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A quantitative analysis of synchronous vs. quasi-synchronous network operations of automated transit systems
Tnmspn Res. Vol. 9. pp. 241-248.
Pergamon Press 1975.
Printed in Great Britain
A QUANTITATIVE ANALYSIS OF SYNCHRONOUS QUASI-SYNCHRONOUS NETWORK OPERATIONS AUTOMATED TRANSIT SYSTEMS+
VS. OF
ALAIN L. KORNHAUSER Department of Civil Engineering, Princeton University, Princeton, NJ 08540,U.S.A. and PATRICK MCEVADDY Departmentof Electrical Engineering, Princetonuniversity, Princeton,NJO854O.U.S.A. (Received 24 September 1974)
Abstract-This paper investigates the performance of the synchronous and quasi-synchronous network control policies proposed for modern automated transit systems. Performance is analyzed from the user point-of-view in terms of the expected travel time delay associated with each policy. Using an idealization of the network layout and uniform demand for service, analytic expressions for the expected delay are derived for each policy in terms of fundamental performance
parameters (line spacing, trip rate, trip length distribution, of each policy are presented in parametric form.
INTRODUCTION The necessity of automated vehicle operation at short headways in order to achieve high capacity in automated transit systems such as Dual-Mode/Personal Rapid Transit (DM/PRT) systems has placed stringent demands on the network control policy. This is especially demanding in large area-wide networks consisting of hundreds of miles of guideway interconnecting hundreds of stations that are serviced by thousands of vehicles. The inadequacy of block headway control systems commonly used for controlling conventional rapid rail systems has led to the proposal of three alternative network controlmanagement policies for DM/PRT: Synchronous, Quasisynchronous and Asynchronous. All three have received extensive qualitative definition and discussion in the literature (Godfrey, 1968; Surber, 1967; Munson, 1972). Briefly, (1) Synchronous establishes virtual slots moving throughout the network. Each slot moves at the nominal speed defined for each line and is synchronized at each intersection. Each vehicle operates within a given slot (“point-follower” headway control). Departure from a station is not permitted until a sequence of synchronized slots can be reserved that will allow uninterrupted travel from origin to destination. (2) Quasi-synchronous relaxes the reservation constraint of the synchronous system by allowing vehicles to merge into the first available slot at interchanges. At points other than at interchanges vehicles can operate within virtual slots (synchronous) or in a vehicle-follower mode (asynchronous). At interchanges the vehicle can maneuver (“slip slots”) to merge into the next available slot or gap. An “interchange controller” surveys the main stream of traffic to find the location of the next available tPresented at IEEE/IAS Conference, Pittsburgh, Pennsylvania: October 9, 1974.
maneuver
region). Comparisons
of the
slot at the time the requesting vehicle arrives at the interchange (Munson, 1972; Brown, 1973a). (3) Asynchronous assumes the non-existence of a network control policy. A vehicle-based “vehiclefollower” controller maintains proper spacing between individual vehicles (Brown, 1973b). If a vehicle-follower headway control system is adopted the quasi-synchronous modification of the asynchronous policy will need to be implemented if conflicts at intersections are to be avoided. However, if a “point-follower” vehicle headway controller is adopted the literature is unclear as to which network control policy is superior, synchronous or quasi-synchronous. Some qualitative arguments couched mostly in terms of computer hardware and software reliability indicate that quasi-synchronous is better (Munson, 1972: Brown, 1973a) and some quantitative results from computer simulation of specific and very limited networks indicate that synchronous is better (Roesler, 1973; Wade, 1973). The basic criteria for choosing between synchronous and quasi-synchronous network policies are: (1) from the operator point-of-view-which policy will result in the “smoother” operation; (2) from the user point-of-viewwhich policy will result in the shorter delay; and (3) from the planner point-of-view-which is most“cost-effective” to implement. In criterion number 1, “smoother” operation is usually discussed in terms of the sensitivity of the network to failures (individual vehicle reliability, computer reliability, etc.) computational complexity, and cost. Qualitative arguments of this type are often forwarded to deemphasize the synchronous policy; however Wade (1973), has simulated a 500-mile network and claimed that the scheduling algorithm is within the capability of present generation computers. This paper focuses on only the second criterion. It 241
242
A. L.
KORNHAUSER
and P.
compares the performance of the quasi-synchronous policy to that of the synchronous policy from the user point-of-view. The analysis is structured to quantify the difference in performance between the two policies for a range of demands and fundamental network design characteristics. The methodology is based on the idealized city approximation that has been used successfully in quantifying the fundamental tradeoffs in the design, economics and environmental impacts of large-scale PRT networks (Kornhauser, Dais, 1973). The network is characterized as a square grid of interconnected lines with off-line stations located at the midpoints between interchanges. This geometry minimizes access distance to and from the automated network for a spatially uniform demand function. The fundamental network design parameter is then the distance between lines, L. The fundamental parameters of the demand are the trip distance distribution, d, and the trip rate, pd.The expected delay due to each network policy can then be estimated in terms of these fundamental parameters. For the synchronous policy a reservation must be made for each link travelled. The line spacing and the trip distance distribution quantify the number of required reservations distribution and the number of equilength paths from origin to destination. The demand rate gives the probability of the existence of an unreserved slot at the beginning of each link as a function of the distance (time) of this link from the origin. We can express the probability of being able to make a reservation on each link for at least one of the equilength paths assuming a departure at time t + n(At) where n(At) are integer multiples of headway time, At. Consequently we have an expression for the expected wait time as a function of the fundamental network and demand parameters and the probability of securing all necessary reservations. Similarly for the quasi-synchronous policy we have derived an expression for the expected number of slots that need to be slipped to achieve a merge at each turn-intersection and the probability of an abort (thus increasing the trip length) as a function of the maximum number of slots that can be slipped at each interchange (a network design parameter). Summation over the expected number of turns gives the expected delay for the quasi-synchronous policy. The paper develops the above models in the ensuing section. Parametric comparison of the expected delay for each policy as a function of the fundamental network design and demand parameters of line spacing, length of maneuver area at interchanges, demand rate and trip length distribution are presented in Section 3. Section 4 discusses the limitations and possible extensions of the models presented herein.
