions of the system to be measuring the

' same thing in the same way. Thus, these data could be meaningfully compared
82

and even co-mingled. As pointed out above, commonly used terms had not always

been defined in the same way by different authors in the past. The PMS program should lead to less ambiguity and more conformity in the literature.
' \ 83

14. Error Sensitivity

The sensitivity of a model can be defined as the relative size of the fluctuation in the output of the model compared to the size of the variations in the

input which cause the fluctuation. Such variations or deviations can be the

result of

• errors in the input data,

• errors induced by simplifying assumptions concerning the workings of

real-world system,

• errors Induced by mathematical approximations.

The sensitivity of a model with respect to any of these errors is a measure of

the confidence one can reasonably have in the output of the model. The sensitivity of a model can be determined either by a mathematical analysis or by

purposely introducing fluctuations in the input, assumptions, procedures, etc,

and then observing the resulting change in the output.

Many authors have pointed out that for large values of the traffic intensity,

the average delay time and the average length of a queue at a lock are very

sensitive to the average arrival rate. (For example, see Page 63 of this

report.) The average delay time and the average length of a queue at a lock

are also sensitive to the probability distribution of the tow arrivals

(Poisson arrivals vs regular arrivals) but the level of sensitivity to the

probability distribution of the tow arrivals is much below the level of

sensitivity to the arrival rate.

As part of the PTTSC simulation program of 1971, Gimbel [39] studied the sensitivity of the version of LOKSIM existing at that time to a number of simplifying assi iptions that had been incoporated into the model. He examined these
84

assumptions and the effect of replacing them by premises more closely

reflecting the actual operations of a waterway. These included

• the introduction of separate locking time distributions for up-stream

and down-stream traffic,

• a reevaluation of Howe's [48] formula for the speed attainable by a

tow in a channel,

• improving the method for determining the type of lockage required by a

given towboat and barge flotilla,

• an examination of tow delays based on channel width, the no-passing

rules, and other channel restrictions,

Gimbel described the level of sensitivity of the model to each of these

changes in the model. For example, he concluded that the delays due to channel restrictions are small in comparison with lockage delays and need not be

Included in the model for short reaches. Unfortunately, Gimbel did not provide specific numerical values documenting the size of changes in the output

data resulting from each of these changes in the model.

As part of Rao's study [69] in 1977, he predicted the year in which traffic

levels through Lock and Dam No. 26 would reach the capacity of the lock. This

prediction was made for the existing two chamber lock configuration and for

two possible lock replacements then under consideration. The basis for these

predictions was the expected annual throughput tonnage, Rao performed a sensitivity analysis of the annual throughput to lock and vessel characteristics.

Unlike the sensitivity analysis of Gimbel described above, the one performed

by Rao was mathematical in nature. Rao had derived a formula for the throughput tonnage in terms of the service rate at the lock, the number of barges of

various types being used by shippers, and the average loading per barge. In

order to find the sensitivity he differentiated the throughput tonnage with
85

respect to each of these variables. This yielded the expected rate of change

which would occur in the tonnage when each of the variables was changed. A

mathematical approach to sensitivity such as Rao’s is a rarity in the literature on traffic on a waterway.

In 1979, Rochberg [72] made an extensive study of the advantages and disadvantages of queueing theory and computer simulation as methods for estimating the

capacity and average delay time at a lock. As part of this study he examined

the sensitivity of each method to the data and assumptions on which it

depends. He also pointed out a number of flaws inherent in each method. For

example, it is a standard procedure in queueing theory to use the average

value for a variable , rather than the individual values that the variable

takes on. Thus, if we have two rectangles, one 3 high by 5 wide and the other

5 high by 7 wide then the average area of the two rectangles is 25 (one half

of 15 plus 35). On the other hand, if the average area of the rectangles is

computed as the product of the average height (4) and the average width (6)

then we obtain an average area of 24, an error of 4%. The same error occurs

when the total tonnage per tow is obtained as the product of the average number of barges per tow and the average tonnage per barge.

Rochberg expressed the opinion that, in general, computer simulations provide

a more satisfactory method for modeling the operations of a waterway than does

queueing theory. However, queueing theory can be used to enhance the effectiveness of a computer simulation. For example, it is universally agreed that

computer simulation runs should be "long" - though the specific advantage of

long runs is seldom stipulated. Using queueing theory it is possible to answer questions such as: How lonj must a computer simulation run be in order to

determine (within a specified degree of certainty) whether the queue at a lock
86

will tend to grow indefinitely or will fluctuate about some finite queue

length?

