NBSIR 77-1253

FILE CO PI

DO NOT REMO'V

Cost/Benefit Analysis of

Automated Transit Information

Systems

Judith F. Gilsinn

Applied Mathematics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D. C. 20234

February 1 977

Final

Issued June 1 977

Technical Report to Urban Mass Transportation Administration

Department of Transportation

Washington, D.C. 20590

Douglas R. Shier

NBSIR 77-1253

COST/BENEFIT ANALYSIS OF

AUTOMATED TRANSIT INFORMATION

SYSTEMS

Douglas R. Shier

Judith F. Gilsinn

I Applied Mathematics Division

Institute for Basic Standards

National Bureau of Standards

Washington, D C. 20234

February 1 977

Final

Issued June 1 977

Technical Report to Urban Mass Transportation Administration

Department of Transportation

Washington, D.C. 20590

U S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Dr. Sidney Harman, Under Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Acting Director

ABSTRACT

This report discusses the costs and benefits associated with

automating the route-finding portion of a telephone transit information

system that responds to telephone inquiries. The various costs of

implementing such a system are categorized and compared with those of a

manual system over an appropriate time span using a present value approach.

A queuing model, described in the report, is used for computing manpower

requirements of the two systems, manual and automated. Outputs of the

queuing model for a wide range of input parameters are tabulated in an

appendix. Benefits from automating transit information route-finding

are discussed, and measures of performance improvement available as

output from the queuing model are provided.

Key Words : Automation; cost /benefit ; models; queuing; telephone systems;

transit information; transportation.

TABLE OF CONTENTS

PAGE

1. Introduction 1

2. Cost Elements 4

2.1

Cost Elements for an Automated Information System 7

2.1.1 Computer Space Preparation

2.1.2 Hardware

2.1.3 Software

2.1.4 Computer Operation

2.1.5 Terminals

2.1.6 Data Base Management

2.1.7 Information Operator Personnel

2.1.8 Furniture

2.1.9 Telephones

2.1.10 Physical Plant Overhead

2.1.11 Training of Operators

2.2

Cost Elements for a Manual System l4

2.2.1 Data Base Management

2.2.2 Information Operator Personnel

2.2.3 Furniture

2.2.4 Telephones

2.2.5 Physical Plant Overhead

2.2.6 Training of Operators

2.3 Aggregation of Cost Elements 17

3. Queuing Analysis 23

3.1 Queuing Model 23

3.2 Input Specifications 25

3.3 Model Assumptions 30

3.4 Mathematical Description 33

4. Manpower Estimates Using the Queuing Model 37

5. Benefits 39

5.1 Improvements Resulting from Automation 40

5.2 Benefits to Transit System Users and Transit

Properties 42

5.3 Measuring the Benefits 45

5.4 Disadvantages to Transit System Users and

Transit Properties 46

5.5 Other Transit Information System Improvements 50
TABLE OF CONTENTS (continued)

PAGE

6. References 51

Appendix A: An Approximation for the Relationship Between

X and X* 53

Appendix B: Tables for Use in Estimating Manpower Requirements.. 58
LIST OF TABLES

PAGE

1. Cost Elements for Automated Information System 5

2. Cost Elements for Manual Information System 6

3. Sample Worksheet for Aggregating Automated System Costs.... l8

4. Sample Worksheet for Aggregating Manual System Costs 19

5. Benefits from Automation 41
LIST OF FIGURES

PAGE

1. Queuing Model of Transit Information Process 2h

2. The Functional Relationship A* = g(A) 55
1 . INTRODUCTION

Transit companies provide prospective riders with information about

transit itineraries and service through the use of several different

media: signs at transit stops, stations and in vehicles; printed schedules

and maps (available in stores or transit facilities, or mailed to prospective riders); advertising messages in newspapers; and possibly a telephone

information facility. It is this latter service, the one provided by

the telephone transit information center, which is the subject of the

present report. Our specific aim is to investigate the costs and benefits

arising from automating such a service. Possible tradeoffs among the

various media used to furnish transit information will not be analyzed;

treatment of that broader topic would require an extensive marketing

study, and the particular mix of media which best supplies information

in a city is very likely to depend on particular characteristics of the

transit system, transit riders, and alternative transportation modes

available in that city.

In the typical telephone transit information center, transit system

employees (whom we will call "operators") answer inquiries from prospective

riders about schedules, routes and particular trip itineraries. Presently,

the operators in most manual systems consult maps and schedules in

piecing together the requested trip information. Operators must be

quite familiar with both the transit system and the regional geography

in order to locate a trip’s origin and destination and to find appropriate

routings for the trip between a transit stop near the origin and one

near the destination. Automation of the route-finding function is under

consideration, and this report is designed to aid in the evaluation of
such proposed systems.

In an automated system the operator would ascertain from the caller the

desired trip and would input this information to a computer, which would

find an appropriate ("best") itinerary and report it back to the operator.

The operator would then relay this itinerary to the caller. Although

the possibility of further automating the response to the caller (through

the use of an automated voice response device) has been proposed, it is

not included in the analysis below, since several problems with its

implementation remain unsolved. Thus the automated systems to be evaluated

here have an operator to converse with the caller, to translate the

request and to relay the answer; these systems include automation only

of the route-finding portion of a call. Such an automated system will

be compared to a similar manual system in which the route-finding function

is performed by operators consulting hardcopy maps and schedules.

An important first step in weighing a (partially) automated transit

information system against a manually oriented system is to survey the

individual cost elements incurred by the two systems. Accordingly, the

following sections will provide a framework for specifying and classifying

in detail the various costs associated with such systems. Moreover, a

general cost model (which employs a queuing analysis to generate appropriate manpower estimates) will be described for combining, over a given

time horizon, the cost elements into an estimated total system cost.

The cost differential between the systems can then be compared with the

net benefits accruing from automation to obtain a final evaluation of

the impact of automation on a transit information system.
The less mathematically oriented reader may wish to skip the detailed

description of the queuing model in Section 3 and Appendix A, concentrating

on the description of the cost elements in Section 2, the use of the queuing

model output for estimating the number of operator positions in Section 4,

the discussion of benefits in Section 5> and the tables in Appendix B.
- h -

2. COST ELEMENTS

The major elements of total cost are identified in Tables 1 and 2

for both an automated and a manual information system. The type of

automation envisioned requires utilizing a computer code to provide

point-to-point trip itineraries, under some appropriate criterion for a

"best" trip or several alternative "good" trips. Several such criteria

are discussed in some detail in a previous report [U]. Those portions

of an incoming transit information request which involve comprehending

the caller’s question and providing the response are assumed here to

remain under manual control.

In Tables 1 and 2 the cost elements are further classified according

to whether they represent initial (capital) costs, recurring (annual)

costs or both. In addition, those cost elements which depend on the

number of transit information operators are identified. Detailed descriptions

of the individual items that constitute these cost elements are provided

in the sections that follow. The costs for an automated system are

presented in Sections 2.1.1-2.1.11, while those for a manual system are

described in Sections 2. 2. 1-2. 2. 6.

Note that although some categories of cost appear to be similar in

the automated and manual systems, the costs associated with those categories

may vary in magnitude between the two systems. For example, telephone

and furniture costs may be higher in computer assisted systems in order

to interface effectively with the terminals or operator activities.

