calculation of performance and fleet size in transit … · web viewdepartment of mechanical...

Journal of Advanced Transportation, 16:3(1982)231-252.

Calculation of Performance and Fleet Size in Transit Systems

J. Edward Anderson1

The paper provides a consistent, analytic approach to the calculation, from the demand ma-trix, of parameters needed to analyze the performance and cost of transit systems. It covers all types of transit systems, including the new automated systems. The basic analysis applies to loop systems, which include those collapsed into line haul systems. We then extend it to ap-ply to all types of network systems in which vehicles may transfer from one line or loop to an-other. The novel features of the paper lie in 1) the layout of the computations in a straight-forward, ordered way, 2) the computation of vehicle dwell times in stations from loading rates, 3) the use of the Poisson distribution to estimate and show how to shorten the passenger wait time in off-line stations, and 4) the simplicity of the means of extending the results to net-work systems.

Contents

Page1 Introduction 12 The Demand and Path-Length Matrices 33 Passenger Flows 44 Average Trip Length 55 Operating Headway 56 Station Dwell Time 67 Minimum Safe Headway 78 Round-Trip Time 89 Fleet Size 910

The Revised Time Headway, Round-Trip Time and Load Factor 10

11

Average Trip Speed, Trip Time, and Stops per Trip 11

12

Vehicle Miles per Hour and per Day 11

13

Average Peak-Hour and Daily Load Factor 12

14

Station Wait Times 13

15

The Travel-Time Matrix 14

16

Extension to Networks 15

1 J. Edward Anderson is professor and director, Industrial Engineering Division. Department of Mechanical Engineering, University of Minnesota. Minneapolis, Minnesota.


17

Summary Remarks 18

18

Notation 18

19

References 20

1. Introduction

Economic analysis of transit systems requires calculation of patronage and direct costs, which, in turn requires calculation of travel times, station wait times, fleet size, and vehicle-miles per day. Represent-ing system performance requires that one calculate additional parameters. The purpose of this paper is to derive and present for easy reference equations needed for the above calculations. The methodol-ogy presented builds on the work of Anderson (1978), but as a result of additional stages of appli-cation to practical problems, extends and simplifies it. The development of the methodology was the re-sult of interest in understanding the characteristics of alternative types of automated guideway transit sys-tems, but it is also applicable to all types of transport systems. Although all detailed analyses of transit systems have required similar calculations (Manheim (1979)), they have usually been pro-grammed on computers in ways that mask their general applicability. Reducing required performance calculations to simple analytical expressions makes application much more rapid, clearer and easier to teach, while simplifying system comparisons—all of which advances the art of analysis of transit systems.

The inputs to the calculations are the demand matrix and the path lengths. Computation of de-mand requires knowledge of travel times and wait times; thus, the process is iterative: we must first esti-mate travel and wait times based on past experience, then calculate and use patronage analysis to refine the travel and wait times, etc. The operating headways computed from line flows must, of course, be greater than the minimum safe headway. The paper summarizes major results of the theory of minimum safe headway, which is treated fully in the literature (Anderson (1978), Irving (1978), McGean (1976)).

Calculation of trip times and minimum safe station headway requires the station dwell time (the time the vehicle dwells in a station). Usually the station dwell time is assumed as a fixed value; however, it depends on station demand and headway. Therefore, I have derived formulas for station dwell based on a more fundamental factor: loading rate per vehicle.

In most systems, the station wait time (the average time a person waits for a vehicle) is simply half the sta-tion headway; however, in a true personal rapid transit system, it is possible to reduce the station wait time substantially by arranging for one or more empty vehicles than needed to meet average rush-period de-mand. Indeed, the way such a system would be operated, each station would store as many empty vehicles as possible throughout the day. Only at the busiest stations, where the station headway is very short, would there be no extra vehicles stored during the rush period. I have estimated the station wait time on the as-sumption that passengers arrive randomly at a fixed frequency, that is, they are Poisson distributed.