There are several important reasons for performing a sensitivity analysis on a

model

• to determine what level of quality is required for the input data,

• to determine whether some portions of the model must be modeled in

finer detail than others,

• to determine what confidence can be placed in the output of the

model

• to determine what length a computer simulation should be and to

determine how many runs should be made before changing the values of

the parameters,

A sensitivity study usually requires a more difficult and more subtle analysis

than did the construction of the model itself. In addition, there is not the

sense of immediacy about a sensitivity analysis as there is for the analysis

of the real-world situation. Unfortunately, it is very tempting to devote all

of one’s energy to the model and ignore the sensitivity completely.
87

15. Other Modes of Transportation

In order to recognize and appreciate the complexities involved in commercial

traffic on a waterway, it is helpful to compare traffic on a waterway with

traffic using other modes of transporation. There is a fundamental similarity

between traffic on a waterway and traffic using any other mode of transportation. The basic scenario describing commercial traffic traveling on a locked

waterway is

• there is a single traffic-way which, for the most part, is used

simultaneously by two streams of traffic traveling in opposite

directions and not interfering with each other,

• there are a number of bottlenecks in the traffic-way at which

• traffic is slowed considerably,

• the two streams of traffic interfere with each other,

• traffic may be permitted in only one direction at a time.

There are several other transportation situations which also fit this general

description to a greater or lesser extent:

• aircraft arriving and departing at an airport, perhaps using the same

or nearby runways

• a single-lane bridge on a two-lane country road,

• a railroad spur as part of a railroad system,

• a drawbridge over a navigable waterway,

• an intersection of two highways or city streets,

• a two-lane road in which a portion of one lane is closed (as for

repairs)

Undoubtedly there are other transportation phenomena which approximate this

basic scenario.
88

The similarity between scenarios describing locks on a waterway and bottlenecks encountered when using other modes of transportation suggests that

research performed in the context of these other modes of transportation might

be applicable to operations on a waterway. On the other hand, it is highly

unlikely that a computer simulation of any other mode of transportation would

contribute substantially to a computer simulation of a waterway. The current

state-of-the-art in computer simulation of waterborne traffic (e.g., the INSA

program) is the culmination of a decade of research and has reached a level of

sophistication, flexibility and accuracy not often found in research devoted

to other means of transportation. On the contrary, it is much more likely

that analyses of other modes of transportation by means of computer simulation

could profit by borrowing from the research performed on waterway traffic. It

appears safe to conclude that if research performed in the context of some

other mode of transportation is to make a contribution to the analysis of a

waterway, then it would necessarily be in the area of mathematical queueing

theory.

One drawback of queueing theory as a tool for modeling is its inflexibility.

Even a relatively minor change in the operation being modeled may result in a

serious complication in the formulation of the queueing model or in its mathematical tractibllity . Thus, Incorporating features found in the operation of

a waterway into a queueing model developed in the context of another mode of

transportation may be a difficult problem. There are several aspects of tow

traffic through a lock for which counterparts at the bottlenecks in other

modes of transportation seldom, if ever, exist. Among these rather unique

aspects of tow traffic are

• the lateral maneuverability of a tow on a wide river,

• the distinction between a long entry and a short entry.

<1
89

• multiple and setover lockages,

• the use of a helper boat,

• the inability of the bottleneck, when empty, to accept traffic from

either direction without substantial preparation,

24 , • the exceptionally long time that a tow actually spends in the lock /.

It is thus clear that to apply to the lock and dam situation research based on

another mode of transportation will generally require extensive, and possibly

prohibitive, adjustments in order to make it relevant. In addition, for many

bottleneck problems, the waiting time is inconsequential (e.g., at a singlelane bridge on a lightly traveled country road) and has not merited the

intensive research effort which has been made for traffic through a lock.

There appears to be no form of transportation in which the length of delay

times at a bottleneck reaches those which are common at a busy lock. Nevertheless, a number of papers have been written concerning the flow of traffic

through a bottleneck on other modes of transportation.