Costs of these apparently related categories may also vary significantly

depending on the particular design of the information center.
- 5 -

TABLE 1

Cost Elements for AutomatedInformation System *

Dependent on Number of

Cost Initial Recurring Information

Description Cost Cost Operators

Computer Space Preparation x

Computer Hardware (Leased) X X

Software Development X

Computer Operation X

Terminals X X X

Data Base Management X

Information Operator Personnel X X

Furniture X X X

Telephones X X X

Physical Plant Overhead X X

Training of Operators X X X

*Items below the dotted line appear in both Tables 1 and 2; those above,

appear only in Table 1.
- 6 -

TABLE 2

Cost Elements for Manual Information System

Cost

Description

Initial Recurring

Cost Cost

Dependent on Number of

Information

Operators

Data Base Management

Information Operator Personnel

Furniture

Telephones

Physical Plant Overhead

Training of Operators

X X

X X

X X X

X X X

X X

X X X
- 7 -

2.1

Cost Elements for an Automated Information System

2.1.1

COMPUTER SPACE PREPARATION

Automation of the transit information facility is expected to

require a computer dedicated to this task. (Use of an existing computer

configuration would require appropriate modification of the costs included

here, which do not consider shared computer usage.) Therefore, additional

space and appropriate preparation of that space will be necessary in

order to provide the proper environment for the new computer. In particular, the following items will contribute to the (initial) expense of

computer space preparation and should be included in this cost element:

a) Routine site preparation

b) Special electrical preparation

c) Installation of air conditioning

d) Installation of sound conditioning

e) Installation of data communication cabling ducts

f) Flooring modification

2.1.2

HARDWARE

These costs refer to the required computer hardware, which is

assumed to be obtained on lease. This assumption is used for convenience

of reference since such leasing costs could be replaced by amortized

value of purchase cost if equipment purchase is deemed more appropriate

for a particular system. The annual leasing cost should be incremented

by the amounts necessary for maintenance and repair (if not provided in

the leasing contract).
- 8 -

To some extent the choice of hardware configuration is dependent on

the number of information operators, inasmuch as sufficient computing

power must be available to offer time-sharing service to the operator

terminals without excessive wait time. Some peripheral equipment may be

purchased, rather than leased as needed.

2.1.3 SOFTWARE

Software costs represent an initial outlay to provide operating

system software and the applications software for algorithms that perform the point-to-point trip processing. In addition, one should include

in this figure the one-time cost of creating for the computer a data

base representing the transit network. A certain amount of supervisory

assistance from the transit authority will be required in creating this

data base, and therefore an appropriate amount for supervisors’ salaries

and fringe benefits should also be included. Programs for editing and

updating the data base are also required. Additional programs to provide

management information on system performance may be included in the

system program package.

2.1.4 COMPUTER OPERATION

This item consists primarily of salaries for the computer system

operators. The number of operators required per shift and the number of

shifts per day and week will depend on the sophistication of the computer operating system, the number of hours per day during which the

transit information center will be answering requests and whether or not
- 9

-

updates are performed during the day or later at night. In addition,

this figure should reflect electricity costs for the computer system and

any additional air conditioning required. Another item falling under

this heading (if not already included in hardware costs) is a contract

assuring maintenance and service for the computer equipment.

2.1.5 TERMINALS

The one-time costs referred to here are expenditures for the initial

purchase of terminals. The number of these purchased will depend on the

number of operator positions required to meet the initial level of

demand; the number of such positions can be estimated through the use of

the queuing model. In addition, it is appropriate to include the cost

of purchasing additional terminals, which are to be used for training, for

maintenance and for performing updates. To the extent that demand

increases (on an annual basis), additional operator positions will be

needed to meet this demand and still maintain a desired level of service.

Recurring costs for these additional positions will be reflected in the

cost of additional terminals.

2.1.6 DATA BASE MANAGEMENT

This item refers to the cost of maintaining and updating the computer data base. In contrast to the activity of creating an appropriate

data base (see Section 2.1.3), this aspect will involve a recurring cost

(in addition to an initial programming expense), the magnitude of which
10

depends on the frequency of updates and the overall complexity of the

underlying transit network. It is possible that such updates could be

entered through one of the information operator terminals during offpeak hours. If instantaneous corrections to schedules or routings are

required, a separate spare terminal could be used to enter such changes

2.1.7 INFORMATION OPERATOR PERSONNEL

An important factor in comparing automated and manual information

systems is the level of staffing required for the transit information

operators. This staffing level depends on the number of operator positions

which are being used at any one time by the systems. In turn, the

number of operator positions reflects the capacity of the system, in

terms of the maximum number of calls which can be simultaneously handled.

Since operator working shifts cannot be immediately and completely

compatible with changes in the level of demand (number of requests for

information) during the day, it is appropriate to assume that a constant

number of operator positions are manned throughout the peak demand

period. Indeed, it is at such times of maximum demand that the information system would most likely exhibit overloading or congestion and a

high incidence of lost calls. It is the ability of an information

system to cope under such circumstances that is a key issue in assessing

how well the system is providing service.

In order to provide a consistent basis for comparing the automated

and the manual systems during periods with a constant number of operator

positions maintained, a queuing model (which is described in fuller
11

detail in Section 3) is used to calculate relative manpower requirements

for the two systems. This queuing model estimates the minimum number of

operator positions required by each system to achieve a prescribed level

of service. For example, one might specify that the percent of calls

which encounter a busy signal during the period should not exceed 1%.

Alternative measures for the "level of service" are: the expected

number of lost calls (i.e., the average number of callers encountering a

busy signal), the average number of persons "on hold" at any time, the

average length of time a person who is not serviced immediately must

wait before an operator is free, and the average time required for the

caller to complete the transaction (including any time spent "on hold"

together with the time spent communicating with the information operator).

The queuing model employed here represents an acknowledged simplification

of a complicated process, but it can provide a useful benchmark for

estimating the relative manpower requirements of alternative information

systems. The formal specification of this queuing model is given in

Section 3, where the types of input required by the model as well as its

underlying assumptions are discussed.

Once the number of operator positions s has been determined for a

given time period (using the tables of Appendix B produced via the

queuing model), the required number n of operators for the automated

system can be calculated from: s , the ratio of demand in peak and

non-peak periods, and the number of hours worked per week by an operator.

The total cost contribution of operator salaries and fringe benefits is

then simply n times the appropriate average salary plus benefit figure
12

for a single operator. In addition, one should include salary and

fringe benefit costs for any supervisory personnel required to coordinate

the information operators.

2.1.8 FURNITURE

The one-time costs referred to here are expenditures for the

initial furnishings of the transit information facility, such as desks and

filing cabinets. The number of desks purchased initially will depend on

the number of operator positions required to meet the initial level of

demand; the number of such positions can be estimated through use of the

queuing model. To the extent that demand increases (on an annual basis),

additional operator positions will be needed to meet this demand and

still maintain a desired level of service. Recurring costs for these

additional positions will be reflected in the cost of extra desks and

cabinets

2.1.9 TELEPHONES

Initial costs will be incurred for installing operator telephones

and for instituting an Automatic Call Distribution System (ACDS), if

such a system is not already present. The ACDS is required for routing

calls to the information system. Most probably the operator terminals

will be connected directly into the mainframe computer system; if not,

then additional fixed costs are incurred in providing data phone lines

between the computer and the terminals. Recurring costs take the form
- 13 -

of operating (rental) costs for telephone service to each of the operator

positions. Such costs depend on the number of operator positions provided and thus would reflect any increase in the number of operator

positions as a result of increased demand.

2.1.10 PHYSICAL PLANT OVERHEAD

These costs are those for preparing the site which houses the

information operators and terminals (e.g., space partitioning) as well

as undertaking routine electrical preparation, installing air conditioning

and installing sound conditioning. Besides such initial expenditures,

there will be a number of recurring expenditures for both the information

operator room and the computer room: namely, the costs for rental of

space, utilities, insurance and janitorial services. It is assumed here

that the space provided for the information operators is sufficient to

allow for any later expansion in the number of operator positions to

meet increased demand.

2.1.11 TRAINING OF OPERATORS

It will be necessary to train the operators in using the terminal,

inputing data about itinerary requests and interpreting output, but

extensive training in city geography and available routes will not be

required. Training emphasis in an automated system could focus more

directly on improved communications skills, including how to elicit

information more efficiently and how to articulate more clearly. Costs
- Ik -

of training will depend on the operator turnover rate, the number of

training hours required per operator, the need for special materials,

and the cost of supervisors who conduct training.

2.2 Cost Elements for a Manual System

2.2.1 DATA BASE MANAGEMENT

Initial costs are incurred here in setting up the data base from

which route and schedule information can be generated for the transit

information operators. This cost will be reduced considerably if an

appropriate data base already exists or if currently available schedules

are already in a form appropriate for use by the information operators.

In any event, recurring costs will be encountered in updating and

maintaining such a data base for use in periodically providing up-to-date

route and schedule information to the operators.

