The rolling horizon scheme of traffic signal control

Transpn Res.-A, Vol. 32, No. 1, pp. 39±44, 1998 # 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0965-8564/98 $19.00+0.00

Pergamon PII: S0965-8564(97)00017-7

THE ROLLING HORIZON SCHEME OF TRAFFIC SIGNAL CONTROL G. F. NEWELL

Institute of Transportation Studies, University of California, Berkeley, 94720, U.S.A. (Received 5 August 1996; in revised form 27 April 1997) AbstractÐA comparison is made between the rolling horizon strategy of trac signal control at an isolated intersection and the strategy of switching the signal when a queue vanishes. It is shown that the former strategy has some undesirable consequences. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: highway trac, trac signals, control strategies, 1. INTRODUCTION

The following is, in part, a response to the review by Michael Bell (1993) of the book on highway trac signals (Newell, 1989). Bell thought the book should have included more discussion of `modern thinking' on trac signal control and, in particular, justi®cation of its criticism of the `rolling horizon' scheme of trac control for an isolated intersection. In this scheme one chooses a strategy at each time t so as to minimize the total delay during some future ®nite time period (the horizon). As time goes on the horizon also moves forward (rolling). Alan Miller (1963) seems to have been the ®rst to propose a strategy of this type at a symposium in London. In the discussion following his presentation, however, it was pointed out that this myopic strategy could lead to some queues never being served even though the signal could operate so as to serve all trac streams. Unfortunately, the criticism of this was never published and the scheme (or variations of it) has since been revived and attracted a large following, Robertson and Bretherton (1974), Chen et al. (1987), Bell (1990), Heydecker (1990), among others. Some authors, recognizing the dangers of Miller's strategy have added an extra delay penalty depending on the state at the end of the horizon. The penalty is evaluated from empirical curve ®tting and seems to be an approximation to the delay that would result over a second horizon. If so, then this is similar simply to doubling the length of the horizon. In this control scheme one tests, at each time point of closely spaced time intervals (a few seconds), whether or not one should switch a signal. The actual calculation of delay can be quite tedious if the horizon covers several cycles but, for most time points, the conclusion is rather obvious. Certainly one would not switch a signal if it is serving trac at the maximum rate. One would not pay a penalty of a `loss time' to switch to another signal phase which would serve trac at the same or a lower rate. (This conclusion should result for any reasonable objective). The issue is whether or not one should continue a phase after the service rate drops because a queue vanishes. If the signal is serving two trac streams simultaneously as at a four-way intersection (with or without turning movements), the service rate may drop after one of the two queues has been served or after both have been served. 2. EXTEND THE GREEN?

Consider ®rst the question of whether nor not one should continue a phase after the last queue has vanished if the signal is serving two trac streams or, after a single queue vanishes, if the signal serves only one stream at a time as on a one-way street. It is tempting to argue that if a vehicle is expected to reach the intersection within say 3 s after the queue vanishes then, by extending the phase 3 s to serve this vehicle, it will save a time equal to the time it takes the signal to serve the cross street and return (probably at least 20 s) plus the cost 39

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G. F. Newell

associated with one stop. The immediate penalty to the cross street is that each waiting vehicle is delayed an extra 3 s. If there are only say four vehicles on the cross street at the time, the total delay to them is only 43=12 s, so it seems that one has saved 8 s of delay plus one stop. The issue is not that simple, however. If one extends the green 3 s, one or more vehicles may arrive on the cross street which will cause the cross street green to be longer (with a vehicle-actuated signal), which, in turn, will increase the next green time of the original direction, etc. Increased delays may persist for many subsequent cycles. To illustrate this, consider a hypothetical intersection with ¯ows which are symmetric with respect to interchange of N±S and E±W. Let Ci be the ith half-cycle, a lost time L for switching the signal plus the green time for the N±S or E±W direction. Even i means N±S green, odd i means E±W green (or vice-versa). The i+lth half-cycle is equal to L plus the time needed to serve the vehicles which arrived in the ith half-cycle (plus those which arrive before the queue vanishes). The time Ci‡l ÿ L required to serve the queue in the i ‡ lth half cycle is proportional to the number of arrivals in the ith plus i ‡ lth cycle. If we neglect stochastic e€ects and interpret the Ci as the expected time, then for some constant  Ci‡l ÿ L ˆ …=2†…Ci ‡ Ci‡l † Ci‡l ˆ