MCEVADDY
square grid network with one-way links. Stations lie between the intersections and are off-line as is shown in Fig. 1. Such a network is characterized by a single parameter, L, the distance between parallel lines. It has been shown that this model adequately characterizes the gross behavior of the interior of large networks (University of Minnesota Task Force, 1972). The demand model Of interest in the comparison of network control policies is their steady state performance under various levels of demand as well as their transient response to time variation in demand. The analysis developed herein only investigates the steady state performance in that the level of demand is assumed to be uniform and constant. The loading of the uniform demand on the idealized network has the following implications: -the steady state loading on each link of the network will be a constant, p -each trip will have an average of 2.5 turns. Trips having one turn and four turns each represent 12.5% of the total trips and those having two and four turns each represent 37.5% of the total. (For a network of infinite extent, trips having zero turns represent an infinitesimal percentage of the total.) The travel distance on the network differs from the airline-trip distance by two factors. The grid increases the travel distance by an average of 30% and the “round-theblock” penalty due to the one-way orientation increases the travel distance by an average of twice the line spacing; therefore, the relationship between average trip length on the network, D, and the airline travel demand, d, is: D= 1.3d+2L.
(1)
MODEL DEVELOPMENT
The network model Since the purpose of this paper is to present a quantitative comparision of synchronous and quasisynchronous network control policies it is important that the results be as network independent as possible. To obtain both analytic insight and generality the network model is an idealization of an area-wide interconnected
i
Fig. I.
LEGEND Gwdeways StatIOnS Service nrea
zzrx 0 _--
A quantitative analysis of synchronous vs quasi-synchronous network operations of automated transit systems Data presented in the paper will be with respect to the network trip length, D. Quasi-synchronous policy As previously mentioned, the quasi-synchronous control policy requires the network stations and interchanges to have maneuver regions at each intersection that enable turning vehicles to delay while waiting for the first available merge slot. Vehicles going straight encounter no delay. The wait process at stations is identical to that at turn intersections. Assuming (1) all slot slipping is done on the divert link, and (2) line loadings are uniform and are uneffected by the merge algorithm; we wish to express (a) the expected number of slots (delay) the vehicle must slip in order to merge, and (b) what is the probability of the slot slipping queue being filled, thus forcing the vehicle to abort its minimum time (distance) path to its destination. The expected wait time at each turn intersection results from the delay of preceding vehicles propagating through the merge queue plus the wait for a merge opportunity once the vehicle reaches the head of the merge queue. The expected length of the queue and the expected delay at each turn intersection is a function only of the contemporaneous probability that a slot is empty in the merging lane and the percentage of vehicles requesting turns. From assumptions of the geographical uniformity of demand and random spacing of empty slots, the expected percentage of vehicles requesting turns at each intersection is related to the average network trip length, D, the line spacing, L, and the average number of turns per trip ((2.5) turns/trip on average) to give % turning,
pr = 2.5L/D.
100%.
because headway slot size does not go to zero in the limit.) It is reasonably accurate to say that events (vehicles occupying slots) are independent and equally probable at any time. The specific choice of models are an M/M/l infinite queue length model for the process of leaving a station and an M/M/l-Mq for the merge process at intersections where Mq is the maximum queue length. From queueing theory the probability of no vehicles in queue, the probability that exactly n vehicles are waiting to egress from the station, P.,, the expected queue length, in the station, L,, and the total expected delay at the station, W,, in terms of p, p, L, D and the headway time between vehicles (in seconds), H are given by: Pas= l-ps/(l-p(l-p~))
(4)
P,s= Pa&/(1 -p(l
(5)
-p))
L~=p,z/((l-p-p)z-p~(l-p(l-p)))
(6)
Ws=H(L,/ps+l/(l-p(l-/A))).
(7)
In the above, p,/H represents the mean arrival rate and (I- p - PP~)/H represents the mean service rate in terms of the arrival rate of empty slots at the merge point. Continuity has been enforced in that the rate of vehicles leaving the station equals the rate entering = pS. Using a finite queue-length model to prepresent the waiting process at intersections provides the following relationships for no vehicles in the turn, PO,, the probability that exactly n vehicles are waiting to turn, P,,, the expected queue length at the intersection, LT, in terms of p, p, L, D, H, and the maximum queue length Mq:
(2) Pot= (1- S)/(l-
Thus, for L = l/2 miles and d = 5 miles, on average 16.7% of the vehicles entering an intersection desire to turn and the rest go straight. Also from the uniformity assumption, the station service rate pS, is given by:
S”*+‘)
P,,=(l-F)“/(l-f?“) L,= S/(1 - Is-(Mq w = pptl(l -p(l-
ps = pLID.
243
(8) (9)
+ l)s”“+‘/(l-6”~+‘) pt)).
(10) (11)
(3)
For one-second headways and p = 80%, L = l/2 miles and D = 6 miles each station on the network would service an average of 240 vehicles per hour. As mentioned the process of merging from a station to the main line is identical to that of merging at an intersection except the queue length for the intersection merge must be considered as finite because guideway construction would only allow short maneuver regions and vehicles can readily be denied access to the intersection (aborted turn). Access to stations is dependent on access to the arrival queue. For egress it is physically possible for the departure queue to extend into the arrival queue, thus, from an analytical standpoint, it is appropriate to consider infinite queue lengths in stations. The process of quasi-synchronous merging can be approximated by a standard queueing theory process of Poisson arrivals and exponentially distributed service times. (Actually we do not have a true Poisson process
In the above, ppt/H represents the mean arrival rate and I- ~(1 - p,)/H represents the mean service rate; their ratio is denoted by S. Synchronous policy The modeling of the sychronous control policy differs greatly from that of the quasi-synchronous case in that: -the probability of being able to confirm a reservation on each link in the journey made at a given request time to is a function of the number of confirmed reservations for that link which were made prior to time to. -reservations must be secured for each link of the journey, not just turn intersections. A passenger arrives at the origin station, 0, on some link L,, at time Q and attempts to schedule a departure for station D, starting at to see Fig. 1. The difference tn - TV can be controlled to yield link loadings about the nominal design value, p. For a given O-D pair there might exist a
244
A. L. KORNHAUSER and P. MCEVADDY
number of minimum paths, [PI,. . . PI]. Each path is uniquely determined by the sequence of links, thus:
contemporaneous rate, p., is:
line loading of p, the station process
Pj = (LjO, Lj,, . . .) 4,).
ps =
The reservation algorithm must verify a matching sequence of vacancies on the entire set of constituent links, i.e. a complete reservation is the m + I-tuple R = (To, T,, . . T,) where II; is arrival time on link L,;. Thus, for each link in the network it is necessary to maintain a list indicating, for each time step, whether a vehicle has reserved to enter onto that link. For example to reserve a path out of station 0, it is necessary to first check the list for link La0 at time t - L/2vt to verify if it has been previously reserved. If this slot is unreserved (or if the previously reserved trip has station 0 as its destination), the next step attempts to make a reservation at intersection #1 at the projected vehicle arrival time of
Therefore, reservation
pk.
the probability of being able to make a at time to to just get out of the station is: l.O-p(l.O-
L/D).