2.2.2 INFORMATION OPERATOR PERSONNEL

As in the case of the automated system, manpower requirements for

the manual system can be estimated using the queuing model of Section 3.

The major difference is that now one of the input parameters to this

model is changed: namely, the parameter describing the average number

of calls which can be serviced per operator per hour during a busy

period. As described in Section 3.2, the difference between the automated and manual systems would be reflected by more rapid servicing of
- 15 -

calls in the former system compared to the latter. That is, an automated

system having fewer operator positions can achieve the same level of

service as a manual one. Once the number of operator positions s

required in the manual system to meet certain minimum performance

levels has been determined, then the total number n of operators can

be found as well as the total cost of such personnel. The cost of

maintaining supervisory personnel, at their appropriate salary and benefit

levels, should also be included in this cost element. A manual system

may require more intensive use of supervisors than will an automated

system, since the system requires greater use of judgement by the individual

operators

2,2.3 FURNITURE

A manual system requires certain furniture for the transit information operators, in particular desks large enough to accommodate the

various maps and schedules required to answer calls effectively. Moreover, there is a need for certain general office equipment (e.g., filing

cabinets) and special equipment (possibly map display cases). As the

number of operator positions expands to meet demand, recurring annual

costs for furnishing these additional positions will be incurred.

2.2.4 TELEPHONES

Initial costs are incurred for installing operator telephones as

well as for instituting an ACDS if one were not already available.
- 1 6 -

Recurring costs will result from charges for ordinary telephone service.

The total charge will increase with any increase in the number of

operator positions (which entails an increase in the number of incoming

lines )

2.2.5 PHYSICAL PLANT OVERHEAD

This cost element will include initial expenditures for preparation

of the site housing the transit information operators, undertaking routine

electrical preparation, installing air conditioning and installing sound

conditioning. There are a number of recurring expenditures for the

information operator room, including the rental cost of the space,

utilities, insurance and janitorial services. Again, it is assumed that

the space allocated for the information operators will be sufficient to

absorb any later addition to the number of operator positions.

2.2.6 TRAINING OF OPERATORS

It is necessary to train new information system operators in city

geography, in transit system operations, routes and schedules, and in

communication. Usually operators receive a period of intensive training

in these skills in a classroom and then serve an "apprenticeship"

period answering transit information requests with a gradually decreasing

level of supervision. The training and apprenticeship periods vary with

the complexity of the transit system, the type of program, and the
- IT -

background, of the information operators (for example, former bus drivers

\ would need, considerably less training in city geography or the transit

system). Training costs include full operators' salaries for the period

of classroom training, partial salaries (reflecting lower productivity)

for the apprenticeship period, salaries of supervisory personnel and

teachers for the time spent involved in training, and expenses for

instructional equipment and materials. Total training costs are affected

by operator turnover rate and the size of staff required.

2.3 Aggregation of Cost Elements

We defer to the next section the procedure for estimating the

number of operator positions, a calculation which influences the magnitude

of several cost elements. (Tables 1 and 2 indicate which cost elements

are so influenced. ) The issue discussed in this section is how to

aggregate the various cost elements into a total cost figure, for

\ either a manual or an automated information system.

The steps of this aggregation process can be illustrated by using

the sample worksheets in Tables 3 and 4, which refer respectively to

automated and to manual systems. Here the use of Table 3 is discussed

in detail; however, a similar procedure also applies to Table U.
- 18 -

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(1) Decide upon (based on estimated life cycle) the number T of time

periods which will constitute the planning horizon and specify the

length of the time period. For example, Table 3 was developed in terms

of one-year periods (so all cost figures to be entered here are treated

on an annual basis) and a ten-year planning horizon (T = 10).

(2) Prepare and enter estimates of each initial cost element, and for

the corresponding cost elements recurring in every time period. Certain

of these estimates will require projections for the number of operator

positions in each time period; such projections can be found using the

queuing model tables of Appendix B. See Section 4 for a description

of the use of these tables. In addition, certain unit costs (e.g.,

operator personnel costs) will increase over time as a result of inflationary

forces. For example, a 6 % annual inflation factor might be assumed

applicable for operator salaries. One simple way to account for such

rises is to assume a constant "inflationary" factor of i percent per

year over the time horizon. Then the unit cost in period k + 1 will

equal that in period k multiplied by 1 + i , and thus the total

stream of costs over time can be successively calculated by beginning

with period 1. For example, an inflation factor of 6% might not be

uncommon for operator salaries and benefits, while a rate of 3% might

apply to telephone costs; on the other hand, computer hardware costs may

remain relatively constant over the time horizon.

(3) For every cost element, combine the initial costs and the recurring

costs. One method of doing this is by converting the stream of recurring
21

costs into a present value, taking into account an acceptable annual

\ discount rate (e.g., 9-10$). More specifically, if d is the discount

rate, then a cost of C dollars in time period 1 is equivalent to a

cost of C/(l+d) dollars in time period 0 , the present. In a similar

way, C dollars in time period t is equivalent to C(l+d) ^ dollars

in time period 0 . Therefore, the stream of recurring costs can all he

converted into equivalent "period 0" costs and then combined with initial

costs. The total (period 0 ) cost TC(j) for cost element j is then

T

-t

TC( j ) = IC + Z RC( t ) [l+d] , t=l

where IC denotes the initial cost of element j and RC(t) is the

recurring cost of element' j in period t=l,...,T. There are other

ways of combining initial and recurring costs (see [12], for example).

However, the present-value method seems most appropriate, since it

allows one to compare total costs (capital, operations and maintenance)

over a specified time period in terms of constant dollars. It is

customary to convert costs to the "present value" to facilitate comparisons

with other different investment opportunities. Also, by considering

total (present value) costs, one can easily determine how many years

will be required before two alternative systems are comparable (if at

all) in terms of total cost.

(4) Compute the total system cost by summing the total cost elements

found in Step (3). Then the total cost for the automated system,

TC ( A )

,

based on the eleven cost elements of Table 1, is given by
- 22 - 11

TC(A) = l TC(j) . j=l

A similar procedure applies when calculating the total cost TC(M) for

the manual system.

Thus the total or aggregated costs can he determined for the two

systems over the selected time horizon. There are several possible uses

for these quantities. First, they give an idea of the magnitude of the

total cost for either system over the selected or planning horizon.

These costs are given in terms of present dollars, and represent how

much the proposed system would cost if all expenditures were made today.

In addition, by considering the difference in total cost between the

systems, one obtains a cost differential (in present dollars) which can

be compared with the benefits accruing from automation. Section 5 discusses

in more detail the nature of such benefits and how a comparison of

benefits and costs can be made. Furthermore, as mentioned earlier,

these cost figures can be obtained over different time horizons, and so

it can be determined when (if at all) the costs of the two systems would

become comparable.
- 23 -

3. QUEUING ANALYSIS

3.1 Queuing Model

The representation of the transit information process using a

queuing model can be described by reference to Figure 1. The physical

system consists of some number s of servers (corresponding to operators

in service) and a queue of maximum length Q (corresponding to a fixed

number of telephone hold positions). Arriving calls randomly enter the

system at an average rate of X per hour; if a server is free, an

arriving call is serviced immediately, but otherwise the call joins the

current queue. (if the queue is full, an arriving call receives a busy

signal, i.e., it cannot enter the system and so the call is "lost".)

While in the queue, a particular caller will renege—i.e., become tired

of waiting and leave the system without service—according to a negative

exponential probability distribution with parameter a . When a server

becomes free, he/she services the next caller (assumed to be chosen from

the front of the queue). The length of time required for service varies

from call to call, and it is assumed to follow a negative exponential

probability distribution with mean l/y . The parameter y can also

be interpreted as the rate (average number per hour) at which a busy

server processes the calls.