2L  ‡ Ci : 2ÿ 2ÿ

…1†

The stationary solution of this equation is C ˆ L=…1 ÿ †:

…2†

The  is interpreted as the `trac intensity' or `degree of saturation'. Suppose now that we were to extend one of the green times, say C0 . The transient solution of (1) is …Ci‡l ÿ C† ˆ …Ci ÿ C† with ˆ =…2 ÿ † ˆ

1 ÿ …1 ÿ † 1 ‡ …1 ÿ †

or Ci ˆ C ‡ …C0 ÿ C† i :

…3†

If q is the arrival rate of all trac in either the N±S or E±W directions, the expected number of vehicles on the cross street at the time the green is extended is qC. These vehicles are each delayed an additional time C0 ÿ C. But this also causes an additional average green of the cross-street of (C0 ÿ C† , etc. The total delay caused by the extension C0 ÿ C is 1 X …qC†…C0 ÿ C† i ˆ qC…C0 ÿ C†=…1 ÿ † iˆ0

which is larger by a factor of 1=…1 ÿ † than the delay in only the ®rst half-cycle. Even if one considers a horizon of several cycles, one may signi®cantly underestimate the delay caused by extending the green in a single cycle if  and are close to 1. If one repeats this strategy by also extending subsequent green times after a queue vanishes, this is equivalent to having a larger L i.e. a longer mean cycle time (and larger average delay). This is not the worst possible consequence of the rolling horizon strategy, however. 3. SHORTEN THE GREEN?

Suppose that the service rate drops towards the end of a green interval of a four-way intersection because the queue vanishes in one of the two directions N or S, or the signal is serving a queue

The rolling horizon scheme

41

from a turn bay with a `lagging left' turn signal after the through trac queues have been served. Again we assume that the intersection and the ¯ows are symmetric with respect to interchange of N±S with E±W. Suppose that prior to time 0 the signal had been serving all trac streams switching the signal from N±S to E±W or vice versa when the queue being served last vanishes. After time 0, however, we decide to terminate each green early in such a way as to permit a residual queue Q…t† to grow at an average rate. Q_ ˆ dQ…t†=dt: _ i per half-cycle. (This is the This is the average rate over a cycle; the residual queue grows by QC total residual queue regardless of direction). We wish to test if the bene®t of shorter subsequent cycle times to the vehicles being served will exceed the penalty of allowing a residual queue to grow over a ®nite horizon. If we had served these residual vehicles they would have been served at some rate s (less than _ i =s. the rate s at the start of green) so the half-cycle has been reduced by QC The new recursion relation for the Ci is the same as eqn (1) but with  replaced by _  0 ˆ  ÿ Q=s with a new equilibrium half-cycle time L=…1 ÿ 0 † We are not so much concerned here with evaluating some `optimal' Q_ as determining if some Q_ > 0 is `better' than Q_ ˆ 0 relative to the delays over some ®nite horizon. _ the new equilibrium cycle would be If we were to choose an arbitrarily small positive Q, C‡

dC 0 CQ_ : … ÿ † ˆ C ÿ  d s …1 ÿ †

…4†

The cycle time, however, would not reach the new equilibrium immediately. The transient solution _ starting from Ci ˆ C for i ˆ 0, is (to ®rst order in Q) Ci ' C ÿ

ÿ
CQ_ 1 ÿ i : s  …1 ÿ †

…5†

Suppose also that  and are suciently close to 1 that the cycle time varies only slightly from one cycle to the next and we can treat the cycle time as a continuous function of time t ˆ iC and i ˆ exp…‰t=CŠ ln † ' exp…ÿ‰1 ÿ Št=C† so eqn (5) becomes C…t† ' C ÿ