(14)
The probability of being able to reserve a slot at each succeeding link of the trip is dependent on the time into the future for which the reservation is being made as well as whether the vehicle desires to turn or go straight. If the vehicle goes straight through an intersection, call it the jth from its origin station, then the probability of securing a reservation based on a departure time at to is:
t,: L tl = to+%.
(12)
Entry onto the next link, L,,, can be blocked only if line p carries a vehicle which has made a reservation to turn onto line u at time tl (although it actually arrives on line a: at t, + (y. L/v)). The remaining intersections are tested in similar fashion. If a reservation cannot be made, the next iteration of the algorithm attempts to make a new complete reservation starting from station 0 at time to+At. Assuming a constant demand rate in both trip length and trip distribution implies that the percentage of slots confirmed by other vehicles on each link in the network decreases linearly: from the contemporaneous line loading at the entrance to the origin station to the station arrival rate at the last link of the trip, see Fig. 2. For a
P,=l-
PtPi l-(l-Pf)Pl
where p, = p(l - (2j - l)L/20). If the vehicle desires to turn at an intersection, the probability of securing a reservation is:
PL D
0
D TRIP LINK
Fig. 2.
tv is the nominal line speed.
k, then
where pk is, again, the effective line loading k intersections into the future. The probability of a departure at time to is then the combined probability of being able to secure a reservation on each link of the minimum path from origin to destination and is given by:
(17) where J is the set of no turn intersections, turn intersections.
F
(15)
K is the set of
Quasi-synchronous case Some quantitative results obtained from employing the queuing model discussed above are presented in Figs. 3-6. Figure 3 represents a composite characterization of system performance, displaying total average delay as a function on the line loading, p. For p = 80% the total delay, averaged over the uniform turn distribution discussed in the section on the demand model, is approximately 20 headways units. This includes delay in stations as well as delay encountered in making turns. It is noted that this delay becomes increasingly sensitive to increases in line loadings greater than p = 65%. Figure 4 demonstrates the sensitivity of average delay in the station to changes in the trip length factor, D/L, for line loadings approaching 80%. As expected decreased line spacings promote ease of egress in attempting a departure. Figure 4, however, suggests that this effect becomes less significant as nominal line loadings drop below p = 70%. This lack of sensitivity to changes in D/L for line loadings under 70% is apparent in the next figure, but the effect is not as pronounced. This figure depicts the
A quantitative analysis of synchronous
vs quasi-synchronous
network operations of automated transit systems
245
25
20 -
0.6 f
4 !I
13 -
w
0.6-
Lz s f
IO -
o.4 _ AT AN INTERSECTION
‘;’
5IN THE o50
, 60
I 70
60
MEAN LINE LOADING (% OF SLOTS OCCUPIED)
0
50
I 60 MEAN
STATION I 70
LINE
60
LOADING
Fig. 6. Fig. 3.
4\
p=7Q% p=60%
0’ IO
I 20 D/L
Fig. 4
‘.Ol-----l
Fig. 5.
33
variation of mean queue length in the station to changes in D/L for several values of the fundamental parameter p. For values of L = l/2 mile, D = 6 miles, p = 80%, the expected queue length is 0.5 whereas a decrease in p to 70% cuts the expected queue length to half that value. It is clear that as the spacings between lines is increased the probability of long backups into the station holding areas is minimized. It should be observed, however, that increases in D/L above 20 result in increasingly smaller improvements in obviating station backups. The implications for station design should not be ignored. Unlike the situation in the synchronous case, each turn on a trip under quasi-synchronous management is identical. This, of course, results from the fact that the contemporaneous line loadings at each turn is just the mean line loading p. Under the synchronous policy reservation for the turn has been made well in advance of the actual arrival time at the turn, consequently, the line loading must be attenuated by a factor depending upon when the turn is to be made. Therefore, no two turns of a trip can have the same probability of success. Since the quasi-synchronous policy does not rely upon the confirmation of reservations, it is sufficient to examine the performance of only one intersection to understand completely the turning behavior. The effect of line loading, the fundamental network parameter, on the expected queue size is exhibited in Fig. 6, demonstrating the same sensitivity as seen before to increases in p above 65%. Data on the behavior of the station queue is included for comparison. Recall that the two models are different; infinite queue lengths have been permitted at station departures whereas the queues at intersections have been restricted by a maximum queue size, in this analysis A4q = 8.0. Data from graphs not included in this paper demonstrate very small changes in performance for queue sizes larger than 5, while decreasing the probability of an abort. Typical delay at an intersection for p = 80% is about Sheadwaysanddropstoabout3.3headwaysforp = 70%. Synchronous case Figures 7-11 summarize the delay incurred for the synchronous network control policy. Mean delay versus contemporaneous link loadings are presented in Fig. 7 for various values of the nondimensional trip length factor,
246
A. L. KORNHAUSER and P. MCEVADDY D/L = I8 D/L : 15 /
j-
I
I
I
I
60 MEAN LINE
LOADING
I
70 OF SLOTS OCCUPIED)
(%
I
80
Fig. I.
18
D/L. The link loading, p, is an indication of the productivity of utilization efficiency of the guideway system. It is noted that: -The penalty for trying to get high productivity (large p) is to incur high average waits at the station, approximately 70 headway units for p = 80%. Even for a one-second headway system this implies a mean wait of about one-minute. The deteriorating performance of larger headway systems is obvious. -The reservation system is rather insensitive to p until p becomes greater than 70%. -The effect of changing trip length characteristic D/L is small except at high values of p.
The effect of D/L is better represented in Fig. 8. For large values of p the effect of D/L is large for small values of D/L and then approaches an asymptotic value for large values of D/L.Actually D/L is a measure of the number or reservations that need to be made for an average trip. For few reservations the addition of one makes a large change in the probability of the sequence of Bernoulli trials; however, for trips requiring a large number of reservations the addition of one makes little change in the probability of verifying a complete sequence of reservations. The individual probabilities of being able to secure a reservation at each succeeding link of the trip is presented in Fig. 9 for several values of D/L and p. Note that individual probabilities of making a reservation are much larger (e.g. 0.885 vs 0.51) for vehicles going straight rather than turning at an intersection, Figure 10 presents the cumulative probability of
A quantitative analysis of synchronous vs quasi-synchronous
247
network operations of qutomated transit systems
SYNCHRONOUS NETWORK CONTROL POLICY D/L
= 12
2 EQUIDISTANT
I
I
I
20
40
60
I
DELAY
I
I
80 100 120 TIME , T ( HEADWAYS)
TURNS
I 140
1
I60
I 0
Fig. IO.