Under the above assumptions, the system will attain an equilibrium

condition, or steady state . The queuing calculations to be described

will all be based on the analysis of this steady state. In particular.
- 2k -

RENEGING

CALLERS

COMPLETED

SERVICES

ARRIVING

CALLS

Figure 1

Queuing Model of Transit Information Process
- 25 -

these calculations make it possible to estimate the probabilities P

m

of finding m calls in the system at any time instant. These probabilities

can be used, in turn, to calculate"

1' important characteristics of the

queue , namely

(a) The percent of calls unable to enter the system because the

queue is full (calls "turned away" by a busy signal)

(b) The average number of calls per hour unable to enter the

system

(c) The average number of callers in the queue at any instant

(d) The average length of time a person who is not immediately

serviced must wait in the queue

(e) The average length of time required for a call to pass through

the system (i.e., the total call length time)

Several of the above outputs have been tabulated for a variety of

different input specifications and are displayed in Appendix B. The use

of such tables is described in Section 4.

3.2 Input Specifications

This section discusses the various input specifications needed for

the queuing analysis and how the necessary parameters can be estimated.

There are four basic types of input data that must be provided to the

queuing model, and these are discussed in turn below.

The mathematical details of such calculations are described in Section 3-4.
- 26 -

The first type of input consists of projected call arrival rates

A for each of the years i = 1,...,T comprising the time horizon (e.g.,

the amortization period of the system). The quantities X are measured

in units of number of calls per hour, and they represent the average

rate at which calls are attempting to enter the system. Since attention

is really focused on the peak period during the day, it is probably

reasonable to assume that over this period the arrival rate is constant.

In order to estimate these quantities X it would be useful to have

data pertaining to the current year: more specifically, data on the

number of calls attempting to enter the system per hour. Two difficulties

may arise with currently available data, and may thus necessitate a

special data collection effort.

First, existing data may only indicate the number of calls per week

of operation. Such quantities can be converted into a peak hour rate by

assuming, for example, that 60 percent of the calls arrive during the

busiest eight hours of a day. Then an estimate for the peak hour arrival

rate A is

A = (R/7) x (. 6 0/8) ,

where R is the measured number of calls per week. Off-peak rates may

be calculated similarly. The factor 7 in the above expression is used

to convert the weekly rate into a daily rate. In some cases a different

factor (such as 5) might be more appropriate, depending on the number of

days per week that the information facility operates.
- 27 -

A second issue that arises in using existing arrival rate data is

that the estimated rates should include all calls attempting to enter

the system, not just those calls which succeed in entering. In other

words, the value for X should include calls that receive a busy

signal. However, the most widely available hardware measures the number

of calls actually received. There are two possible approaches for

dealing with this difficulty. First, one could assume that all callers

receiving a busy signal will call back again at another time, and so the

total number of calls received will ultimately be the same as the number

of callers attempting to utilize the system. This approach suffers from

the drawback that while the number of callers serviced will not be

affected by this procedure, the total number of calls will be affected,

as will the statistical distribution over time of arrivals, and so the

input stream can no longer be guaranteed to follow strictly the Poisson

process [3], assumed above. Moreover, a caller who finds a busy signal

may not return the call during the same period; since the analysis

treats periods within a day separately, such a caller would be effectively "lost" to our representation of the system.

An alternative approach is to estimate the proportion of calls -lost

(or assume a reasonable proportion for calls receiving a busy signal).

The measured arrival rate during any period can then be "inflated" by

this factor. The above method is only approximate. However, if a more

accurate determination were required, then an iterative refinement

process could be used to estimate this factor accurately. Namely,

an initial guess for the factor is made, then the queuing model is

employed (using data appropriate for current operations) to produce an
- 28 -

estimate of the proportion of lost calls. This new estimate is used to

modify the arrival rate and the model is again used, resulting in an

improved estimate for the proportion. The procedure is iteratively

applied until successive estimates agree, whereupon an appropriate

estimate for the proportion p of lost calls will he available. The

* measured arrival rate A , which excludes lost calls, can then be

modified to produce an estimate X , which will include lost calls;

* namely, A = A /(l-p).

Alternatively, if one assumes that the reneging rate a is effectively zero, a mathematical expression can be derived which gives a good

approximation to the true value A , based on the observed arrival rate

A . This analytical expression (the origin and use of which is described

further in Appendix A) approximates rather closely the actual relationship

* between A and A over the range 1 <_ s <_ 60, where s denotes the

number of servers (operator positions) in the system.

Finally, recall that estimates must be provided for A^ in each of

the years to be used in the analysis. If service-demand projections for

future years are not available, it may not be unreasonable to assume'

that the call rates increase at some rate a% per year (or time interval)

for the duration of the time horizon being considered.

The next type of input information needed for the queuing analysis

consists of the service rates y^ for each of the years. It is probable

that these service rates do not change significantly from year to

year 5 and so only their common value y =

y_^ needs to be
- 29 -

estimated. (if, however, there are strong reasons to believe that the

service rate will change appreciably over time, then appropriate estimates

y^ should be provided for each year. ) The required value y describes

the service rate per busy server (in units of number of calls serviced

per hour). Since l/y gives the average duration of a service in

hours, it is probably easier to estimate this quantity than to estimate

y directly. Existing data on the average duration of time to service a

call can then be used to give an estimate of l/y , and thereby a

service rate y^ for the manual system. In order to estimate the

service rate y^ for the automated system, it will be assumed (see [4]

for justification) that the computerized system can reduce the duration

of a call by 20% compared to a manual system. Therefore y^ can be

estimated using y = 1.25 y^ . This relation was used in developing

the tables in Section 4, but is not a basic requirement of our analysis

method.

A third input parameter is the reneging rate a , also assumed to

be constant from year to year. The value a measures in a sense the

number of reneging calls per hour. More precisely, 1/a represents the

average length of time (in hours) a person will wait on hold before '

leaving without service.

The tables in Appendix B correspond to the case in which a = 0.0

(that is, no reneging occurs). Similar tables can be provided for other

values of a , but have not been included here. Indeed, the case

a = 0.0 is computationally simpler, the volume of tabular material

becomes excessive if several values of a are used, and no data are

available on the range of reasonable values for this parameter. Moreover,
30 -

the system size figures resulting from the use of the tables for a = 0.0

overestimate manpower requirements and thus provide a conservative

overestimation of costs.

In addition to the input parameters described above, a value needs

to be provided for the maximum length of queue (number of hold positions)

that can be accommodated in the system. This value Q, may be specified

by the hardware of the Automatic Call Distribution System being used.

For ease of display and use, the tables produced in Appendix B have as

input the total number of lines (Q + s) with Q varying usually

between s and 2s, the range found in [7] and used in [12].

3. 3 Model Assumptions

Use of the queuing model described in the preceding sections

requires that certain assumptions be fulfilled, if not exactly then at

least approximately. Certain other simplifying assumptions have been

made for the purpose of computational and expository convenience, and

these can be changed without affecting the model's validity. In this

section the intrinsic assumptions of the model (assumptions which must

hold, at least approximately, for the model to be valid) are categorized

under the following four components of the queuing system.

(l) Input Calls . It is assumed that the input calls form a

Poisson process (see [3] for a mathematical description of this process).

Such a process describes a stream of arrivals that occur in a "purely

random" manner, i.e., the time until the next arrival is completely

uninfluenced by when the last arrival occurred. In practice, it has
31 -

leen found [2], [5] that the pattern of arrivals at a telephone system

is quite closely approximated by a Poisson process. Accordingly, such

a process would also seem to be appropriate for arrivals at a transit

information facility. It should be noted that in a Poisson process, the

arrival rate X is assumed to remain constant over the period of interest.

Since the present analysis concentrates on the period of peak demand for

the information facility, this requirement of a constant arrival rate is

likely to be fulfilled.

(2) Queue Discipline . It is supposed here that the queue operates

on a "first come, first served" basis; that is, earlier arrivals to the

system will always be served before later arrivals. Such an

assumption will be satisfied for any reasonable policy of handling

incoming calls. In particular, the use of ACDS will ensure that this is

the case. Employing ACDS -also has the advantage of allowing a prerecorded message to be played to callers as they enter the system. This

message can, for example, serve as a screening device for callers

(routing callers to other transit property phone numbers if they are

seeking other than itinerary information) or to alert callers as to the

format for information requests.

