Interactions Between Road Traffic Assignment and Signal Timing: A Simplified Model

Copyrighl © I FAC COl1lroi in Transportalioll Syslems. Vienna. Austria. 1986

INTERACTIONS BETWEEN ROAD TRAFFIC ASSIGNMENT AND SIGNAL TIMING: A SIMPLIFIED MODEL P. H. Fargier Cm/re dE/lldp,1 de,1 Trall,lpo/'/; L'rbllill; (CLTL'R), 8 Al 'fllIlI' Ari,/idl' Brialld, 92220 Baglll' lIx, Frall(e

Abstract. This paper deals with the following problem: a matrix of origin-destination demands of road traffic being given , determine signal timings minimizing the total travel time , knowing that each driver mini mizes his own travel time (first principle of Wardrop). It is known that minimizing delay for the observed traffic pattern may increase delay after establishment of the new traffic equilibrium . A simplifying assumption is used: for each link , the travel time is constant as long as the traffic is below capacity . Three types of problem are considered restrictions of capacity on certain links (in some cases this can improve the overall performance) , cycle splits into phases , and offsets between intersections . As a rule , this study does not allow to get the opt i mal solution . Some typical cases are considered , and guidelines to improve a solution and get local optima are given . Keywords. Traffic control; road traffic; game theo r y; research; optimisation; linear programming.

o pe r ati o ns

The first principle corresponds to an equilibrium of a game theory situation where each driver is a player . The second principle supposes a centralized un i que decision center (or a road pricing at a suitable "marginal cost " ). Here we dea l with the first case .

INTRODUCTION On a road network , a modification in signal control is likely to change the routes of some road users . However signal timings are usually chosen for a given assignment of traff i c (e . g . the well - known TRANSYT prog r am) . Of course , if changes of traffic f l ows are observed , these timings can be recalculated periodically . However , this process may not improve the overall performance . In fact , when choosing signal control , the reassignment of traffic result i ng of it should be taken into account . A similar problem arises in assignment programs using an iterative procedure between assignment and signal timing calculations .

The difficulties arise from the differences between the situation correspnnding to these two principles. If the second principle were realistic, an iterative procedure between traffic assignment and signal timing computation for a given flow would improve the performance and would be very likely to converge to an overall optimum . A first interesting difference between system and user optimized patterns is that , for the latter , a reduction of capacity on a link may be beneficial. Access control on freeways may be looked at as an applica tion of this property . But the main diffe rencies should concern signal timings at intersections. Let us illustrate this with the two following examples .

The purpose of this paper is to help to solve these difficulties. It deals with the following problem : a matrix of origindestination demands being given , determine signal control minimizing the total travel time , knowing that each driver minimizes his own travel time (first principle of Wardrop). As it is usually done, we consi der on l y static situations : flows and travel times are independant from time .

In the simple network shown in Fig . 1 (from Smith , 1978), the demands are FAB from A to Band FDE from D to E ; there is a signal at intersection I . Capacity at I is not sufficient to take care of the whole traffic FAB and FDE ' The route AGB has infinite capacity , no intersection , and a running time 2 minutes higher than run ning time on AIB .

Let us recall the first principle of Wardrop for traffic assignment (Wardrop , 1952) : " The journey time on all routes actually used are equal , and less than those which would be experienced by a s i ngle vehicle on any unused route" . It can be opposed to the second principle " The average journey time is minimum" .

Then , as long as signal I is set to allow FDE to flow through it , there will be a waiting time of 2 minutes at I on AI (equa lization of travel times on AIB and AGB) . Usual methods , tending to equalize satura tion on AI and DI , will create some satura tion on DI without any benefit for AI .

An equilibrium which satisfies the first princ i ple (resp . the second principle) is ca l led a user optimi7.e~ oattern (resp . a system optimized pattern) .

227

228

P. H. Fargier

principle allows us to eliminate discont inuities and to find directions of improve ments . Within this framework we study some typical situations .

G

D

11----1-----A ' -_ _----=.

B

E

Fig.l . Example of network In Fig . 2 , the traffic demand is FAB from A to Rand FBA from B to A ; ADB and AEB are two - way roads, AEB having an infinite capacity and no intersection . The intersection on ADB have a given cycle split into phases , and the resulting capacity for AB and for BA is c . Suppose that FBA < c < FAB and that running time on AEB (or BEA) is 5 minutes greater than on ADB (or BDA) .