SYNCHRONOUS
37.5% 2-turns, 37.5% 3-turns, 12.5% Cturns) we get a system average wait for reservation to be 82 headway units for D/L = 12 and p = 80%. It is also important to note that the calculations made for the synchronous case neglect the establishment of departure queues in stations, i.e. vehicles can be dispatched as soon as a sequence of reservations is confirmed. If queues were to form in stations then the data presented yield only the delay once the vehicle has reached the head of the queue. Total delays would be much larger. This has implications on station design in that a synchronous network control policy requires a parallel bay station configurations so that conflicts be e1iminated.t (The parallel bay design provides no advantage over the sequential bay design in the quasi-synchronous case).
-
_
QUASISYCHRONOUS
I
I
70
60 LINK
LOADING
Fig. II.
securing a complete sequence of reservations in less than T headways as a function of T for several values of D/L and p. Note that the 0.95 probability of securing a sequence of reservations is achieved after only I5 headway times for p = 50%, while it requires an order of magnitude more time (0.80 headway times) to achieve a 0.95 probability for p = 80% (D/L = 12). The above data has been presented for a fixed number of turns (either zero or two) equally spaced along the trip. Figures 7 and 9 indicate that the expected delay is very dependent on the number of turns. If we take the distribution of turns for a uniform demand (12.5% l-turn,
tExcept
equal time departures.
Comparison
Figure 11 summarizes the difference in performance between the two network management policies for a two turn trip with D/L = 15. Both demonstrate sensitivity to increases in line loading, but the delay experienced under synchronous control dominates for p greater than 70% (near 4 times as great at p = 80%). This relationship holds for other values of the parameter D/L. For large values of p the delay under the synchronous policy is very sensitive to changes in the trip length factor, as discussed above. This is not the case in the quasi-synchronous model, where changes in D/L effect only small changes in the expected delay in the station queue (leaving, of course, intersection queues unchanged); a change in D/L from IO to 30 increases the mean delay from 5.7 to 8.3 headway units (p = 80%). Thus the difficulty in obtaining a complete sequence of reservations in the synchronous case seems to out-weigh the advantages of the decrease in contemporaneous line loadings at intersections distant (in time) from the origin station. Further it has been shown above that there is a lower probability of successfully making a reservation to turn, particularly in the early stages of the trip. The sequence of turns is not significant
248
A. L. KORNHAUSERand P. MCEVALIDY
under the quasi-synchronous scheme. In examining the queueing model, the parameter to study is the ratio of mean arrival rate to mean service rate. For small values of this parameter the expected delay remains small, as does the expected queue length. As shown above this parameter remains much smaller than 1.0 for small values of p, the average percentage of those attempting to turn. Under nominal conditions it appears that p will remain relatively small (-20%). Design of the divert links to accomodate reasonable queue sizes minimize the probability of an abort. Even if an abort is necessary, the greater flexibility of the quasi-synchronous approach has an advantage over the synchronous scheme which is forced to reschedule the complete sequence of link reservations. (This is clearly the case if the aborted turn does not increase total travel distance and, consequently, total travel time.) EXTENSIONS
The preceeding discussion has established the capability of a quasi-synchronous control policy to provide higher levels of productivity for the assumed network/demand configuration. It was also shown that this scheme was better from the user point of view in that the average delays for a complete trip were lower. However, further analysis is warranted. In particular no attempt was made to quantify the effect of multiple paths in improving the performance of the synchronous scheme. Clearly, there exists a significant percentage of trips having several equilength and even equilength-equiturn paths (recall that reserving a turn is less probable than passing straight through an intersection). Hence the performance of the synchronous scheme can be improved. At present it is hard to say by how much. For an assumed number of equivalent paths, iV,the solution is obtained from a series of Bernoulli trials by calculating the probability of at least one successful reservation in N trials. This will be true only when the trials are independent and equiprobable. It will usually be the case, however, that the multiple paths will overlap to some extent and hence would not be disjoint events. Until this topological problem of the network is resolved the best that could be done would be to study the sensitivity of system performance for an assumed maximum possible number of equivalent paths and to verify what conditions will insure that system performance approaches that of the quasi-synchronous scheme. One possible variation of the synchronous scheme offers some promise of improving performance. Under synchronous control a reservation is thwarted if one slot
is occupied at one specific time. The new scheme, called tran-synchronous, would permit the reservation algorithm to resolve the conflict by looking for a vacancy that would arrive A headway units later at the merge point. The control system would then make the reservation and command the vehicle to slip A slots on the divert link while waiting for the reserved vacancy to arrive. (It should be noted that the waiting vehicle could potentially block a following vehicle from merging into its own reserved slot unless all maneuver slots have been reserved). Finally, it should be remarked that this paper has investigated only the steady state performance of the proposed models, ignoring the effect of relaxing the constraint on the assumed network/demand configuration. In particular the impact of letting demand level vary as a function of time should be investigated, particularly the effect on maximum queue size needed in the quasisynchronous scheme. In addition there remains the question of relaxing the assumption of geographical uniformity, permitting spatial distribution in the demand function.
REFERENCES
Brown S. J. (1973a) Merge control in automated transit system networks. Presented at Intersociety Transportation Conference, Denver, Colorado. AMSEpaper No. 73-ICT-109.
BrownS. J. (1973b)Designof car-followertype control systems with finite bandwidth plants. Proceedings Princeton Conference on Information Sciences and Systems. Godfrey M. B. (1968) Merging in automated systems: MIT Sc.D. Thesis (unpublished).
transportation
Kornhauser A. L. and Garrard W. L. (1973)Design of ootimal feedback systems for longitudinal control of automated transit vehicles. Transpn Res. 7, 125-144. Kornhauser A. L. and Dais J. L. (1973) Economic, environmental and design aspects of large-scale PRT networks. Highway Research Record No. 425, 2637. Munson A. V. (1972) Quasi-synchronous control of high-capacity PRT network, 325-350 of Personal Rapid Transit (Edited by Anderson, Dais, Garrard and Kornhauser). Dept. of Audio Visual Ext., Univ. of Minnesota. Roetler W. L. et al. (1973). Comparison of synchronous and asynchronous PRT vehicle management and some alternative routing algorithm. Personal Rapid Transit II (Edited by J. E. Anderson). Dept. of Audio Visual Ext., Univ. of Minnesota. Surber M. A. (1967) Feasibility of automatic guideway control. DIR-751, HUD Contract No. H.-777. General Research Corporation. Task Force on New Concepts in Urban Transportation, Planning for Persona/ Rapid Transit (1972). University of Minnesota. _ Wade R. W. (1973) A cost oriented control system for a large PRT network. Personal Rapid Transit II (Edited by J. E. Anderson). Dept. of Audio Visual Ext., Univ. of Minnesota.