(3) Service Times . The queuing model assumes that the service

times (times for an operator to answer completely a request) are distributed according to a negative exponential distribution with parameter

y . This particular form of probability distribution (as well as the

form of the input call distribution) is the "traditional" form that has

been assumed in order to facilitate mathematical analysis. In fact,

under these assumptions the steady-state probabilities for the queue can
32 -

be given in an explicit form (see Section 3.M* With other assumptions

about the arrival and service time distributions, the analysis becomes

more difficult, and it is unlikely that there are closed-form expressions

for measures of system performance, such as queue length, waiting time

and lost calls. While the arrival pattern of calls conforms quite

closely to a Poisson process, the assumption of a negative exponential

distribution for service times requires careful consideration. In

general, a negative exponential distribution is appropriate when a large

number of calls require short service times, and a smaller number of

calls require longer service times [2], In the present circumstances,

this may only be approximately true, since there is a minimum service

time for each call; for example, it does not appear possible for a call

to require less than (say) 15 seconds of service time. By contrast, the

negative exponential distribution would predict that short calls of

(say) 0-15 seconds are quite probable. Perhaps a more accurate (but

less tractable) assumption about service times would be that the length

of a service in excess of, say, 15 seconds follows a negative exponential

distribution.

Since the queuing analysis is to be used mainly for comparing the

two systems, strict adherence of service times to a negative exponential

form may not be necessary. In fact, the assumption of exponentially

distributed service times provides in a sense a conservative approach to

comparing the systems, since it tends to underestimate the contribution

of an automated system. The reason for this is that the exponential

distribution puts relatively more emphasis (higher probability) on calls

of a short duration compared to calls requiring a longer service time.
- 33 -

But the automated system ought to perform better relative to the manual

\ system when there are more time-consuming (i.e., more complicated)

calls. Finally, it should be noted again that it is necessary to confine

the queuing analysis to a period during which the parameter y for the

service time distribution will be (approximately) constant over the time

span of interest.

( k ) Reneging . The general queuing model assumes that the time an

individual caller will wait before reneging follows an exponential

distribution with parameter a . This assumption is imposed mainly for

mathematical tractability , and verification would require the collection

of data on actual customer behavior. Again the assumption of exponentially distributed reneging times implies that shorter reneging times

are more likely than longer ones. Therefore, this situation will obtain

in practice to the extent that early impatience of callers outweighs

their patience. A convenient baseline for comparing the manual and

automated systems is the case a = 0: that is, when no reneging is

"allowed” by the model. This case provides worst-case estimates for

many of the quantities characterizing the level of service. For example,

average waiting time will be longest when reneging is not allowed, since

reneging callers reduce the waiting times of all callers following them

in the queue.

3.4 Mathematical Description

This section will detail the mathematical calculations used to

analyze a transit information facility. Specifically, the facility is

modeled as a multiple server queue having a maximum queue size. Arrivals
- 34 -

are assumed to be generated by a Poisson process, while service times

and reneging times are assumed to be governed by an exponential distribution. For notation, we define

A = arrival rate (number/hr),

y = call service rate (number/hr),

a = reneging rate (number/hr),

s = number of servers,

Q = maximum queue size,

M = s+Q = maximum number allowable in system,

r = A/y .

Then the steady-state probability P^ of finding m callers in the

system can be calculated using the recursive relations

for 1 < m < s

P

Pm-1 for s < m < M

m y s + (m-s )a

and the fact that the probabilities must sum to l:

M

E P. = 1 . _ l

The proportion of callers that find the system full (i.e., the proportion

of lost calls) is just P

M

and the expected number of calls lost per
- 35 -

hour is then AP^ . In addition, it is straightforward to calculate the

expected number L of calls in the queue as

Q.

M

L = Z (i-s)P. . q. i

Once has been found, then the average waiting time for a

caller who does not find a server free on arrival can be determined [ 1 ]

W

q.

L

5

M

where r = £ P. is- the probability that all servers are busy.

i=s

Finally, the average length of time W spent by a caller in the system

(including both queuing and service times) is given by

W = I + a

y a(i-pm

)

It should be noted that the values for and W include the queuing

times for reneging callers, as well as callers who are ultimately served.

These calculated quantities (P.., AP.., L , W and W) can serve as ^ M M q q

measures of the level of service provided by the queuing system (transit
- 36 -

information facility). Therefore, such measures can he used either

individually or collectively to determine the minimum number of servers

that will be required in order to meet certain service level standards.

Further details on this procedure are given in the following section.
- 37 -

4. MANPOWER ESTIMATES USING THE QUEUING MODEL

Once the necessary inputs to the queuing model have been specified

(Section 3.2), the mathematical calculations of Section 3.4 can he

employed to determine values for various characteristics of a queuing

system with those input parameters. The particular queue characteristics detailed in Section 3*1 provide measures of the amount of congestion

in the system, with special emphasis on the number of lost calls, waiting

time in the queue and total transaction time. If in addition certain

minimum performance standards are prescribed for these measures, then

estimates can be made for the number of operator positions required to

achieve such standards.

For ease of application, tables have been prepared which give queue

characteristics for transit information facilities with various input

specifications. These tables are listed in Appendix B for the case a

= 0.0, according to arrival rate X , service rate y , total number

of telephone lines L and number of servers s . From these latter

two parameters one can calculate the maximum queue length Q = L-s

Three different queue characteristics are listed in each table. For

example, the first table refers to an arrival rate X = 200 calls/hour

and a service rate of 15 calls per hour. The entries corresponding to 40

lines and 13 servers are

(a) number of hold positions = 27

(b) percentage of lost calls = b%

(c) average waiting time = 286 seconds

(d) average time in the system = 500 seconds
- 38 -

If we require the proportion of lost calls to he at most 1%, then for

the given specifications s = l4 operator positions will he required.

If, in addition to this requirement, we insist on an average waiting

time in the queue of no more than 100 seconds, then s = l6 operator

positions will he needed.

A user of this model can choose any single criterion or set of

criteria for minimum performance levels, and such choices will enable

the determination of a required number of operator positions s . Such

values for s can he determined for both the manual and automated

systems (which will differ only in that service rates follow y^ = 1.25 y^),

and these values can then he entered into the appropriate cost calculations

described in Section 2. Since the difference in number of operator

positions between the two systems is of major interest, the exact

performance levels which are set may not be crucial. However, levels

should be set which are reasonable in light of current operating policy

and desired quality of service. As guidelines here, we suggest as

typical using a maximum of 1% calls lost and a maximum waiting time of

one to two minutes. The sensitivity of the number of operator positions

to various design factors, such as the arrival rate, the service rate,

and the number of hold lines should also be taken into account, so that

variation from the design level will not adversely affect the system

performance.
- 39 -

5 • BENEFITS

The previous sections have described how to estimate and compare

the costs of an automated and a manual transit information system. It

is not at all clear that automation will necessarily cost less, over its

useful lifetime, than a comparable manual operation. In addition, even

if an optimal automated system would save money, in the long run other

considerations such as union contracts or tight budgets may necessitate

selection of a less efficient system, in which case the theoretical

savings predicted by the cost model would not be attained. On the other

hand, there are benefits associated with an automated transit information

center other than direct cost savings; there are also potential disadvantages to this type of automation. This section explores these concepts

in detail.

We will first describe, and quantify where possible (using the

queuing model of Section 3), the system improvements resulting from

automation. Next we will discuss the benefits from these improvements

and note who is benefited by each improvement, an important consideration

since a transit company may be unwilling to fund automation if it does

not perceive adequate benefit to the transit property. Also the public

may be reluctant to see tax dollars used to pay for a system which does

not provide much public benefit. Automation must either provide sufficient

benefit to the transit company to underwrite its cost, or provide additional

benefits to the public which justify public subsidy.

Although we will indicate methods for quantifying the levels of

system improvements, we will not attempt to relate them directly to
- 1+0 -

system costs by assigning dollar values to the benefits, since such

values could differ widely from system to system and would tend to be

arbitrary. Rather, for decision-making purposes, we prefer to provide

the levels of system improvement; these then can be evaluated directly

on their own separate merits.