The parameters derived and presented are the following:

Station flows

Total system demand

2


Line flows and weighted average line flow

Passenger-miles per hour

Average trip length

Operating headway

Station dwell time

Minimum safe headways

Round trip time

Fleet Size

Average trip speed

Average trip time

Average number of stops per trip

Vehicle miles per hour, per day

Average peak-hour and daily load factors

Station wait times

Travel-time matrix

The basic transit-system unit is a two-way line or collapsed loop. Therefore, I have derived the parameters for a general loop system (which may be collapsed or not collapsed) in which the stations may be on line or off line, the service may be scheduled or on demand, and the vehicles may be designed to accommodate a group or a single party traveling together. When the formulas differ, I make distinc-tions. Extension of the formulas to use in network systems is the final step in the analysis. The networks considered are of two types: 1) those requiring manual transfers from line to line, and 2) those in which the vehicles transfer from line to line to permit direct travel from origin to destination.

2. The Demand and Path-Length Matrices

Let the stations or stops of a transit system be numbered in an ordered way, and let n be the number of stations. We can represent demand for service between all station pairs as a matrix:

(1)

3


The symbol represents the number of people per unit of time traveling from station to station that is, let us adopt the convention that the first index represents an origin station, and the second in-

dex a destination station. Usually, the units of are people per hour, and the main quantity of interest is

averaged over the busiest period of the day, for those values determine the capacity requirements of the system.

Similarly, let represent the distance from station to station If we number the stations con-secutively,

(2)

in which If the system is a single loop, we can express the length of the loop as

(3)

For simplicity of notation, we will, without ambiguity, let the length of the link from station to station

be represented by

(4)

3. Passenger Flows

We can represent the total number of people per hour originating travel at station as

(5)

4


where is summed over all destination stations, and represents the demand for round trips.

Similarly, the total number of people per hour terminating travel at station is

(6)

The quantity

(7)

represents the total system demand in the hour for which we have found the The values of of in-terest are for the peak-flow period, such as the peak hour or the peak 15 minutes. The peak period used in system design determines rush-period waiting times—the shorter the period, the shorter the wait times but the larger the fleet size.

We denote the flow on link in people per hour by is the sum of all the for which the

trip passes along the link Once we find it in one link, we may find it in the next link from the equation

(8)

In a line-haul system, the computation of the is begun by noting, if the end station is numbered 1,

that We can make the following calculations from knowledge of only the and

in one link—the individually are no longer needed. Thus, in making these calculations on the basis of travel counts, it is necessary to count only the total flows into and out of each station, and, in a non-col-lapsed loop, the flow in one link.

The number of passenger-miles of travel per hour is

5


(9)

where, once we have found the , it is clearly preferable to use the single sum to find PMPH. The weighted average passenger flow is

(10)

where is the round-trip length of the loop.

4. Average Trip Length

From equation (9), the demand-weighted average trip length is

(11)

Using equation (10),

(12)

Equation (12) has this simple physical interpretation: consider a loop system. If, in a fictitious case, there were only one station and the flow of people enters the station, travels around the loop, and

egresses at the same station, so i.e., the total system flow would be the same as the

total line flow. If were the situation would correspond to the case of a two-station loop

system, in which

6


5. Operating Headway

The headway between vehicles, is the time measured at a fixed point between passages of a

given point (e.g. the nose) on the vehicles. On a given link is the number of people per vehicle av-

eraged over the time period for which is averaged. In a loop system, continuity requires that aver-aged over the loop time be the same in each link of the loop.

Let be the design capacity in seated and standing passengers of each connected unit, whether it

be a single vehicle or a train. Let be the average load factor (the ratio of people per connected unit to

) in the link for which is maximum. Denoting the maximum link flow by the operating headway is

(13)

Choose the design value of based on behavior of people unloading and loading at stations and the excess ca-pacity for surge loads to be designed into the system. The fleet size will be inversely proportional to the

system-design value of . Then compare the value of computed from equation (13) with the minimum

value of T permissible from considerations of safety. If then we must increase until increases either by using larger vehicles or by training vehicles of a given size. If we train vehicles, we must understand

in equation (13) to be the train capacity. Later, we adjust T to accommodate an integral number of operat-

ing vehicles and then recompute from equation (13).