To justify this simplification , remark it is when capacity is reached on some links that interactions between traffic control and route choice are likely to be important. Therefore the model should allow to under stand the main features of the phenomenon , and remain of interest in actual situations . Moreover , if necessary , the simplify i ng assumption could be discarded in a more elaborate theory . Next section deals with the properties of user optimized flows on a linear capacited network and of exponential assignment , and formulates the control problem in such a context . In following sections , applica tions to capacity restrictions , cycle splits into phases , and offsets between intersections are considered. ASSIGNMENT AND TRAFFIC CONTROL ON A LINEAR CAPACITED NETWORK Consider a network where X is the set of nodes and E the set of links. As mentioned above , in a " linear capacited network ", on each link e f E the travel time t(e) is equal to a constant b(e) provided that the traffic f(e) using e is strictly less than capacity c(e) (see Fig . 3).

Fig.2 . Example of network In these conditions, whatever the offsets between the signals may be , the sum of waiting times on the intersections from A to B will be 5 minutes (equalization of travel times on AEB and ADB). Therefore a coordination from A to B would be of no effect . The best solution is to coordinate from B to A, on the contrary of the natural tendancy to coordinate in the direction of the heavier traffic . Interaction between traffic control and route choice have interested researchers for a long time . First , studies proved that, in some cases , reducing capacities or increasing delays improve the overall performance of a network (Murchland , 1970 ; Smith , 1978 ; Fonlupt and others , 1980). Concerning the choice of cycle split into phases , Smith (1980, 1984) showed that instabilities may occur when interections with assignment are neglected , and that delays may even tend to infinity. He suggested some policies avoiding these instabilities. Other authors looked for methods optimizing a given criterion . Fisk (1984) presented various known formulations of the problem, added some new ones , and suggested two algorithms based on penalty functions . Here we use a simplifying assumption : on each link the travel time is constant as long as the traffic flow is below capacity. Thus we get what we call a "linear capaci ted network" . Moreover , the use of exponential assignment as a n approximation of Wardrop

t(e)

b(e) r-_ _ _~

+-- ---,.:-,---------.-

f (e ) c(e) Fig . 3 . Relation between flow and travel time on a link 0

First we must give a few precisions on the use of Wardrop first principle in such a case . If we assume that the travel time is still b(e) when e is saturated, it is usually impossible to find an equilibrium situation which satisfies this principle . In fact we assume that for f(e) = c(e) , t(e) can take any value greater or equal to b(e). This is quite natural : when on a bottleneck, the arriving flow exceeds the capacity , a queue is forming; then the travel time increases , but the leaving flow remains equal to capacity; eventually when an equilibrium is reached (static case), the flow entering also becomes equal to capacity , and we have f(e) = c(e) and t(e) > b(e) . Futhermore in some cases the vertical line can also be seen as an appro ximation of a convex function. Therefore we have the condition e f E, t(e) b(e) i f f(e) < c(e) , t(e) > b(e) c(e) , - if f(e)

for any

229

Road Traffic Assignment and Signal Timing

Traffic demand is defined by a set of values Fi', representing the amount of traffic w~ich has to go from node i to node j. We note fij(e) the flow going from i to j and using lLnk e. Therefore f (e)

.r J. l

L,

X

f i J' (e) .

System and User Optimized Assignments The assignment on such a network is studied in particular in Fargier and Fonlupt (1983). It is well known that the system optimized assignment is the solution of a linear program where the total travel time (in that case rE b(e) f(e)) is minimized . Let S be that fuinimum , and let u(e) be the value of the dual variable of the capacity constraint on e . It can be proved that system optimized and user optimized flow are identical , but the travel times are different : - for the system optimized solution(s) t(e) = b(e) - for the user optimized solution(s) t(e) = b(e) + u(e). The property gives a way of determining the user optimized pattern (from the solution of a linear program) and shows that the difference with the system optimized pattern consists only in the "unnecessary" delays u(e) on congested links . The total travel time of the user optimized pattern is T = S + tlEc(e) u(e) . Usually there is a single solution for the u(e) ' so But for some "critical" sets of values of the c(e) ' s (elE), there may be some indetermination in the u(e) ' s , and consequently in T . Then control policy should aim at choosing the best u(e) 'So When the c(e)'s vary without reaching any above mentionned critical set of values, the routes used by each relation, and the values taken by the u(e) 's and T remain constant . But discontinuities occur when crossing these critical sets. To avoid these discontinuities, it may be useful to use an exponential assignment as an approximation of Wardrop first principle (see Dial, 1971, and Fonlupt , 1981) . Exoonential