The Science Citation Index and the On-Line Bibliographic Retrieval System were

used to search the literature to locate those papers which treat the problem

of two streams of traffic passing through a bottleneck where only a single

stream can pass at a time, but which were not concerned specifically with

waterway traffic. In 1977, Eshcoli and Adiri [36] considered a single-lane

bridge on a two lane road. Their aim was to obtain the steady-state distribution and the expectation of the flow time as a function of the bridge’s length, or equivalently, of the time spent crossing the bridge. They defined the

flow time as the interval between the arrival of a car at the

24

/In other modes of transportation, the time acutally spent in the hotter ^k,

rather than in a queue leading to the bottleneck, is so short as to be

Insignificant
90

bridge and the departure of the car at the other end. This differs from the

service time (which they did not consider) since, in their model, they assumed

that several cars traveling in the same direction can be crossing the bridge

at the same time. They further assumed that traffic would continue to flow in

one direction unitl no waiting cars in that direction remained to be serviced.

Traffic would then immediately begin flowing in the other direction. Also,

they assumed that cars traveling in the same direction maintain the same (constant) speed and a fixed safety distance from each other. No comparison was

made between results obtained theoretically and either real-world or simulated

data.

A related model using vehicle-acutuated traffic lights controling the

intersection of two roadways was discussed by Darroch, Newell and Morris [26]

in 1964. In this model, once the light at the Intersection has turned green

in either direction, it remains green until there occurs a time period of a

pre-assigned length during which there is no queue in that direction. At that

point the light turns red in that direction and green in the opposite

direction. Otherwise this scenario is very similar to the one studied by

Eshcoli and Adiri. The literature dealing with bottleneck problems, either in

the abstract or in terms of some other mode of transportation, appears to be

very limited. The two papers discussed above are representative of the

current state-of-the-art in modeling traffic bottlenecks, not in the context

of waterway traffic, as exemplified in the published literature. Although

these studies were performed in terms of automobile traffic, the general

approach and the mathematical tools used are very similar to those commonly

employed in studying tows passing through a lock. However, the scenario of

tows passing through a lock seems to have received much more attention than

other bottleneck problems. Non-waterway research would appear to have very
91

little to contribute to the treatment of the corresponding waterway problem in

the areas of simulation construction and the application of queueing theory.

Nevertheless, there are several papers in the transportation research literature (written in a context other than traffic on a waterway) which presented

ideas that might be carried over profitably to studies of waterway traffic.

For example, in 1979, Newell [63] presented a survey and critique of the literature dealing with airport delay and airport capacity. In this paper, Newell

made no attempt to actually calculate either capacity or delay. Instead, he

developed careful descriptions of the operations which take place at either a

single runway airport or at one with several parallel runways. In terms of

these operations, he formulated precise definitions of airport capacity, delay

time, arrival rate, service rate, etc. for arriving and departing aircraft

serving a mix of several different types of planes. Finally, he pointed out

several possible areas for research (e.g., the effect of FAA safety regulations on airport capacity and delay times the possible use of incentives to

smooth out periods of peak demand, the desirability of sequencing planes in

takeoffs and landings, etc.).

In 1978, Carson [17] considered techniques for reducing the variance of variables Involved in a computer simulation of a queueing process. Reducing the

variance of a variable results in a decrease in the length of computer runs

needed to attain a desired level of confidence in the output. In a subsequent

paper in 1980, Carson and Law [18] showed that there were several different

orders in which a computer simulation of a G/G/s queueing system can estimate

the variables being sought (delay time, queue length, arrival rate, etc.). It

was then shown that the best results are obtained when those variables with

the smallest variances are estimated first and the remaining variables are

estimated in terms of these.
92

In 1978, Rosenbloom [73] listed 22 major classes of promising techniques for

reducing congestion in highway traffic and analyzed each of them singly and in

a number of combinations. In some cases, implementing several techniques simultaneously results in an improvement in traffic flow greater than the sum of

the improvements accrueing from the Individual techniques. Such reinforcement

does not always occur. Occasionally, different techniques for improving traffic flow cancel each other out when they are used simultaneously. Unfortunately, many of the techniques considered are relevant for highway traffic,

but not for a waterway (e.g., the use of telephone calls in place of automobile trips). Nevertheless, the procedure is novel and might be carried over

to waterway research.

In 1980, Nihan and Holmesland [64] explored the possibility of using a

recently developed mathematical technique called "time series analysis" to

forecast short term traffic volume. Monthly data on the volume of traffic

using a particular freeway segment were collected for the period 1968-1976.