In on-line-station systems, station headway and line headway are the same; however, in off-line-station

systems, the station headway at station depends on the maximum of and denoted

7


If the average load factor of occupied vehicles passing through station is in analogy with equation (13),

(14)

If the system is a personal rapid transit system as opposed to a group rapid transit system, we can assume to be

a constant equal to

6. Station Dwell Time

Let be the loading or unloading rate per vehicle or train in people per unit of time. Obviously,

depends on the width and number of doors. If is the station-dwell time at station that is, the

time each vehicle or train is stopped at station is the total number of people who can unload and load during the time the vehicle is standing in the station. The number of people who must load onto and un-

load from each vehicle or train at station is proportional to in on-line-station systems) and de-pends on whether the unloading and loading occurs in sequence or simultaneously. If the process occurs in sequence, i.e., if people first unload and then load through the same doors,

(15)

If loading occurs simultaneously with unloading, as would be the case if people load through one door and unload through another (e.g. past the driver at the front of the vehicle and out through middle doors),

(16)

In both of these equations is a dimensionless factor greater than or equal to one included to account for uneven loading through several doors and for surge loading of more than the average number of people onto a given vehicle or train.

8


Note, in both cases, that must be large enough so that If we do not satisfy this inequality, we must in-

crease by adding more vehicles to each train, or by using more or wider doors on each vehicle.

With on-line-stations, from equation (13) Thus, for sequential unloading and loading

(17)

and for simultaneous unloading and loading

(18)

With off-line stations, sequential unloading and loading will be almost always used because this case almost always applies to automated, single-door vehicles. Thus, substituting equation (14) into equation (15),

(19)

7. Minimum Safe Headways

The minimum safe time headway between vehicles or trains stopping each at the same position in a station and then passing through the station is

(20)

or

9


(21)

in which is the line speed, is the length of the vehicles or trains, is the normal acceleration and deceleration, and

(22)

where is the emergency braking rate and k is the safety factor, that is, the ratio of minimum distance between vehicles to the stopping distance of one vehicle. These equations are from Anderson (1978).

In a conventional train system, it is common to take and in which case In this case, equation (20) is applicable. In small-vehicle automated systems in which all passengers are seated, it is possible to

take . Then and usually Always use equation (21) in such an instance.

The end stations of a line-haul rail system are often back-up stations. In this case, Anderson (1978) showed that when a = 1 the minimum safe headway is

(23)

in which case is the track length needed for one incoming train to clear the last outgoing train.

The simplest expression for minimum line headway, i.e., minimum headway along a line away from stations, is

(24)

10


in which is the time required to initiate braking at a rate and is the emergency braking rate. Anderson (1978) or McGean (1976) supply more complex equations accounting for velocity differences and jerk; however, equation (24)

accounts for the principal effects. The brick-wall stopping criterion is obtained by taking

There can be only one value of system headway throughout the loop. It is therefore the maximum of the various

With a given line speed, equation (23) gives a larger value than equation (20) and determines the system minimum headway. However, if we reduce the speed approaching the end stations, then it is possible to reduce the back-up station

to the straight-through station given by equation (20). In on-line-station systems, the values of station are

greater than the line , given by equation (24). However, in off-line-station systems, by use of multiple berths and batch loading as described by Irving (1978), it is possible to increase station throughput to almost any desired value.

8. Round-Trip Time

The general formula for round-trip time is

(25)

in which is the number of stops per round trip, is the excess time required for each stop at a com-

fort level of acceleration and comfort level of jerk , and the third term is the sum over the station delays at the stations at which the vehicles or trains stop.

For an on-line-station system in which the vehicles stop at all of n stations

(26)

where

11


(27)

is the average excess time required to stop at a station with the comfort level of acceleration , comfort level of jerk and aver-age dwell time

(28)

For off-line-station systems in which all trips are nonstop,

(29)

and equation (25) becomes

(30)

in which equation (27) gives

9. Fleet Size

In a loop system, the required number of vehicles in operation in the peak period is

(31)

where we find from the above section and from equation (13). Since is an integer, we must round off the value

given by to an integer. If at the next integer take as that integer. If this value produces

12


a line headway less than take as the value of rounded to the next lower integer.

The total fleet size is

(32)

where is a number of extra vehicles that may be required at some stations of a personal rapid transit system to

minimize wait time (for other types of systems and is the maintenance float. If the rush period time is

if the vehicle mean time to failure is and if the mean time to restore a vehicle into service is less than the time between rush periods,

(33)

10. The Revised Time Headway, Round-Trip Time and Load Factor

In equation (31), comes from equation (13) based on an estimate of and then for on-line-station systems we

computed from either equation (17) or equation (18) based on this value of Since in equation (31) must be an

integer, we must modify and, hence To do so, consider equation (25). Let where is the integer

determined below equation (31) and the correct value of is to be determined. Then in equation (25), substitute

a known value, and the value computed as indicated above. Then equation (25) becomes

13


or

(34)

This is the operating headway. The round-trip time is then

(35)

and, from equation (13), the peak-link load factor is

. (36)

In off-line station systems, line headway does not affect the Therefore, after rounding in equation

(31), equation (30) still gives Thus the operating headway is

(37)

where is the integer found from equation (31). Using equation (37) for equation (36) gives the peak-link load factor.