~ssignment

The traffic demand is defined by a set l Fij; i, jlX I , where Fij is the amount of traffic which has to go from node i to node j. Let Ki' be the set of possible routes i to j (klK is a subset of E) , and t(k) the travel time of route k . We have an "exponential assignment" if, for each relation (i,j) the flow fij(k) using route k is proportionnal to e - rt(k) , r being a positive parameter fiJ'(k)

=

F' . e-rt(k) / r e - rt(m) LJ m,Kij

Remark this can be considered as a logit model, with a uti-lit y (-rt(k)) . I f r is small, traffic is parted almost equally between all the routes of the relation . If r is great , traffic is more cODcentrated on routes of lowest travel times . We have the following pro p erties (given here without proof) - the assignment can be obtained by an iterative procedure,

- when r tends to infinity, the exponential assignment tends to the user optimized pattern, 1 - let V = r f(e) t(e) + - P where e lE r P = . r r fij(k) Log fij(k) L ,J lX k,Kij If r tends to inf ini ty, V tends to the total travel time T, when the travel times vary, we have (with notations of differential calculus) (1)

dV = efE f(e) dt(e) df i j (k)

=

r f i j (k )

(da,~ J' - d t ( k) )

(2 )

where Control Problem in a Linear Capacited Network The control policy allows to modify the c(e)'s, u(e)'s and b(e) ' s to some extent . Among the feasible values of these varia bles , we want to choose those which minimize the total travel time T. If we use the exponential assignment as an approximation, we will try to minimize V instead of T. The problem can be stated as choose the u(e) ' s and b(e) ' s such as the f(e)'s meet the capacity constraints . Formula (2) gives the consequences on f(e)'s of small variations of u(e) ' s and b(e) ' s , and allows to determine the feasible directions of variations . Among these directions , formula (1) allows to choose those decreasing V. Some examples of possible applications of these princi ples are given in the following paragraphs . USE OF LIMITATIONS OF CAPACITY As said in the introduction , it is known that, in some cases , reductions of capacity are beneficial . Let us start with the simple example illustrated Fig . 4 where - the traffic demand is FAE from A to E and FBE from B to E - c (DE) has a value c , the other capacities being infinite ; FAE < c < FAE + FBE - b(BGE) - (b(BD) + b(DE)) = 8b

A

B

G E

Fig.4. Example of network This may represent a freeway AE with an access in D. The user optimized solution gives : u(DE) = 8b, f(BD) = c - FAE' But by reducing the capacity of BD to c(BD) = c - FAE' we have

P. H. Fargicr

230

a possible solution with the same flows and u(DE) = 0 , u(BD) = b. Then the total trave l time T decreases by FAE . b .

be i ng reluced to 15 . Therefore the capacity of AB should be restricted to 2 .

We can make two remarks - wi th the capacity reduction on BD there is an indetermination between u(BD) and u (DE) ; i n fact the optimal solution (the flows being fixed) is given by the linear program :

But if we take the same example and a dd link OED (see Fonlupt and others , 1 980) , c(OF.D) =(1), b(OED) = 16 , the same method wi ll not \lork : in the user optimized solution , the r.raffic uses two routes OED a.nd OA BD , wi th a travel time equal to 16. However , by reducing c (AB) to 2 , we would get the previous solution which is better (travel time of 15) but with modified flows.

u(BD) + u(DE) Min

=

(j

b

f(BD) u(BD) + f(DE) u(DE)

- to make sure that , on the field , u(BD) = hb and u(DE) = 0 , either capacity on BD is rest r icted slightly below its theore t i cal value , or , better , queue lengths are controlled on line . Th i s examp l e shows that, in some cases , with our simplified model , the network performance can be improved by modifying th e u(e) ' s without changing the flows . Then the constraints on the u(e) ' s are g i ven by writing the equality of travel t i mes of routes effectively used. Thus , as above , we get a linear program for the u(e) ' s . In particular this method can be u sed for freeway corridor control i n cases mo r e comp l ex than the example of Fig . 4 . The same method allows to solve the problem p r e s ented by the " classical" lay - out of Fi g . 5 afte r removing link OED (see Murchland , 1970) . This problem may not be very r ealistic, but is interesting on a t heo r etical viewpoint . Supposes that : b(OA) b(AD) b(AB) c(OA) c(OB)

b(BD) b(OB)