Pergamon Press 1975.
Printed in Great Britain
A QUANTITATIVE ANALYSIS OF SYNCHRONOUS QUASI-SYNCHRONOUS NETWORK OPERATIONS AUTOMATED TRANSIT SYSTEMS+
VS. OF
ALAIN L. KORNHAUSER Department of Civil Engineering, Princeton University, Princeton, NJ 08540,U.S.A. and PATRICK MCEVADDY Departmentof Electrical Engineering, Princetonuniversity, Princeton,NJO854O.U.S.A. (Received 24 September 1974)
Abstract-This paper investigates the performance of the synchronous and quasi-synchronous network control policies proposed for modern automated transit systems. Performance is analyzed from the user point-of-view in terms of the expected travel time delay associated with each policy. Using an idealization of the network layout and uniform demand for service, analytic expressions for the expected delay are derived for each policy in terms of fundamental performance
parameters (line spacing, trip rate, trip length distribution, of each policy are presented in parametric form.
INTRODUCTION The necessity of automated vehicle operation at short headways in order to achieve high capacity in automated transit systems such as Dual-Mode/Personal Rapid Transit (DM/PRT) systems has placed stringent demands on the network control policy. This is especially demanding in large area-wide networks consisting of hundreds of miles of guideway interconnecting hundreds of stations that are serviced by thousands of vehicles. The inadequacy of block headway control systems commonly used for controlling conventional rapid rail systems has led to the proposal of three alternative network controlmanagement policies for DM/PRT: Synchronous, Quasisynchronous and Asynchronous. All three have received extensive qualitative definition and discussion in the literature (Godfrey, 1968; Surber, 1967; Munson, 1972). Briefly, (1) Synchronous establishes virtual slots moving throughout the network. Each slot moves at the nominal speed defined for each line and is synchronized at each intersection. Each vehicle operates within a given slot (“point-follower” headway control). Departure from a station is not permitted until a sequence of synchronized slots can be reserved that will allow uninterrupted travel from origin to destination. (2) Quasi-synchronous relaxes the reservation constraint of the synchronous system by allowing vehicles to merge into the first available slot at interchanges. At points other than at interchanges vehicles can operate within virtual slots (synchronous) or in a vehicle-follower mode (asynchronous). At interchanges the vehicle can maneuver (“slip slots”) to merge into the next available slot or gap. An “interchange controller” surveys the main stream of traffic to find the location of the next available tPresented at IEEE/IAS Conference, Pittsburgh, Pennsylvania: October 9, 1974.
maneuver
region). Comparisons
of the
slot at the time the requesting vehicle arrives at the interchange (Munson, 1972; Brown, 1973a). (3) Asynchronous assumes the non-existence of a network control policy. A vehicle-based “vehiclefollower” controller maintains proper spacing between individual vehicles (Brown, 1973b). If a vehicle-follower headway control system is adopted the quasi-synchronous modification of the asynchronous policy will need to be implemented if conflicts at intersections are to be avoided. However, if a “point-follower” vehicle headway controller is adopted the literature is unclear as to which network control policy is superior, synchronous or quasi-synchronous. Some qualitative arguments couched mostly in terms of computer hardware and software reliability indicate that quasi-synchronous is better (Munson, 1972: Brown, 1973a) and some quantitative results from computer simulation of specific and very limited networks indicate that synchronous is better (Roesler, 1973; Wade, 1973). The basic criteria for choosing between synchronous and quasi-synchronous network policies are: (1) from the operator point-of-view-which policy will result in the “smoother” operation; (2) from the user point-of-viewwhich policy will result in the shorter delay; and (3) from the planner point-of-view-which is most“cost-effective” to implement. In criterion number 1, “smoother” operation is usually discussed in terms of the sensitivity of the network to failures (individual vehicle reliability, computer reliability, etc.) computational complexity, and cost. Qualitative arguments of this type are often forwarded to deemphasize the synchronous policy; however Wade (1973), has simulated a 500-mile network and claimed that the scheduling algorithm is within the capability of present generation computers. This paper focuses on only the second criterion. It 241
242
A. L.
KORNHAUSER
and P.
compares the performance of the quasi-synchronous policy to that of the synchronous policy from the user point-of-view. The analysis is structured to quantify the difference in performance between the two policies for a range of demands and fundamental network design characteristics. The methodology is based on the idealized city approximation that has been used successfully in quantifying the fundamental tradeoffs in the design, economics and environmental impacts of large-scale PRT networks (Kornhauser, Dais, 1973). The network is characterized as a square grid of interconnected lines with off-line stations located at the midpoints between interchanges. This geometry minimizes access distance to and from the automated network for a spatially uniform demand function. The fundamental network design parameter is then the distance between lines, L. The fundamental parameters of the demand are the trip distance distribution, d, and the trip rate, pd.The expected delay due to each network policy can then be estimated in terms of these fundamental parameters. For the synchronous policy a reservation must be made for each link travelled. The line spacing and the trip distance distribution quantify the number of required reservations distribution and the number of equilength paths from origin to destination. The demand rate gives the probability of the existence of an unreserved slot at the beginning of each link as a function of the distance (time) of this link from the origin. We can express the probability of being able to make a reservation on each link for at least one of the equilength paths assuming a departure at time t + n(At) where n(At) are integer multiples of headway time, At. Consequently we have an expression for the expected wait time as a function of the fundamental network and demand parameters and the probability of securing all necessary reservations. Similarly for the quasi-synchronous policy we have derived an expression for the expected number of slots that need to be slipped to achieve a merge at each turn-intersection and the probability of an abort (thus increasing the trip length) as a function of the maximum number of slots that can be slipped at each interchange (a network design parameter). Summation over the expected number of turns gives the expected delay for the quasi-synchronous policy. The paper develops the above models in the ensuing section. Parametric comparison of the expected delay for each policy as a function of the fundamental network design and demand parameters of line spacing, length of maneuver area at interchanges, demand rate and trip length distribution are presented in Section 3. Section 4 discusses the limitations and possible extensions of the models presented herein.