11. Average Trip Speed, Trip Time, and Stops per Trip

In a loop system, the average speed is

(38)

The average trip time is then

14


(39)

But, in analogy to equation (26), is also

(40)

Hence the average number of stops per trip is

(41)

For nonstop trips, That the equations are consistent with this result may be seen by substituting from

equation (30) into equation (38) and then the resulting value of into equation (41).

12. Vehicle Miles per Hour and per Day

In a loop system, the number of vehicle-miles in the peak hour is

(42)

In a demand-responsive system, the number of vehicle-miles per day is directly proportional to demand. Therefore,

(43)

15


in which the ratio of peak hour to daily travel, ranges, from data reported by Pushkarev and Zupan (1978), from about 0.08 for fully demand-responsive systems up to the range of 0.10 to 0.24 for scheduled sys-tems.

In a scheduled system, can be written in the form

in which is the number of hours per day the vehicles run at headway But, from equation (35), the peak headway is

and , using equation (38), we have

(44)

where

(45)

If the data is available, it may be more convenient to find VMPD from the equation

(46)

13. Average Peak-Hour and Daily Load Factor

The average number of people per vehicle in the peak hour is the passenger-miles of travel per peak hour di-

vided by the vehicle-miles of travel per peak hour. Thus, if is the average peak-hour load factor,

16


where the right-hand equality is found by substitution from equation (10) and (42). Then

(47)

Equation (47) also gives the daily average load factor in a loop system if the headways throughout the day are adjusted according to demand, i.e., according to equation (13), and the ratio of average to maximum line flow remains the same. We can expect this in a completely demand-responsive system such as a personal rapid transit system. In a scheduled system, however, headway cannot be adjusted exactly according to demand because: (1) the schedules must be published and adhered to; and (2) increasing headway as demand decreases according to equation (13) increases waiting time and, thus, reduces demand more than would be the case if shorter headway were maintained. Thus, in a scheduled loop

system we express the number of vehicle-miles per day by either equation (44) or equation (46). From

equation (10), the number of person-miles per day is

(48)

where is the weighted average daily link flow in people per day. Dividing by equation (46), the daily average load factor for scheduled loop systems is

(49)

It is instructive to find by observing that the operating fleet size can be expressed as

17


(50)

where the middle expression is from equation (31), and the right-hand expression from equation (13). Using equation (12),

Using equation (38), equation (47) follows, which shows that the equations we developed are consistent.

14. Station Wait Times

In an on-line-station system, the average station wait time is one half the line headway, In an off-line-

station system in which no extra vehicles are assigned to stations, the average station wait time is, similarly, but varies from station to station depending on the station flow. In a personal rapid transit system, if we determine that the wait time at a specific station is too long, we can decrease it by arranging requisitions for empty vehicles in such a way that an extra empty vehicle is nominally present at the station. (This is not practical with group rapid transit since people on ve-hicles may be required to pass through the station without getting off. Thus, the extra vehicle would frequently be pushed through empty.)

The average wait time in the station of a personal rapid transit system that is assigned one more empty vehicle than needed to meet demands for service can be estimated by assuming that vehicles arrive at the station at the rate of

vehicles per unit of time, where is the arrival rate needed to meet demands for service, but that the pas-senger arrivals are randomly distributed. (For purposes of this analysis, the passenger is either one individual or a

small group traveling together.) If one passenger arrives within the time interval since the last vehicle left, the pres-

ence of the empty vehicle insures that there is no wait. If there are two passenger arrivals in the first party

need not wait, but the second party must wait an average of Thus, the average wait time considering both parties

18


is If there are three passenger arrivals in the second party waits an average of but the third party

must wait an average of Thus, the average wait time of the three parties is

If there are n arrivals in , the average wait time for these n parties is

With randomly distributed passenger arrival rates occurring at an average arrival frequency of passen-ger groups per unit of time, the probability of n arrivals in time t is given by the Poisson distribution function:

Thus, the probability of n arrivals in time is

The average wait time assuming one extra empty vehicle is therefore

With no extra vehicles, the wait time is Therefore, addition of one additional empty vehicle reduces the average wait time to 22% of its value with no extra vehicle.