CYCLE SPLITS INTO PHASES The choice of cycle splits seems to be the ma i n type of possible app l ications of th i s study . To formalize the problem , consider a junction with n phases. We wi ll f i rst assume that on each phase i there is on l y one critical flow , arriving on link e i f E . We denote : - S(ei) the saturation flow of lin k e i (capacity i f it is always on green phase) , - Pi proportion of effective green of pha.se i (with respect to cycle length) , - mi the minimum possib l e for Pi' n

Then we have c(ei) = Pi s(ei) and i £ 1 P i = P (P < 1 because of lost time between effective greens). Therefore the equat i ons are

5

10

2

c(BD) c(AD)

Ca l culus shows that the use of expone n t i a l assignment allows to go toward the optimum in this case : increasing u(AB) al l ows t o decrease V as long as OA and BD are used at capacity . This kind of method could be used in more complex networks but may g i ve a local optimum only .

3

c(AB)

n ~

'"

;; 1

a(e i ) c(ei) •

and c(ei)

•

~

=

1 , with a(ei)

=

P/s(ei)

mi s(ei)

A

D

o

E I-_~M +

_ _- i G

E

A Simple Example B

Fig.5

The problem is to find the c(ei) ' smeeting these relations and minimizing T . If on some phases several links must be cons i dered, we will have several relations of the same type .

Consider the example Fig. 6 (sketch of a ring road and two radial roads inter sectiong at M) slightly more complex than the one of Fig . 1 in the introduction. We have two O- D relations from A to Band from E to G with

Fig . 6

There is only one relation, from 0 to D, FOD 4. The user optimized pattern gives u(OA) u(BD) = 3 , f(OA) = f(BD) = 3, f(OB) = f(AD) = 1 , f(AB) = 2, the travel time from 0 to D is 18 .

-

If we want to minimize T, the flow on each link being given, we get : u(OA) + u(AB) = 3 u(AB) + u(BD) = 3 Min 2u(AB) + 3u(OA) + 3u(BD) The optimal solution is u(OA) = u(BD) u(AB) = 3 , the travel time from 0 to D

0 ,

FAB = FEG = F at junction M, c(AM) + c(EM) = c b(AM) + b(MB) = b(EM) + b(MG) = bl b(AE) + b(EB) = b(EB) + b(BG) = b2 > bl capacity on AEBGA is infinite .

It is easy to find the optimal control of junction M, as a function of F : if F < c/2 , c(AM) = c(EM) = c/2 is an optimal solution, all the flow using the direct roads AMB and EMG - if c/2 < F < c , wi th c(AM) = c(EM) = c/2 , a part of each OD flow would have to use

Road Traffic Assignment and Signal Timing

the ring road, and there would be a waiting time (b2 - bl) at M on AM and on EM ; the optimum consists in favouring one of the streams, for instance by giving enough capacity to AMB (c(AM) > F) to make the travel time AB equal to bl - if F > c, for any feasible solution, the travel times of both relations are equal to b2 For c / 2 < F < c the traditionnal methods tending to equalize the saturations would probably lead to a solution where both relations would have a travel time b2' In fact, as long as c - F < c(AM) < F, variations of c(AM) have no effect on the overall performance ; it is therefore difficult to find a direction of improvement. To get over this difficulty, we can use the exponential assignment. Then V has a maximum for c(AM) = c(BM) = c/2 and there are two local minima at the points where u(AM) or u(BM) becomes equal to O. Networks with only one Saturated Intersection We consider a network with several 0-0 relations, but where only one intersection is saturated, i.e. has some entrance links working at capacity. To simplify the presentation, we suppose the intersection has only two entrance links el and e2' and that we have c(el) + c(e2) = c When c(el) varies from 0 to c, u(el) (resp. u(e2» is a non-increasing (resp. non decreasing) step function of c(el) (see Fig. 7). For a given value v of c(el), T is equal to a constant minus the sum of the two hatched areas of Fig. 7.