Using these data, a forecast was made of the volume of traffic for 1977. This

forecast turned out to be very accurate. The scheme described by Nihan and

Holmesland is simple to describe, flexible and easy to use. It might prove to

be an improvement over methods currently used in navigational research to

forecast tonnage levels of various commodities.
93

16. Conclusions and Recommendations

A great deal of effort has been expended by a large number of researchers in

modeling the flow of tow traffic through a portion of an inland waterway.

This effort can be divided into two distinct categories according to the main

tool used in the construction of the model, namely; computer simulations or

mathematical queueing theory. Both computer simulations and queueing theory

models have been studied in great detail in the last 15 years. Substantial

progress has been made in that period in identifying many of the areas within

each model which did not adequately reflect the operation of the waterway. In

many cases it has been possible to make modifications in the model which

resulted in greater accuracy in the output.

It is generally agreed that computer simulations of any kind have definite

25

drawbacks and limitations /. However, within these recognized bounds, the

large scale simulations which have been directed to the operation of an

extended inland waterway have been quite successful so that results now

obtained (e.g., from the INSA program) enjoy a great measure of acceptance

within the research community. With the availability of large quantities of

data from the PMS, it became possible to determine the accuracy of computer

packages for different portions of the inland waterways system. Thus, any

cases in which the computer simulation models still do not adequately describe

the real-world operations may be Identified and, hopefully, corrected.

Undoubtedly refinements of one sort or another in computer simulation techniques will continue to be made in the future, but the extensive testing to

which models currently available have already been subjected suggests that

there is only a limited amount of room remaining for major improvements.

25

/See pp 21-22.
94

The same sanguine picture cannot be presented for the current state-of-the-art

with regard to the application of mathematical queueing theory to even a single lock, much less to an extended waterways system. In the current mathematical literature devoted to tow traffic through a lock, there is still significant disagreement concerning the proper probability distributions and queue

discipline parameters to be used in order to provide an accurate model. This

is particularly true for

o a lock with two (possibly unequal) chambers,

o a lock which is not isolated from other locks and whose input is

therefore influenced by the outputs of adjacent locks.

However, even in the simplest case (that is, an isolated lock with one chamber) , the complicated scenario describing the passage of a tow through a lock

has presented difficulties which queueing theory has not completely overcome.

A large portion of the queueing theory research now in the literature centers

around a specific lock and has been made to accomodate the peculiarities found

there. Such research has enjoyed a certain degree of success in producing

output data about as accurate as that produced by computer simulation. However, the attempts to use queueing theory to describe a "general" lock (so

that the results would be universally applicable) have not been successful,

and the current state-of-the-art leaves much to be desired.

It is generally agreed that when a queueing theory model of an operation is

available which is as accurate as the state-of-the-art simulation of that

operation, then the advantages of queueing theory outweigh those of simulation. Unfortunately, such a queueing theory model for an operation as complex

as passing a tow through a lock usually does not exist. In a standard text on

operations research, Hilller and Lieberman [46]' have written:
95

AJ.though considerable progress has been made in developing analytical

queueing results, there still remain many potentially useful queueing

models which are extremely difficult or impossible to analyze mathematically. It appears that simulation... offers the best hope of eventually

obtaining usable results for such models. Simulation has already been

used many times to analyze unique queueing-type problems as they have

arisen in practice.

Choosing between queueing theory research and research devoted to computer

simulation is not an easy task. The likelihood that a queueing theory study

will produce a usable model is significantly smaller than the likelihood that

a computer simulation would produce usable results but when a usable queueing

theory model is produced, it has numerous advantages over a simulation model.

Selecting either alternative over the other is a calculated risk. Certainly,

the ideal would be to have both kinds of research. If a choice is unavoidable, the nod would seem to go to computer simulation. Perhaps even better

than choosing one or the other would be to follow Rochberg's [72] suggestion

2 6 of combining computer simulation with queueing theory /.

It is understandable that editors of scientific journals must limit the space

available to a single research report. In these reports, authors emphasize

the results they have obtained and downplay (or completely ignore) the methods

used, the data required, the sensitivity studies performied, the number and

27

lengths of their computer runs, etc /. It is important that authors (especially those supported by the Corps of Engineers) prepare not only a paper

suitable for publication in a technical journal but also a longer, more

detailed report to appear in mimeographed form only for the use of serious

scholars. This report should contain all of the important details of the work

that could not be included in the published version. Even when such a report

is prepared the problem remains of publicizing its existence and getting it

into the hands of research workers.