19


A similar calculation shows that with two, then three extra vehicles in the station, the average wait time is

(52)

15. The Travel-Time Matrix

Determination of patronage requires knowledge of the time patrons wait at stations and the travel time between

all station pairs. Let be the travel time between station and station Then, if the trip is nonstop, i.e., there are no intermediate stops,

(53)

where is given by equation (27). Strictly, is the average of the time delays at stations and but it is often usually as the average for the system. For an all-stop system such as a rail transit system,

(54)

in which is the number of stops, assuming the stations are numbered consecutively in the direction of flow. Equation

(41) gives the average value of

We make the initial calculation of with values of assumed from experience. Having then gone through all of the above calculations, we find a new trip time matrix and a new set of station wait times, which form the basis of a revised esti-

mate of the The process must be repeated until it converges.

20


16. Extension to Networks

Conventional bus systems consist of a group of interconnected, usually line-haul or collapsed-loop system or routes, each designated by its own route number. The formulas derived apply without modification if we keep in mind the fact that the link flows, station flows, headways and wait times apply specifically to each route. For patronage analysis, the only additional fac-tor is transfer time. Guideway systems, such as conventional rail transit system or the newer automated group rapid transit sys-tems, also operate vehicles on designated routes that sometimes overlap. We can treat these systems in the same way.

In true personal rapid transit systems, however, each vehicle trip occurs over the most direct route nonstop from an ori-

gin to a destination The vehicles are not confined to fixed routes. Because of this operational procedure, we must modify some of the above formulas, as we discuss in the following paragraphs.

In analysis of network systems, links between stations cannot be used to calculate trip times because there is not a unique link between each station pair. As shown in Figure 1, the flow from station 1 splits into two flows downstream of a branch point. We define, instead, the link as used in network personal rapid transit systems as the link between line-to-line branch points, as shown in Figure 2. We will not refer to the points of divergence into and convergence out of off-line stations as branch points. A link between branch points defined in this way may contain any number of stations includ-

ing zero. If the length of link is the sum of the is the total length of guideway not counting off-line-station guide-way.

If the flow just downstream of the branch point initiating link is known, we can find the flows downstream

of stations on link from equation 8. Let be the flow just downstream of the branch point initiating link

Calculation of the requires specification of the path through the network as well as the If none of the links is over-loaded when the minimum-length paths are in use, the minimum-length path is the logical choice and is always the logical choice for the first iteration. Within each link between branch points, designate the sublink from the first branch point to the first

station as sublink , the sublink between the first and second stations as sublink etc. Then designate the path between each set of station pairs as the sequence of links and sublinks traversed. If we set up a group of initially zeroed memory regis-

ters in a computer program corresponding to the links and sublinks, we can find the sublink flows by adding for every

pair to each of the link-sublink registers included in the path from station to station This procedure substitutes for equation

8. Equation 9 is valid if becomes the sublink corresponding to flow Then becomes the sum of the and with this meaning, equations (10) and (12) are valid.

21


We can determine the fleet size (counting empty vehicles required to balance the flows) based on the maximum total system demand D, which for this purpose we can assume to be constant in time. The first step in determining fleet size is to identify the maximum flows of people per unit of time in all of the links between branch points. The total flow of vehicles

per unit of time in link must be at least as high as where is the operating headway given by equation (13) with

the maximum flow in link and lfm the average load factor of an occupied PRT vehicle. In general these are different in different links. Find the actual flows in the links including empty vehicles by adjusting the flows upward by adding empty vehicle flows so that, in each link such as shown in Figure 2, the sum of two merging vehicle flows equals the sum of the two

downstream diverging flows, in analogy to Kirchhoff s Law of electric circuits. Once we find these adjusted values by applying this continuity principle throughout the network, we can find the fleet size.