5 • 1 Improvements Resulting from Automation

Table 5 provides a list of improvements resulting from a well-designed

and implemented automation of the transit information center. They

provide benefits both to the transit company and to transit system

users. These include improved service and productivity resulting from

the faster response of the computer, increased reliability and consistency

because the computer always supplies the same answer to the same question,

a reduction in the training required for operators, the ability to

rapidly incorporate changes in the transit system into the data base,

the capability of automatically gathering statistics about the operation

of the information center, and the development of a data base which can

be used for such other purposes as scheduling and automated printing of

schedules.

These improvements, to be discussed more fully in the following

sections, bring benefits to the transit property in the form of increased

efficiency and goodwill, to the transit system user in the form of

easier access to information and greater confidence in the response, and

to society in general to the extent that easier access to transit information increases transit ridership. There are also benefits to society

as a whole from the automation of a transit information facility, but
- hi -

TABLE 5

Benefits from Automation

Improvements Benefits

To Transit

Company To Users

Shorter service time increased productivity quicker response

Shorter wait time better service less time used

Fewer waiting fewer lines less frustration

Fewer lost calls better service less frustration,

fewer recalls

Increased reliability better service more confidence

Increased consistency better service more confidence

Less operator training cheaper, can use new

people sooner more confidence

Rapid response to changes better service,

flexibility

more confidence

Management Information

System

better evaluation

of performance —

Data Base available for

other users increased efficiency

and productivity
- 42 -

they are somewhat diffuse and difficult to measure and assess. They

include reductions in road congestion, pollution and energy usage

resulting from any increased patronage because of better availability of

information on transit service. Improvements in information dissemination

which lead to increased patronage may also result in expanded transit

service instituted to meet the larger demand, thus providing better

service to all users, both new and old customers. Improvements in the

access to transit information may aid local business, such as the tourist

industry or downtown stores, and might also attract new business. All

of the benefits to society suggested here depend upon a demonstrated

relationship between transit system patronage and the availability of

transit information, a relationship which has not yet been proved.

Thus, although we note these possible benefits to society as a whole, we

make no attempt to quantify them, and they are not emphasized in this

analysis. Transit information dissemination is a part of the broader

subject of marketing mass transportation.

5.2 Benefits to Transit System Users

and Transit Properties

Benefits to the transit system user are more tangible than those

to society. They include a reduction in customer frustration in obtaining

information on proposed trips: information is more accessible since the

telephone is busy less often and the wait time is shorter. A second

benefit is the time savings in obtaining information because of quicker

response and shorter wait time. Finally , increased reproducibility and

consistency of computer responses enhance public confidence and perhaps

even reduce the need for confirmatory calls. All of these advantages
- 1+3 -

make it much easier and less frustrating to obtain desired information

\ concerning transit travel, allowing more people to have access to information

and perhaps actually encouraging rather than discouraging potential

riders to seek such information. More consistent answers, while not

necessarily "better" than manually produced routes, may be perceived as

better simply by virtue of their consistency.

Benefits to the transit company from automation are many. The

first is increased productivity in a very labor-intensive operation.

Since personnel salaries have risen faster than most other costs, companies

are anxious to automate any activity which can be automated without

overwhelming expense. A second benefit from a well-designed computerized

system is ease of making changes to the data base. Either new sets of

routes and schedules or temporary changes in them, as well as special

services, can be incorporated easily and quickly, and old data can be

readily discarded (by being overwritten with new). A third benefit is

the improvement in customer relations because of the better service.

When potential riders can obtain with relative ease needed information

about desired trips, they have a better opinion of transit system service

in general. Contributing also to the improved service is the increased

reliability and consistency of responses. Callers will no longer be

disconcerted by receiving a different routing if they get a different

operator or call at a different time. The first response given is guaranteed

to be the "best" route, according to the chosen criterion; the computer

has considered, and rejected, many alternatives.

Another benefit to the transit company from automating its information

facility is a reduction in training costs, since extensive training in
- hk -

city geography and transit routes is no longer necessary. Training for

the operators of an automated center would emphasize dealing with callers,

eliciting the information to specify information requests, articulating

clearly and using methods of responding to the requests to aid the

caller's retention of information. While these functions are important

in a manual system also, training for an automated system can focus more

specifically on them, and personnel recruiters can emphasize these

characteristics in hiring operators.

Two ancillary benefits resulting from automation of the transit

information center are (a) the availability of continuing statistics

about the operation and (b) the availability of the data base for other

uses within the transit company. The computer program package should

include, at very little additional expense, programs to collect and

print, upon demand, various statistics about the operation of the center,

such as how many calls are answered at each station during specified

intervals of time, the size of the hold queue (perhaps average, maximum

and minimum or fraction of the time the queue is full), and the length

of calls. In addition, statistics can be generated on the operation of

the transit system, such as the trips requested, what areas (origins and

destinations) are represented, the routes chosen, and the times of day

requested. Such statistics can be used to evaluate performance of the

operators, to examine causes of any problems, to suggest geographic

areas in which to focus information campaigns, to provide a partial

basis for establishing new routes or changing old ones, and to indicate

new names or addresses to be added to the geographical location data

base. Once the data base has been encoded for use by the transit information

system, it is available for other uses such as scheduling of vehicles
- 1+5 -

and crews, or automated printing of schedules. It is usually the case

that the computerization of one activity leads naturally to automation

of other related functions, and the availability of a common data base

facilitates this process.

A final hoped-for benefit from automating the transit information

center is increased patronage resulting from the greater accessibility

of schedule and routing information. This is based on the assumption

that some potential transit trips are not now being made (at least by

mass transit) primarily because of lack of easy access to routing and

|

schedule information, and that the improved accessibility of that

information will encourage the trips to be made by mass transit. Although

researchers have attempted to establish a positive relationship between

the investment in improved telephone transit information provision and

increased ridership, they have been unable to do so [ 7]Thus we note

j- increases in patronage as a potential benefit of automation, but do not

focus on it as a prime justification for implementation.

I

5. 3 Measuring the Benefits

1 As noted in the discussions above, many of the benefits or possible

disbenefits of automation are difficult to measure because of the

subjectivity of the assessment and the differences among transit properties

and cities. We can, however, use the queuing model in Section 3 to

quantify the degree of automation-induced improvement m such factors as

the number of lost calls, the average waiting time in queue, the queue

length and the time to service a request. The levels of improvement can

then be compared to any cost differential which was calculated using the
- 46 -

cost model of Section 2. Subjective assessments by the local transit

people and any other interested parties (local Department of Transportation,

citizen groups, etc.) can then be used in assessing the value of the

improvements as compared with any incremental costs of the automated

system over a manual system. If the automated system can be justified

solely on the basis that it costs less than a manual system handling

the same number of calls, quantification of the levels of performance

improvement will not be as important in the evaluation as if the automated

system is more costly, or the breakeven point is in the distant future,

or the cost difference though favorable is smaller than the uncertainty

in the values of the cost estimates.

5 . U Disadvantages to Transit System Users

and Transit Properties

There are, however, some potential drawbacks for the transit system

user from automation. The computer routing is based on one criterion

(or at the most a small set of criteria), and some special requests may

either have to be ignored or be answered manually. Examples of such

requests are requests for routes which avoid certain areas of the city

because of fear of crime, or for "triangle" routes in which the user

wants to stop at intermediate points to run an errand. The operators

could come to rely on the computer to the point that they will be unable

to supply intelligent variances to the routes provided, for example

advice to walk up two blocks to save an additional fare.

Easier access to the automated information system may increase

demand on the telephone information system, even if improvements are
-hi -

\ . made in other methods of information dissemination. (In a similar vein,

the telephone company notes persisting calls to the information operator

despite the availability of telephone books and even charges for the

service.) Initial improvements in access to information may thus lead

only to a newly saturated system with all the long waits of the old one,

though more customers are served.

A final drawback to a computer-aided information system may be that

the routings it provides depend on better on-time performance by transit

vehicles than is actually attained. This may be somewhat alleviated by

using minimum transfer times in computing the routes. However, the

public may still perceive deviations from schedule more vividly when

computerized transit information responses are very explicit. Some of

these disadvantages, such as the improvement- induced demand, may also

apply to an improved manual system. Although all of these (lack of

flexibility in route selection, increased demand for telephone information,

and heightened perception of system delays) are potential drawbacks to

automation, we do not know how to measure their effects, and their

magnitudes are likely to vary greatly from city to city and from transit

property to transit property. Effective management may be able to

minimize or eliminate some of these disadvantages. Potential benefits,

however, appear to outweigh the possible disadvantages.