MCEVADDY
square grid network with one-way links. Stations lie between the intersections and are off-line as is shown in Fig. 1. Such a network is characterized by a single parameter, L, the distance between parallel lines. It has been shown that this model adequately characterizes the gross behavior of the interior of large networks (University of Minnesota Task Force, 1972). The demand model Of interest in the comparison of network control policies is their steady state performance under various levels of demand as well as their transient response to time variation in demand. The analysis developed herein only investigates the steady state performance in that the level of demand is assumed to be uniform and constant. The loading of the uniform demand on the idealized network has the following implications: -the steady state loading on each link of the network will be a constant, p -each trip will have an average of 2.5 turns. Trips having one turn and four turns each represent 12.5% of the total trips and those having two and four turns each represent 37.5% of the total. (For a network of infinite extent, trips having zero turns represent an infinitesimal percentage of the total.) The travel distance on the network differs from the airline-trip distance by two factors. The grid increases the travel distance by an average of 30% and the “round-theblock” penalty due to the one-way orientation increases the travel distance by an average of twice the line spacing; therefore, the relationship between average trip length on the network, D, and the airline travel demand, d, is: D= 1.3d+2L.
(1)
MODEL DEVELOPMENT
The network model Since the purpose of this paper is to present a quantitative comparision of synchronous and quasisynchronous network control policies it is important that the results be as network independent as possible. To obtain both analytic insight and generality the network model is an idealization of an area-wide interconnected
i
Fig. I.
LEGEND Gwdeways StatIOnS Service nrea
zzrx 0 _--
A quantitative analysis of synchronous vs quasi-synchronous network operations of automated transit systems Data presented in the paper will be with respect to the network trip length, D. Quasi-synchronous policy As previously mentioned, the quasi-synchronous control policy requires the network stations and interchanges to have maneuver regions at each intersection that enable turning vehicles to delay while waiting for the first available merge slot. Vehicles going straight encounter no delay. The wait process at stations is identical to that at turn intersections. Assuming (1) all slot slipping is done on the divert link, and (2) line loadings are uniform and are uneffected by the merge algorithm; we wish to express (a) the expected number of slots (delay) the vehicle must slip in order to merge, and (b) what is the probability of the slot slipping queue being filled, thus forcing the vehicle to abort its minimum time (distance) path to its destination. The expected wait time at each turn intersection results from the delay of preceding vehicles propagating through the merge queue plus the wait for a merge opportunity once the vehicle reaches the head of the merge queue. The expected length of the queue and the expected delay at each turn intersection is a function only of the contemporaneous probability that a slot is empty in the merging lane and the percentage of vehicles requesting turns. From assumptions of the geographical uniformity of demand and random spacing of empty slots, the expected percentage of vehicles requesting turns at each intersection is related to the average network trip length, D, the line spacing, L, and the average number of turns per trip ((2.5) turns/trip on average) to give % turning,
pr = 2.5L/D.
100%.
because headway slot size does not go to zero in the limit.) It is reasonably accurate to say that events (vehicles occupying slots) are independent and equally probable at any time. The specific choice of models are an M/M/l infinite queue length model for the process of leaving a station and an M/M/l-Mq for the merge process at intersections where Mq is the maximum queue length. From queueing theory the probability of no vehicles in queue, the probability that exactly n vehicles are waiting to egress from the station, P.,, the expected queue length, in the station, L,, and the total expected delay at the station, W,, in terms of p, p, L, D and the headway time between vehicles (in seconds), H are given by: Pas= l-ps/(l-p(l-p~))
(4)
P,s= Pa&/(1 -p(l
(5)
-p))
L~=p,z/((l-p-p)z-p~(l-p(l-p)))
(6)
Ws=H(L,/ps+l/(l-p(l-/A))).
(7)
In the above, p,/H represents the mean arrival rate and (I- p - PP~)/H represents the mean service rate in terms of the arrival rate of empty slots at the merge point. Continuity has been enforced in that the rate of vehicles leaving the station equals the rate entering = pS. Using a finite queue-length model to prepresent the waiting process at intersections provides the following relationships for no vehicles in the turn, PO,, the probability that exactly n vehicles are waiting to turn, P,,, the expected queue length at the intersection, LT, in terms of p, p, L, D, H, and the maximum queue length Mq:
(2) Pot= (1- S)/(l-
Thus, for L = l/2 miles and d = 5 miles, on average 16.7% of the vehicles entering an intersection desire to turn and the rest go straight. Also from the uniformity assumption, the station service rate pS, is given by:
S”*+‘)
P,,=(l-F)“/(l-f?“) L,= S/(1 - Is-(Mq w = pptl(l -p(l-
ps = pLID.
243
(8) (9)
+ l)s”“+‘/(l-6”~+‘) pt)).
(10) (11)
(3)
For one-second headways and p = 80%, L = l/2 miles and D = 6 miles each station on the network would service an average of 240 vehicles per hour. As mentioned the process of merging from a station to the main line is identical to that of merging at an intersection except the queue length for the intersection merge must be considered as finite because guideway construction would only allow short maneuver regions and vehicles can readily be denied access to the intersection (aborted turn). Access to stations is dependent on access to the arrival queue. For egress it is physically possible for the departure queue to extend into the arrival queue, thus, from an analytical standpoint, it is appropriate to consider infinite queue lengths in stations. The process of quasi-synchronous merging can be approximated by a standard queueing theory process of Poisson arrivals and exponentially distributed service times. (Actually we do not have a true Poisson process
In the above, ppt/H represents the mean arrival rate and I- ~(1 - p,)/H represents the mean service rate; their ratio is denoted by S. Synchronous policy The modeling of the sychronous control policy differs greatly from that of the quasi-synchronous case in that: -the probability of being able to confirm a reservation on each link in the journey made at a given request time to is a function of the number of confirmed reservations for that link which were made prior to time to. -reservations must be secured for each link of the journey, not just turn intersections. A passenger arrives at the origin station, 0, on some link L,, at time Q and attempts to schedule a departure for station D, starting at to see Fig. 1. The difference tn - TV can be controlled to yield link loadings about the nominal design value, p. For a given O-D pair there might exist a
244
A. L. KORNHAUSER and P. MCEVADDY
number of minimum paths, [PI,. . . PI]. Each path is uniquely determined by the sequence of links, thus:
contemporaneous rate, p., is:
line loading of p, the station process
Pj = (LjO, Lj,, . . .) 4,).
ps =
The reservation algorithm must verify a matching sequence of vacancies on the entire set of constituent links, i.e. a complete reservation is the m + I-tuple R = (To, T,, . . T,) where II; is arrival time on link L,;. Thus, for each link in the network it is necessary to maintain a list indicating, for each time step, whether a vehicle has reserved to enter onto that link. For example to reserve a path out of station 0, it is necessary to first check the list for link La0 at time t - L/2vt to verify if it has been previously reserved. If this slot is unreserved (or if the previously reserved trip has station 0 as its destination), the next step attempts to make a reservation at intersection #1 at the projected vehicle arrival time of
Therefore, reservation
pk.
the probability of being able to make a at time to to just get out of the station is: l.O-p(l.O-
L/D).
