Journal of Advanced Transportation, 16:3(1982)231-252.
Calculation of Performance and Fleet Size in Transit Systems
J. Edward Anderson[1]
The paper provides a consistent, analytic approach to the calculation, from the demand matrix, 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 apply to all types of network systems in which vehicles may transfer from one line or loop to another. The novel features of the paper lie in 1) the layout of the computations in a straightforward, 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 network systems.
Contents
Page1 / Introduction / 1
2 / The Demand and Path-Length Matrices / 3
3 / Passenger Flows / 4
4 / Average Trip Length / 5
5 / Operating Headway / 5
6 / Station Dwell Time / 6
7 / Minimum Safe Headway / 7
8 / Round-Trip Time / 8
9 / Fleet Size / 9
10 / 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
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. Representing 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 methodology presented builds on the work of Anderson (1978), but as a result of additional stages of application to practical problems, extends and simplifies it. The development of the methodology was the result of interest in understanding the characteristics of alternative types of automated guideway transit systems, 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 programmed 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 demand requires knowledge of travel times andwait times; thus, the process is iterative: we must first estimate 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 station 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 demand. 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 assumption 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
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 distinctions. 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)
The symbol represents the number of people per unit of time traveling from station to stationthat is, let us adopt the convention that the first index represents an origin station, and the second index 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 consecutively,
(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)
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 interest 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 theend 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-collapsed loop, the flow in one link.
The number of passenger-miles of travel per hour is
(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
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 averaged over the time period for which is averaged. In a loop system, continuity requires that averaged 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 theoperating headway is
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Journal of Advanced Transportation, 16:3(1982)231-252.
(13)
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Journal of Advanced Transportation, 16:3(1982)231-252.
Choose the design value ofbased on behavior of people unloading and loading at stations and the excess capacity 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 operating 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 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
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Journal of Advanced Transportation, 16:3(1982)231-252.
6. Station Dwell Time
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Journal of Advanced Transportation, 16:3(1982)231-252.
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 unload from each vehicle or train at station is proportional to in on-line-station systems) and depends 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. Note, in both cases, thatmust be large enough so that If we do not satisfy this inequality, we must increase 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)
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Journal of Advanced Transportation, 16:3(1982)231-252.
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
(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 kis 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)
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 batchloading asdescribed 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 comfort level of acceleration and comfort level of jerk , and the third term is the sum over the station delaysat 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
(27)
is the average excess time required to stop at a station with the comfort level of acceleration , comfort level of jerk and average 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 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 Sincein 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 ofis to be determined. Then in equation (25), substitutea known value, and the value computed as indicated above. Then equation (25) becomes
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