_  CQ=s 1 ÿ exp…ÿ‰1 ÿ Št=C† : …1 ÿ  †

…6†

The vehicles which arrive and are served in the cycle with half cycle C…t† are delayed on average half the `red time', ‰C…t† ‡ LŠ=2, provided that the service rate remains constant during most of the green time. If the signal is serving two trac streams during a half cycle and the queue of one of the streams vanishes causing the service rate to drop, the average delay is still ‰C…t† ‡ LŠ=2 for those vehicles which su€er a positive delay, but some vehicles which arrive when a queue is zero su€er no delay. In this case one could represent the average delay per vehicle as K‰C…t† ‡ LŠ=2 for some K, 0 < K < 1. If the arrival rate of all vehicles (N, S plus E, W) is 2q, the average delay per unit time is qK‰C…t† ‡ LŠ. _ If Q_ is kept constant, the average The vehicles which are not served accumulate at a rate Q. _ The total average delay per unit time at delay per unit time of these vehicles at time t is Q…t† ˆ Qt. time t is therefore  _  KqCQ=s t…1 ÿ †s _ ÿ1 ‡ ‡ exp…ÿ‰1 ÿ Št=C† : …7† qK‰C…t† ‡ LŠ ‡ Qt ˆ qK…C ‡ L† ‡ …1 ÿ † KCq

42

G. F. Newell

We note ®rst that if we were serving the intersection of two one-way streets with a constant saturation ¯ow s ˆ s, then K ˆ 1 and the condition that the signal be undersaturated is that 2q=s ˆ  < 1. But if this is true then the quantity in the braces of eqn (7) is positive for all t; the linear term in t dominates the other two terms which means that a Q_ > 0 causes an increased delay. This is consistent with the intuitively obvious fact that it is not advantageous to terminate a signal phase while the signal is serving trac at the maximum rate, s. One would not choose Q_ > 0. Suppose, however, that this is a four-way intersection and that the queue of southbound vehicles vanishes before the northbound queue. Then s=2q could drop from approximately 1 to s  =2q ˆ 12 when the ®rst queue vanishes. Even worse, one could have a left turn lane plus two through lanes in each direction so that the s  =2q drops to approximately 14 while serving the turning trac. Now the quantity in the braces is negative for small values of …1 ÿ †t=C; there is a short-term bene®t from having Q_ > 0. For …1 ÿ †t=C much larger than 1, however, the exponential term decays rapidly and the positive linear term dominates. The penalty for the growing residual queue eventually dominates the bene®t from a shorter cycle time. The rolling horizon strategy evaluates (and minimizes) the total delay over some time T; in our approximation this is the integral of eqn (7) over …0; T†. We do not consider here the strategy which minimizes delay but certainly if the integral with respect to t of the quantity in the braces of eqn (7) is negative, this would imply that it is advantageous (relative to the horizon T) to allow a residual queue to grow. This condition is ÿT ‡

T2 …1 ÿ †s C  ‡ 1 ÿ exp…ÿ‰1 ÿ ŠT=C† < 0 2KCq 1ÿ

or if we let

this becomes

T  ˆ …1 ÿ †T=C

…8†

ÿ 
1 ÿ 1 ÿ eÿT =T  > T  ;

…9†

with ˆ

…1 ÿ †s : …1 ÿ †2Kq

A graph of the left hand side of eqn (9) is shown in Fig. 1. If we also draw lines of various slopes  for the right hand side of eqn (9), one can see immediately the range of T  for which (9) is true

Fig. 1. Evaluation of eqn (9).