If there are no station stops in link the number of vehicles found in link at any instant of time during the pe-

riod of peak demand is the adjusted vehicle flow, multiplied by the time required to traverse the link:

22


For every station in the link, the extra vehicles needed as a result of requiring an excess time (see equation (27)) to

pass through the station is where is given by equation (14). Thus, the total fleet size (see equation (32)) is

in which the first sum is over all links between branch points, and the second sum is over all stations. Using equation (14) the total fleet size is

(55)

By analogy with equation (53), in a personal rapid transit system the average trip time is

(56)

Then, using equation (39), the average speed is

(57)

In a network system, it is simplest to compute the vehicle-miles per peak hour from the equation

(58)

in which, from equation (55), Since the system is completely responsive to demand, we can compute the vehicle-miles per day from equation (43).

Finally, the rush-period average load factor counting empty vehicles follows from the first form of equation (50):

23


(59)

in which equation (56) gives In a fully demand-responsive system, varies throughout the day in direct proportion to D. Therefore, equation (59) applies at any time of day and gives also the daily average load factor. In a PRT system,

passengers occupy vehicles with an average load factor unless they are empty. Denoting the occupied vehicles by

and the empty vehicles by

and

Thus (60)

where we find from equation (59) and is the average size of a group of people traveling together.

17. Summary Remarks

The methodology this paper develops reduces the performance analysis of transit systems to simple analytical procedures that can be readily programmed on a computer or worked through by hand. The resulting equations are particularly useful in estimating the cost per trip from an equation derived by Anderson (1981). A number of appli-cation studies have found hand calculation with the method to be simple and instructive. To obtain simplicity, I have based the methodology on the assumptions that, to calculate the required fleet size and potential bottlenecks where capacity limits are reached, the maximum flow can be treated as steady state and the spacing between vehicles is uniform, having no stochastic component. The maximum steady-state flow can be determined over any period desired such as an hour or fifteen minutes; the shorter the period, the larger will be the fleet size calculated, but the closer will the conditions of maximum crowding in vehicles approach a pre-set requirement. Since the actual flow in the system will gen-erally not remain at the maximum for very long and may occur at different times in different parts of the system, the actual number of vehicles required will not be more than computed by the method. Stochastic variations in vehicle headways around the calculated expected values will cause some vehicles in all but the PRT systems to be more crowded than calculated, and the wait times in PRT systems to fluctuate more widely than calculated. Thus the method is use-ful as a preliminary calculation of performance, but is not a substitute for a comprehensive computer analysis of the behavior of the system throughout the day or in the rush period with unavoidable fluctuations in demand and vehicle

24


spacing. We can, however, justify comprehensive computer analyses only in detailed stages of system planning. It is useful to guide them by an analytical procedure such as the one described.

18. Notation

A Comfort level of acceleration

Vehicle load-uneveness factor

Design capacity of a vehicle or train, people

D Total demand, people per hour

Total demand entering system at station people per hour

Total demand leaving system at station people per hour

Demand from station to station people per hour

Link flow just downstream of station i, people per hour

Average link flow, people per hour

Maximum link flow, people per hour

J Comfort level of jerk

L Length of vehicle or train

Loading rate of vehicle or train

25


Average load factor

Load factor in i-th link

Maximum load factor

Distance from station to station

Length of link between branch points

Length of loop transit system

Average trip length

Total fleet size

Number of operating vehicles

Size of maintenance float

Number of extra vehicles at off-line stations

Number of stations in a loop

Number of stops per round trip

26


Average number of stops per trip

Passenger miles of travel per hour

Passenger miles per day

Time headway between vehicles (front to front)

Time headway through i-th station

Minimum safe time headway

Excess time required to stop at a station

Time delay at i-th station

Trip time around a loop transit system

Average speed

Line speed

Vehicle-miles per hour

Vehicle-miles per day

19. References

27


Anderson, J. E., Transit Systems Theory, Lexington Books, D. C. Heath, 1978.

Anderson, J. E., "An Energy Saving Transit Concept for New Towns," Journal of Advanced

Transportation, Vol. 15, No. 2, Summer 1981.

Irving, J. H., Harry Bernstein, C. L. Olson and Jon Buyan, Fundamentals of Personal Rapid

Transit, Lexington Books, D. C. Heath, 1978.

Manheim, Marvin L., Fundamentals of Transportation Systems Analysis, The M. 1. T. Press, 1979.

McGean, T. J., Urban Transportation Technology, Lexington Books, D. C. Heath, 1976.

Pushkarev, B. S. and Zupan, J. M., Public Transportation and Land Use Policy, Indiana

University Press, 1977.

28

calculation of performance and fleet size in transit … · web viewdepartment of mechanical...

Documents