Automation of the transit information center may also have disadvantages to the transit company . First among these is that increasing

the accessibility of transit information may increase the demand for

that information. As more potential riders discover that telephone
- 48 -

information service has improved, they may be encouraged to use the

service more often for more types of trips. The result could be that

the new automated system quickly becomes saturated, and service deteriorates

to a level similar to that existing before automation. More people

are being served, but they have to wait as long and face a busy signal

as often as in the old manual system.

A second disadvantage of the automated system is that operators can

tend to rely on the computer to supply answers to simple questions

which, in the manual system, they might have answered "off the top of

their heads". The result of such actions may be lengthening of the

response to some short requests. Automation is expected to speed significantly the longer calls which require more complicated routes, but may

increase the length of shorter calls, thus reducing the variation in

call length, making it depend less on the complexity of the requested

route and more on the ability of the caller to formulate his question.

Increased operator reliance on the computer will also be a great

disadvantage during periods in which the computer is not operational. To

take advantage of the reduced training costs available with an automated

system, the operators will not receive the intensive instruction in city

geography and the transit system that would be required to deal efficiently

with queries in a manual setting. Even operators previously adept in

the manual system can become so accustomed to using the computer that

their skills in a manual operation become rusty.
- U9 -

Another possible effect of automating the transit information

center is a greater public perception of delays because more patrons

know at what time the vehicle is to arrive or depart. With information

less accessible, fewer people know actual scheduled stop times, so the

system's on-time performance (or misperformance ) is not as widely known.

When told specific departure or arrival times by the telephone information

operator, riders may expect the system to adhere more precisely to the

scheduled times than actually occurs. Wider dissemination of precisesounding information about an inherently imprecise system may call

attention to system variability.

A final difficulty which may arise from computerizing the transit

information function is that changes in the on-line procedures or process

(as opposed to those in the data base) may require altering a computer

program, which can be time-consuming and expensive. The need for such

changes can be minimized by careful system design, but all possibilities

cannot be foreseen initially and some future program changes will

probably be necessary.

Again it is believed that the benefits to the transit company from

automation outweigh potential disadvantages, but no attempt has been

made here to quantify the relative merits and disutilities involved

because of their subjective nature and the differences among cities and

transit properties.
- 50 -

5 . 5 Other Transit Information System Improvements

The cost-benefit analysis presented here has focused on a comparison

between a manual and an automated transit information system, in which a

computer is used to select a "best" route. Other modifications in

methods of providing information are possible and should be considered

by any transit property planning to upgrade its information service.

Among these are several methods of improving the telephone information

service, short of using a computer. They include the use of microfiche

to speed retrieval of schedule and routing data, improvements in the

workspace of the operators and the organization of their materials, the

inclusion of a message to those waiting on hold aiding them in formulating their requests, and the channeling of incoming calls from particular

originating zones to operators with special knowledge of the transit

service in those zones. Besides considering improvements in the telephone

information service, a transit property should also examine the broad

spectrum of methods to enhance the provision of information. Examples

include wider availability of printed schedule and routing information,

more informative signs, and use of mail for specific requests or for

dissemination of schedule and route updates or changes. All of the£e

methods of improving the public's access and awareness of transit

information should be studied in the context of the specific system

being analyzed. This report has focused on evaluating a computer-aided

telephone information service, because such a change represents a wide

departure from previous attempts at improved service, but this focus is

not meant to suggest that such an approach is either the only or the

best plan for any particular situation.
51 -

6. REFERENCES

[ 1 ] Ancker, C. J. , and Gafarian, A. V. , Queuing with Reneging and

Multiple Heterogeneous Servers, Naval Research Logistics Quarterly,

10, No. 2 (June 1963 ).

[2] Cox, D. R. , and Smith, Walter L. , Queues (Methuen and Co. Ltd.,

London, 1961).

[3] Feller, William, An Introduction to Probahility Theory and Its

Applications, 1_, Third Edition (John Wiley and Sons, Inc., New

York, 1968).

[4] Gilsinn, Judith F. , Saunders, Patsy B. , and Pearl, Martin H.

,

Path Finding Algorithms and Data Structures for Point-to-Point

Trip Management, NBS Report Number NBSIR 75-676 (January 1975).

[5] Gross, Donald, and Harris, Carl M. , Fundamentals of Queuing Theory

(John Wiley and Sons, New York 1974).

[6] Gulbrandsen, 0., Cost-Benefit Analysis in Transportation, in CostBenefit Analysis, Ed. M. G. Kendall, pp. 295-302 (American Elsevier,

New York, 1971)

[ 7 ] Kienstra, William G. , and Minnick, Daniel J. , Point-to-Point Trip

Management Program (Preliminary Analysis), NBS Report Number

NBSIR 75-665 (February 1975).

[8] Marquardt , D. W. , An Algorithm for Least-Squares Estimation of

Nonlinear Parameters, J. Soc. Indust. Appl. Math., 11 ( 1963 ).

[ 9 ] Mishan, Edward J. , Cost-Benefit Analysis; An Introduction (Praeger,

New York 1971).
- 52

[10] Nicholson, R. L. R. , The Practical Application of Cost-Benefit

Analysis to R & D Investment Decisions, in Cost-Benefit Analysis,

Ed. M. G. Kendall, pp. 167-182 (American Elsevier, New York 1971)

[11] Olver, F. W. J., Asymptotics and Special Functions (Academic

Press, New York, 197^)*

[12] Wells, William R. , Public Transit Information Systems: What

Level of Sophistication? (Stanford University, May 1973)-
53 -

APPENDIX A

AN APPROXIMATION FOR THE RELATIONSHIP BETWEEN A and A*

As noted in the text, care has to he exercised in describing arrival

rates associated with the queuing system under study. The use of the

queuing model to provide manpower estimates is based on obtaining an estimate A of the rate at which calls attempt to enter the system. In

practice, however, it is more likely that one is able to measure the

* rate A at which calls actually enter the system (i.e., calls which are

not turned away by encountering a busy signal).

The fundamental relationship between these two quantities is the

equation

(1) X* = a(i-p

m

).

where P^ denotes the probability that the system will be found to be

full (i.e., all hold positions are occupied) at any given instant of

time. It is possible [5] to express P^ analytically in terms of the

parameters r = A/y, s and q (see Section for their respective

definitions) using

This expression is obtained under the assumption that reneging does not occur , i . e. , a = 0.

( 2 ) P

M

s+q

s^* s

s— 1 . q

/{ Z ri ri Z i=0 i! +

s! j=0

Given, therefore, A (along with other parameters such as s, y, q)

* it is possible using (l) and (2) to find A . In fact, these equations

* provide a direct functional relationship of the form A = g(A). However,

it is really the inverse of this relationship that addresses the major

* issue here: namely, given A , calculate the corresponding value of A. There appears to be no simple analytical way to invert the given relation-

* * ship A = g(A) in order to obtain the relationship A = f(A ) that is

sought. The object of this Appendix, then, is to present an approximation

to this latter relationship that will be reasonably accurate over appropriate ranges for the parameter values.

* / \ To begin, it is useful to study the relationship A = g(A) which is

defined by (l) and (2). Certain observations relevant to this relationship

can be readily established. First, as A -> 0 it can be shown, using (l)

* and (2), that g(A) ^ A. In other words, for "small" values of A, A and

A will be nearly identical; intuitively, this is reasonable since for
small X, the turning-away of calls will be rare. Second, as X -> °° it can be verified that g(x) -*• sy. That is, for very large arrival rates

* the system essentially becomes saturated, and the rate X at which

calls can enter the system approaches the maximum effective rate

Sy at which calls (from s busy servers) can leave the system. Third,

it can be verified that the function g(X) is a (strictly) monotone

increasing function of X, and g(X) <_ X holds for all X. The above

* observations show that the relationship expressing X as a function

* of X, X = g(X), has the general form shown in Figure 2. This figure

* displays the horizontal asymptote at X = sy as well as the (dashed)

* line of equality X = X.