(14)
The probability of being able to reserve a slot at each succeeding link of the trip is dependent on the time into the future for which the reservation is being made as well as whether the vehicle desires to turn or go straight. If the vehicle goes straight through an intersection, call it the jth from its origin station, then the probability of securing a reservation based on a departure time at to is:
t,: L tl = to+%.
(12)
Entry onto the next link, L,,, can be blocked only if line p carries a vehicle which has made a reservation to turn onto line u at time tl (although it actually arrives on line a: at t, + (y. L/v)). The remaining intersections are tested in similar fashion. If a reservation cannot be made, the next iteration of the algorithm attempts to make a new complete reservation starting from station 0 at time to+At. Assuming a constant demand rate in both trip length and trip distribution implies that the percentage of slots confirmed by other vehicles on each link in the network decreases linearly: from the contemporaneous line loading at the entrance to the origin station to the station arrival rate at the last link of the trip, see Fig. 2. For a
P,=l-
PtPi l-(l-Pf)Pl
where p, = p(l - (2j - l)L/20). If the vehicle desires to turn at an intersection, the probability of securing a reservation is:
PL D
0
D TRIP LINK
Fig. 2.
tv is the nominal line speed.
k, then
where pk is, again, the effective line loading k intersections into the future. The probability of a departure at time to is then the combined probability of being able to secure a reservation on each link of the minimum path from origin to destination and is given by:
(17) where J is the set of no turn intersections, turn intersections.
F
(15)
K is the set of
Quasi-synchronous case Some quantitative results obtained from employing the queuing model discussed above are presented in Figs. 3-6. Figure 3 represents a composite characterization of system performance, displaying total average delay as a function on the line loading, p. For p = 80% the total delay, averaged over the uniform turn distribution discussed in the section on the demand model, is approximately 20 headways units. This includes delay in stations as well as delay encountered in making turns. It is noted that this delay becomes increasingly sensitive to increases in line loadings greater than p = 65%. Figure 4 demonstrates the sensitivity of average delay in the station to changes in the trip length factor, D/L, for line loadings approaching 80%. As expected decreased line spacings promote ease of egress in attempting a departure. Figure 4, however, suggests that this effect becomes less significant as nominal line loadings drop below p = 70%. This lack of sensitivity to changes in D/L for line loadings under 70% is apparent in the next figure, but the effect is not as pronounced. This figure depicts the
A quantitative analysis of synchronous
vs quasi-synchronous
network operations of automated transit systems
245
25
20 -
0.6 f
4 !I
13 -
w
0.6-
Lz s f
IO -
o.4 _ AT AN INTERSECTION
‘;’
5IN THE o50
, 60
I 70
60
MEAN LINE LOADING (% OF SLOTS OCCUPIED)
0
50
I 60 MEAN
STATION I 70
LINE
60
LOADING
Fig. 6. Fig. 3.
4\
p=7Q% p=60%
0’ IO
I 20 D/L
Fig. 4
‘.Ol-----l
Fig. 5.
33
variation of mean queue length in the station to changes in D/L for several values of the fundamental parameter p. For values of L = l/2 mile, D = 6 miles, p = 80%, the expected queue length is 0.5 whereas a decrease in p to 70% cuts the expected queue length to half that value. It is clear that as the spacings between lines is increased the probability of long backups into the station holding areas is minimized. It should be observed, however, that increases in D/L above 20 result in increasingly smaller improvements in obviating station backups. The implications for station design should not be ignored. Unlike the situation in the synchronous case, each turn on a trip under quasi-synchronous management is identical. This, of course, results from the fact that the contemporaneous line loadings at each turn is just the mean line loading p. Under the synchronous policy reservation for the turn has been made well in advance of the actual arrival time at the turn, consequently, the line loading must be attenuated by a factor depending upon when the turn is to be made. Therefore, no two turns of a trip can have the same probability of success. Since the quasi-synchronous policy does not rely upon the confirmation of reservations, it is sufficient to examine the performance of only one intersection to understand completely the turning behavior. The effect of line loading, the fundamental network parameter, on the expected queue size is exhibited in Fig. 6, demonstrating the same sensitivity as seen before to increases in p above 65%. Data on the behavior of the station queue is included for comparison. Recall that the two models are different; infinite queue lengths have been permitted at station departures whereas the queues at intersections have been restricted by a maximum queue size, in this analysis A4q = 8.0. Data from graphs not included in this paper demonstrate very small changes in performance for queue sizes larger than 5, while decreasing the probability of an abort. Typical delay at an intersection for p = 80% is about Sheadwaysanddropstoabout3.3headwaysforp = 70%. Synchronous case Figures 7-11 summarize the delay incurred for the synchronous network control policy. Mean delay versus contemporaneous link loadings are presented in Fig. 7 for various values of the nondimensional trip length factor,
246
A. L. KORNHAUSER and P. MCEVADDY D/L = I8 D/L : 15 /
j-
I
I
I
I
60 MEAN LINE
LOADING
I
70 OF SLOTS OCCUPIED)
(%
I
80
Fig. I.
18
D/L. The link loading, p, is an indication of the productivity of utilization efficiency of the guideway system. It is noted that: -The penalty for trying to get high productivity (large p) is to incur high average waits at the station, approximately 70 headway units for p = 80%. Even for a one-second headway system this implies a mean wait of about one-minute. The deteriorating performance of larger headway systems is obvious. -The reservation system is rather insensitive to p until p becomes greater than 70%. -The effect of changing trip length characteristic D/L is small except at high values of p.
The effect of D/L is better represented in Fig. 8. For large values of p the effect of D/L is large for small values of D/L and then approaches an asymptotic value for large values of D/L.Actually D/L is a measure of the number or reservations that need to be made for an average trip. For few reservations the addition of one makes a large change in the probability of the sequence of Bernoulli trials; however, for trips requiring a large number of reservations the addition of one makes little change in the probability of verifying a complete sequence of reservations. The individual probabilities of being able to secure a reservation at each succeeding link of the trip is presented in Fig. 9 for several values of D/L and p. Note that individual probabilities of making a reservation are much larger (e.g. 0.885 vs 0.51) for vehicles going straight rather than turning at an intersection, Figure 10 presents the cumulative probability of
A quantitative analysis of synchronous vs quasi-synchronous
247
network operations of qutomated transit systems
SYNCHRONOUS NETWORK CONTROL POLICY D/L
= 12
2 EQUIDISTANT
I
I
I
20
40
60
I
DELAY
I
I
80 100 120 TIME , T ( HEADWAYS)
TURNS
I 140
1
I60
I 0
Fig. IO.