231

seems difficult to explore all the possibilities. Here again, the use of exponential assignment would allow to find directions of variation of the u(e) 's decreasing V and meeting capacity constraints. Of course the minimum reached may be only a local minimum. Note that this method would give a solution including the possible benefits of capacity reductions studied abo v e. OFFSETS BETWEEN INTERSECTIONS As an approximation, a change of offset between two successive intersections can be considered as modifying the value of b(e) of link e between them, without changing the value of c(e). In traditionnal methods, these offsets are chosen as if they did not interact with the assignment of traffic on the network. When there is no saturation (i.e. the u(e)'s have zero values) this is probably acceptable. But in saturated situations, the expected advantages may be suppressed by increases of waiting times in queues. The situation presented in the introduction and illustrated in Fig. 2 is an example where variations of b(e)'s on links on ADB directed from A to B have no effect on T, while in the other direction we have dT = ~DA f(e) db(e). On more complex situa1:'ions, we have dT = efEh(e)f(e) db(e), where the h(e) 's are depending on the general traffic pattern. These h(e) 's should be taken into account in the choice of offsets. To determine them, one must study the user optimized pattern or make use of the exponential assignment. CONCLUSION

u(e)

o

At this point of the study, some questions may arise about the possible applications and continuations of this work.

v

c

Fig.7. Capacities and waiting times at the intersection of two links Three remarks can be made : there ma y be several local minima - usually a small variation of c(el) around a given v alue does not show a direction of improvement ; the use of exponential assignment would allow to smoothe the graphs of u(el) and u(e2), and at least get a local minimum; but the only systematic wa y to look for the overall optimum is to study the whole range of possible variation of c(el) - the graphs of u(el) and u(e2) could be obtained experimentally by testing several values of c(el)' General Case If several intersections are congested, it

First, is the model sufficiently realistic? It seems to be relevant in saturation periods when some links works at capacity, and some users modify their routes because of the congestion. In fact, if needed, more complex features could be inserted in the model : limitations of queue lengths, variation of travel time as a function of flow, etc ... Another question concerns the methods. Work has still to be done in order to define them more precisely (problem of local minima, value of r, steps of variations, ... ). Moreover these methods should be compared to those given by Fisk (1984). On a more practical viewpoint, it would be necessary to study some actual situations in order to evaluate the stake of such an approach. The applications could be off line, by using programs integrating traffic assignment and signal control, or on line, by controlling signals and guiding users, knowing traffic flows and lengths. A drawback is the necessity of having some knowledge of OD demand. A comparison should be made with Smith's (1981) approach which avoids this necessity.

P. H . Fargier

232

In fact , a useful result of this kind of study may be to show the interest of thinking about traffic assignment when defining a signal control policy , even if this is not done in a formalized way. REFERENCES DIAL , R.B. (1971) . A probalistic multipath traffic assignment model with obviates path enumeration. Transp . Res ., Vol . 5 , 83-111 . FARGIER, P.H., and J. FONLUPT (1983) . Affectation de trafic selon le principe de Wardrop dans des reseaux avec capa cites. Universite Scientifique et Technique de Grenoble, RR 205 . FISK, C.S . (1984). Optimal signal controls on congested networks. Ninth Inter national Symposium on Transportation and TraffLc Theory . FONLUPT , J. , A.R . MAHJOUB, and J.P . UHRY , (1980). Amelioration de l ' ecoulement du trafic routier par des restrictions de capacite . Universite Scientifique et Technique de Grenoble , RR 205. FONLUPT, J., (1981) . L ' affectation exponen tielle et le probleme du plus court chemin dans un graphe. RAIRO Recherche Operationnelle , Vol . 15, N° 2 , 165 - 184 . MURCHLAND , J.D . (1970). Braess ' s paradox of traffic flow . Transp . Res ., Vol. 4, 391-394 . SMITH , M. J. (1978). In a road network , increasing delay locally can reduce delay globally. Transp . Res . , Vol . 12 , 419-422. SMITH , M.J . (1980) . A local traffic contro l policy which automatically maximises the overall travel capacity of an urban road network . Traf . Eng . and Control , 21 , 298 - 302. SMITH , M. J. (1981). A theoretical study of traffic assignment and traffic control. Eight International Symposium on Transportation and Traffic Theory . WARDROP, J . G. (1952) . Some theoretical aspects of road traffic research . Proceedings of the Institute of Civil Engineers , Part 11 , 1, 325-378) .