Consider now the functional value g(sy) at X = sy

Then, using

(l) and (2), it is straightforward to deduce that

(3) g( sy ) = sy [l-{q + <}>(s)}

1

],

where

(4)

We first find an approximation to <j>(s), as a function of s, and this

will provide in turn an approximation to g(sy).

<f>(s) =

s

s i =0 i

An asymptotic expansion 1 of <j>(s) as s

00 can be obtained by using

the Euler-Maclaurin formula [ll]. Essentially this asymptotic expansion

takes the form

<f>( s % ? 1

+ ^ 2

S 3 + ^4 S

with specific positive values assigned to This particular

approximation is quite good as s becomes large, and is accurate for

values of s even as small as 10. In order to obtain a best approximation

over the range s = 1 , 2,...,60 (which covers reasonable values for the

number of servers), a nonlinear least squares fit was obtained using

a computer routine2 developed at NBS. This approximation A(s) is of

the specific form

(5) A(s) = .606 + 1.26 s' k " + .133 s “* 251

^We are indebted to Dr. F. J. Olver (University of Maryland, and NBS)

for providing a derivation of this expansion.

This data-fitting routine has been implemented by Dr. James Filliben

of the NBS Statistical Engineering Laboratory. This implementation is

based on a computer program of A. J. Miller (CSIRO, Sydney, Australia)

and incorporates features of the Levenberg-Morrison-Marquardt method [ 8 ]
- 55 -

Figure 2 * The Functional Relationship X = g(A)
and yields a residual standard deviation of 0.000013; such a small value

for the residual standard deviation indicates an extremely good fit of (5)

to the function (4) over the range of interest.

Given equation (5), it is now possible to tackle the original problem of

* approximating the relationship X = f(X ). What is required then is a

* * * function h(X ) which is asymptotically equal to X as X -* 0, and which

* approaches °° as X approaches sy from below (see Figure 2). In addition,

such a function could be made to have the ordinate X = sy when the

* abscissa Is X = g(sy). Since the calculation of g(sy) requires knowing

<J>(s) and since the latter is somewhat involved to calculate, we use instead the approximation A(s) for <j>(s). Accordingly, we require that

h(x) have the value sy when x = sy[l-{q + A(s)} ^ ]One of the simplest

functional forms for h that will meet the above requirements is given by

(6) h(x) = x + B(s,q)x

2

/(sy-x)

,

where

(7) B(s,q) = [q - 1 + A(s)]"2

.

It is direct to show that h(x) does in fact satisfy the three properties

mentioned above. For example, it is clear from (6) that h(x) x, as also

required.

In order to illustrate this procedure, consider the case when s=l4,

q=20 and y=38. Direct calculations give

A(l4) = 5.377,

B(l4,20) = (24.377)

-2

h(: = x + 0.001683x'

532-x

0.001683,

* Suppose for example that one observes the value X = 518. The last

equation above can then be used with x = 518 to estimate the true

arrival rate X: namely, X - h(5l8) = 550. As a matter of fact, the true

* value of X corresponding to X = 518 is found to be X = 549. As

* another example, when X = 472 then X ^ h(472) = 478, while the true

value is X = 475*

In the above cases, the approximation provided by (5)-(7) gives

quite close agreement to the true values for X. One caveat must be

borne in mind, however, when using this type of approximation to the

* * *

X = f(X ) relationship. Namely, since X-*» as X +sy, the function f(X ) becomes extremely steep in the vicinity of sp . That is to say, small
- 57 -

* changes in X in this vicinity will produce disproportionately large

changes in the corresponding value of X.' The approximation that has

* been given here is made to agree closely with the true X = f(X ) relation

for values of X that are not too large. It cannot, however, approximate

* * to any reasonable accuracy the steep behavior of f(X ) as X becomes

close to sy.

In practical terms, this means that the approximation defined by

* (5)-(T) should not be used when X - sy. More specifically, it has been

* found that if X > (.95)sy — i.e., if X is within 5 l of sy — the above

* approximation will not be accurate. As a matter of fact, if X is really

* as close to sy as this, then the true value f(X ) is so sensitive to

* changes, in X that accurate estimation of X is in principle very difficult.

* The reason is that if X is sufficiently close to sy then just the

* uncertainty in the measurement of X is enough to create an extremely large

uncertainty in the true value of X. In such a case (which corresponds to an almost complete saturation of system capacity), the true value of X may be

unobtainable to any reasonable accuracy. Fortunately, this type

of situation (near-complete saturation) is not expected to occur in

actual applications.

To summarize, a reasonably accurate approximation (over the range of

parameter values of interest) has been provided here. The basic procedure

is as follows.

1. Determine appropriate numerical values for s, y and q.

2. Calculate A(s) using (5), and then B(s,q) using (j).

*

3. Given any observed arrival rate X for calls which successfully

* * enter the system, compute h(X ) using (6). The value h(X ) then

provides an estimate of the true arrival rate X for calls attempting

to enter the system.

Finally, it should be emphasized that the above approximation is based

upon the assumption of no reneging in the queuing system. Unfortunately,

it does not appear possible in any straightforward way to extend the

procedure given here to the case with reneging: i.e., when a > 0.
- 58 -

APPENDIX B

TABLES FOR USE IN ESTIMATING MANPOWER REQUIREMENTS
TRUE

ARRIVAL

RATE

(LAMBDA)

=

200

PER

HOUR

VALUES

IN

EACH

CELL

ARE:

(1)

NUMBER

OF

HOLD

POSITIONS

(2)

PERCENTAGE

OF

LOST

CALLS

(3)

AVERAGE

QUEUEING

TIME

(SECONDS)

(4)

AVERAGE

TIME

IN

SYSTEM

(SECONDS)

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ARRIVAL

RATE

(LAMBDA)

=

200

PER

HOUR

VALUES

IN

EACH

CELL

ARE:

(1)

NUMBER

OF

HOLD

POSITIONS

(2)

PERCENTAGE

OF

LOST

CALLS

(3)

AVERAGE

QUEUEING

TIME

(SECONDS)

(4)

AVERAGE

TIME

IN

SYSTEM

(SECONDS)

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60

60

60

60

60
TRUE

ARRIVAL

RATE

(LAMBDA)

=

200

PER

HOUR

VALUES

I

NEACH

CELL

ARE:

(l)

NUMBER

OF

HOLD

POSITIONS

(2)

PERCENTAGE

OF

LOST

CALLS

(3)

AVERAGE

QUEUEING

TIME

(SECONDS)

(4)

AVERAGE

TIME

IN

SYSTEM

(SECONDS)

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NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 77- 1253

2. Gov't Accession

No.

3. Recipient’s Accession No.

4. TITLE AND SUBTITLE

Cost/Benefit Analysis of Automated Transit Information

Systems

5. Publication Date June 1977

6. Performing Organization Code

7. AUTHOR(S)

D. R. Shier, J. F. Gilsinn

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Pro ject/Task/Work Unit No. 2050^02

11. Contract/Grant No.

12. Sponsoring Organizatityi Name and Complete Address (Street, City, State, ZIP)

Urban Mass Transportation Administration

Department of Transportation

Washington, D.C. 20590

13. Type of Report & Period

Covered

Final

14. Sponsoring Agency Code

i 5. SUPPLEMENTARY NOTES

6. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

This report discusses the costs and benefits associated with automating the

route-finding portion of a telephone transit information system. The various costs

of implementing such a system are categorized and compared with those of a manual

system over an appropriate time span using a present value approach. A queuing

model, described in the report, is used for computing manpower requirements of the

two systems, manual and automated. Outputs of the queuing model for a wide range

. of input parameters are tabulated in an appendix. Benefits from automating transit

information route-finding are discussed^ and measures of performance improvement

available as output from the queuing model are provided.

i

!j7. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Automation; cost/benefit ; models; queuing; telephone systems; transit information;

transportation.

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(THIS REPORT)

21. NO. OF PAGES

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22. Price

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