The rolling horizon scheme

43

for various values of a. In particular this is true for T  < 2
5 if  ˆ 14 and for T  < 7 if  ˆ 18. For  close to 1, 1 ÿ  2…1 ÿ † so  ' s  =4Kq and  ˆ 14 would be typical for a signal serving two through trac movements (N and S or E and W). This means that for a four-way intersection with no turning trac and somewhat heavier trac N than S (and E than W), a rolling horizon strategy would switch a signal before the N bound queue vanishes allowing a residual queue N and E to grow if the horizon T is less than about (5/4) C=…1 ÿ †, about 2.5 full cycles for  ˆ 0
75, but about 6 full cycles for  ˆ 0
9. The cycle time itself increases as  ! 1, C ˆ L=…1 ÿ †, but this critical horizon length is proportional to L…1 ÿ †ÿ2 . In the rolling horizon strategy one does not actually follow the `optimal' strategy over the horizon, one only implements the strategy at the start of the horizon. At the next time point one implements the strategy for the new (rolling) horizon. If by virtue of the above arguments one would allow a Q_ > 0 starting from an equilibrium state, one would implement this strategy by terminating a phase at the ®rst opportunity just before a queue vanishes. The condition that one would allow a Q_ > 0, however, is based upon successive values of the Ci , not on the initial queue lengths. If one initially had some residual queues, but had been using a half cycle C , and the above conditions are true, then one would still allow Q_ > 0 by choosing a shorter cycle time. This would cause the residual queue to become even larger. The signal would, forever, operate on some cycle time less than C and be `oversaturated'. On the other hand, if the horizon is suciently large, so that eqn (9) is not true, one would not allow a residual queue to grow. If one had a residual queue initially one would choose Q_ < 0 and eventually eliminate the residual queue. 4. SOME SIMPLE ARGUMENTS

There are simple arguments that usually imply that, at an isolated intersection, one should switch a signal when the (last) queue vanishes, i.e. this minimizes the delay over an in®nite horizon for a stationary (possibly stochastic) arrival process. A trac intersection can be modeled (approximately) as a Markov process, which means that the optimal strategy at any time should depend on the `state' of the system at that time. Suppose then that the queues in the N±S direction have cleared. If one extends the green the queues will still be zero in the N±S direction at a later time but the queues will be larger in the E±W direction. They will continue to grow as long as one extends the N±S green. Clearly there must be some critical size of the queue such that one will switch the signal when the queues reach this size. If, however, the queues are already larger than this critical size when the N±S queues ®rst disappear (the typical situation), one should switch immediately. There is an even stranger argument why one should not allow a residual queue (if the intersection is undersaturated). If one switches a signal before a queue vanishes, the residual vehicles must be served sometime. It is advantageous to serve them later only if there is some time in the future when they can be served at less cost than now, but at the end of the next cycle or any cycle thereafter the cross trac queue will be no less than now. One would postpone serving the residual queue only if the expected arrival rate were decreasing so that there would be shorter queues at some future time. In the rolling horizon strategy one can allow a residual queue to grow because there is no penalty for having a residual queue beyond the horizon. This strategy does not recognize that one must serve the residual queue sometime with the same penalties as the present no matter how long one postpones it.

REFERENCES Bell, M. G. H. (1990) A probabilistic approach to the optimization of trac signal settings in discrete time, Proceedings of the 11th International Symposium on Transportation and Trac Theory, Tokyo, pp. 619±631. Elsevier, New York. Bell, M. G. H. (1993) Book Review. Transportation Research A 27, 158±160.

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Chen, H., Cohen, S. L., Gartner, N. H. and Liu, C. C. (1987) Simulation study of OPAC; a demand responsive strategy for signal control, Proceedings of the 10th International Symposium on Transportation and Trac Theory, Boston, pp. 233± 249. Elsevier, New York. Heydecker, B. G. (1990) A continuous time formulation for trac-responsive signal control, Proceedings of the 11th International Symposium on Transportation and Trac Theory, Tokyo, pp. 599±618. Elsevier, New York. Miller, A. J. (1963) A computer control system for trac networks, Proceedings of the 2nd International Symposium on the Theory of Trac Flow, London, pp. 200±220. OECD, Paris. Newell, G. F. (1989) Theory of Highway Trac Signals. Institute of Transportation Studies, Berkeley, CA, ISSN 0192-5911. Robertson, D. I. and Bretherton, R. D. (1974) Optimal control of an intersection for any known sequences of vehicle arrivals, Proceedings of the 2nd IFAC-IFIP-IFORS Symposium of Trac Control and Transport Systems. Monte Carlo.