/ See p. 85

/ See p. 47
96

Finally, in order that these unpublished reports meet the needs of future

researchers, a set of documentation standards should be prepared by the Corps

of Engineers for use by its own research personnel and by researchers it supports (and, hopefully, by other workers in the area, as well).

There is a very large mathematical literature dealing with the theory of

queues in an abstract manner, that is, not in the context of any particular

application. As a part of this survey, a substantial portion of that literature was examined with the hope of finding specific research which might be

applicable to the problem of tow traffic on an inland waterway. In reading

the recent queueing theory literature it soon became clear that the problems

inherent in modeling waterway traffic are quite different from those studied

in the large majority of mathematical papers. For example, the assumption of

independence in arrivals, service times, etc. is extremely common in the literature because without it, the problem becomes intractable. However, this

assumption is seldom valid in the real-world, and certainly not at a lock.

Also, the mathematical literature deals with parameters that are fixed over

long periods of time. At a lock, parameters change according to the time of

day, the weather conditions, the lock crew, the tow crew, even according to

the amount of debris in the water. In the majority of the mathematical literature, given queueing systems are studied to obtain their properties.

Researchers dealing with traffic through a lock (especially a multiple chamber

lock) are still in disagreement as to which (if any) of the classical queueing

systems are appropriate. Finally, the mathematical literature deals with many

2 8 subjects that are not relevant to a lock (e.g. queues in network /,

2 8 /In a network the customer has the choice among several routes between his

origin and destination. '
97

infinitely many servers, balking /, etc.). The papers on modeling the traffic at a lock by means of queueing theory that we have examined in Section 8

of this report have recognized the difficulties that have just been described.

The queueing theory used in these papers (especially [40] and [72]) is at the

state-of-the-art level. Basically, the obstacle is not with the mathematics

of queueing theory, but with the deviations of the operations at a lock from

the common assumptions of the theory. Nevertheless, mathematical queueing

theory still has a contribution to make, primarily as an adjunct to computer

simulation. However, it is reasonable to expect that as additional queueing

theory research is performed in the future, addressing the difficulties

described above, new models will be devised which capture more accurately the

features of waterway traffic. Such research should be supported, but with the

understanding that the problems are extremely difficult and are likely to be

solved only after long, painstaking work.

Mathematics is not a static science. New and powerful mathematical tools are

regularly developed and added to the literature. Several of these hold promise for contributing to our ability to model waterway traffic. For example,

time series analysis [7] has been used successfully in predicting trends in a

wide range of applications. Markov renewal and semi-Markov processes are two

other recently developed tools dealing with probabilistic situations which

reflect many of the traits of the operations at a lock. Whether or not any of

these disciplines can contribute to either our understanding of the operations

at a lock or to our ability to construct better models remains to be seen.

29

/Balking refers to the option of the customer to leave the queue without

service
98

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I
NBS»n4A (REV. 2 «QC) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

81-2411

2. Performing Organ, Report No. 3. Publication Date

March 1982

4. TITLE AND SUBTITLE

An Examination of the State of the Art in Inland Waterways System Lock Research,

5. AUTHOR(S)

Martin H. Pearl

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covere

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Navigation Division

Institute for Water Resources

Kingman Building

Ft. Belvoir. VA 22060

10.

SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a si gnifi cant

bi bliography or literature survey, mention it here)

Locks which pass waterborne traffic through dams act as bottlenecks and interfere

with the free flow of traffic. Two complementary mathematical tools have been

used to determine the maximum capacity of a lock and the operating procedures

which achieve this capacity. These are i) computer simulations of traffic flows,

and, ii) the mathematical theory of queues. A large-scale, multi-purpose simulatioi

package was developed at the Pennsylvania State University and expanded by the

Army Corps of Engineers. Several researchers have attempted to model the operation

at a lock using mathematical queueing theory. In addition, queueing theory has

been applied to analyze traffic flow through a bottleneck on other modes of

transportation. Also, queueing theory has been used to study problems which arise

in the use of computer simulations.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) capacity; dam; lock; queue; simulation; waiting time

13. AVAILABILITY

I y i Uni i mi ted

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20402.

Iy I Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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PRINTED PAGE'

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