SYNCHRONOUS
37.5% 2-turns, 37.5% 3-turns, 12.5% Cturns) we get a system average wait for reservation to be 82 headway units for D/L = 12 and p = 80%. It is also important to note that the calculations made for the synchronous case neglect the establishment of departure queues in stations, i.e. vehicles can be dispatched as soon as a sequence of reservations is confirmed. If queues were to form in stations then the data presented yield only the delay once the vehicle has reached the head of the queue. Total delays would be much larger. This has implications on station design in that a synchronous network control policy requires a parallel bay station configurations so that conflicts be e1iminated.t (The parallel bay design provides no advantage over the sequential bay design in the quasi-synchronous case).
-
_
QUASISYCHRONOUS
I
I
70
60 LINK
LOADING
Fig. II.
securing a complete sequence of reservations in less than T headways as a function of T for several values of D/L and p. Note that the 0.95 probability of securing a sequence of reservations is achieved after only I5 headway times for p = 50%, while it requires an order of magnitude more time (0.80 headway times) to achieve a 0.95 probability for p = 80% (D/L = 12). The above data has been presented for a fixed number of turns (either zero or two) equally spaced along the trip. Figures 7 and 9 indicate that the expected delay is very dependent on the number of turns. If we take the distribution of turns for a uniform demand (12.5% l-turn,
tExcept
equal time departures.
Comparison
Figure 11 summarizes the difference in performance between the two network management policies for a two turn trip with D/L = 15. Both demonstrate sensitivity to increases in line loading, but the delay experienced under synchronous control dominates for p greater than 70% (near 4 times as great at p = 80%). This relationship holds for other values of the parameter D/L. For large values of p the delay under the synchronous policy is very sensitive to changes in the trip length factor, as discussed above. This is not the case in the quasi-synchronous model, where changes in D/L effect only small changes in the expected delay in the station queue (leaving, of course, intersection queues unchanged); a change in D/L from IO to 30 increases the mean delay from 5.7 to 8.3 headway units (p = 80%). Thus the difficulty in obtaining a complete sequence of reservations in the synchronous case seems to out-weigh the advantages of the decrease in contemporaneous line loadings at intersections distant (in time) from the origin station. Further it has been shown above that there is a lower probability of successfully making a reservation to turn, particularly in the early stages of the trip. The sequence of turns is not significant
248
A. L. KORNHAUSERand P. MCEVALIDY
under the quasi-synchronous scheme. In examining the queueing model, the parameter to study is the ratio of mean arrival rate to mean service rate. For small values of this parameter the expected delay remains small, as does the expected queue length. As shown above this parameter remains much smaller than 1.0 for small values of p, the average percentage of those attempting to turn. Under nominal conditions it appears that p will remain relatively small (-20%). Design of the divert links to accomodate reasonable queue sizes minimize the probability of an abort. Even if an abort is necessary, the greater flexibility of the quasi-synchronous approach has an advantage over the synchronous scheme which is forced to reschedule the complete sequence of link reservations. (This is clearly the case if the aborted turn does not increase total travel distance and, consequently, total travel time.) EXTENSIONS
The preceeding discussion has established the capability of a quasi-synchronous control policy to provide higher levels of productivity for the assumed network/demand configuration. It was also shown that this scheme was better from the user point of view in that the average delays for a complete trip were lower. However, further analysis is warranted. In particular no attempt was made to quantify the effect of multiple paths in improving the performance of the synchronous scheme. Clearly, there exists a significant percentage of trips having several equilength and even equilength-equiturn paths (recall that reserving a turn is less probable than passing straight through an intersection). Hence the performance of the synchronous scheme can be improved. At present it is hard to say by how much. For an assumed number of equivalent paths, iV,the solution is obtained from a series of Bernoulli trials by calculating the probability of at least one successful reservation in N trials. This will be true only when the trials are independent and equiprobable. It will usually be the case, however, that the multiple paths will overlap to some extent and hence would not be disjoint events. Until this topological problem of the network is resolved the best that could be done would be to study the sensitivity of system performance for an assumed maximum possible number of equivalent paths and to verify what conditions will insure that system performance approaches that of the quasi-synchronous scheme. One possible variation of the synchronous scheme offers some promise of improving performance. Under synchronous control a reservation is thwarted if one slot
is occupied at one specific time. The new scheme, called tran-synchronous, would permit the reservation algorithm to resolve the conflict by looking for a vacancy that would arrive A headway units later at the merge point. The control system would then make the reservation and command the vehicle to slip A slots on the divert link while waiting for the reserved vacancy to arrive. (It should be noted that the waiting vehicle could potentially block a following vehicle from merging into its own reserved slot unless all maneuver slots have been reserved). Finally, it should be remarked that this paper has investigated only the steady state performance of the proposed models, ignoring the effect of relaxing the constraint on the assumed network/demand configuration. In particular the impact of letting demand level vary as a function of time should be investigated, particularly the effect on maximum queue size needed in the quasisynchronous scheme. In addition there remains the question of relaxing the assumption of geographical uniformity, permitting spatial distribution in the demand function.
REFERENCES
Brown S. J. (1973a) Merge control in automated transit system networks. Presented at Intersociety Transportation Conference, Denver, Colorado. AMSEpaper No. 73-ICT-109.
BrownS. J. (1973b)Designof car-followertype control systems with finite bandwidth plants. Proceedings Princeton Conference on Information Sciences and Systems. Godfrey M. B. (1968) Merging in automated systems: MIT Sc.D. Thesis (unpublished).
transportation
Kornhauser A. L. and Garrard W. L. (1973)Design of ootimal feedback systems for longitudinal control of automated transit vehicles. Transpn Res. 7, 125-144. Kornhauser A. L. and Dais J. L. (1973) Economic, environmental and design aspects of large-scale PRT networks. Highway Research Record No. 425, 2637. Munson A. V. (1972) Quasi-synchronous control of high-capacity PRT network, 325-350 of Personal Rapid Transit (Edited by Anderson, Dais, Garrard and Kornhauser). Dept. of Audio Visual Ext., Univ. of Minnesota. Roetler W. L. et al. (1973). Comparison of synchronous and asynchronous PRT vehicle management and some alternative routing algorithm. Personal Rapid Transit II (Edited by J. E. Anderson). Dept. of Audio Visual Ext., Univ. of Minnesota. Surber M. A. (1967) Feasibility of automatic guideway control. DIR-751, HUD Contract No. H.-777. General Research Corporation. Task Force on New Concepts in Urban Transportation, Planning for Persona/ Rapid Transit (1972). University of Minnesota. _ Wade R. W. (1973) A cost oriented control system for a large PRT network. Personal Rapid Transit II (Edited by J. E. Anderson). Dept. of Audio Visual Ext., Univ. of Minnesota.