Synchronization of traffic lights for high flow
It is assumed here that the flow of traffic is so heavy that it can be described qualitatively by existing theories wherein each car follows another with no passing and that platoons formed by traffic lights spread so much between lights that the time between the arrival of the first and last cars of a platoon at a light exceeds the green period. Under such conditions it is shown that the best pairwise synchronization of lights gives the best overall synchronization for one-directional traffic. The synchronization which minimizes the total delay for all cars is one for which the tail car of a platoon suffers no delay but the lead car is stopped. For two-directional traffic, heavy in both directions, the best synchronization is very likely to be that which is optimal for the direction of traffic with the lower flow. Introduction. It is an unfortunate fact that in many urban areas sequences of traffic lights are synchronized to minimize the delay for a driver traveling at the local speed limit. This usually works well when the traffic density is low but not during rush hours when the optimal synchronization is needed most and no one is able to maintain the speed for which the lights are timed. In a previous paper [1] some theory was described relating to the synchronization of lights when the volume of traffic was so low that interactions between cars could be neglected. Here we consider the opposite case where the traffic is dense and the flow through the intersections near saturation. Under such conditions it is reasonable to assume that passing is rare and that cars behave approximately as described by various continuum [2, 3] or car-following theories [4-7]. The object is only to describe a few qualitative conclusions. The general problem of how one should synchronize traffic lights is difficult for several reasons. First, very little is known about the dynamics of traffic flow. The only reasonably satisfactory theories proposed so far are those in which the effects of passing can be neglected either because the density is so low that cars seldom interact or the density is so high that there are few opportunities to pass. Secondly the effects of traffic lights on even simple flow patterns can be complicated because generally the pattern of arrivals of cars at one light depend not only upon the synchronization of the nearest previous light but the second, third, etc. lights as well. Finally in any real two dimensional network of roads one can not synchronize the lights on one highway independent of the others since the optimal synchronization for the network may not be the optimal for any given highway. Here we avoid the first problem by assuming that the density is very high and we avoid the last problem by disregarding the cross-traffic. We shall see that the second problem disappears if the flow is high enough and the lights not too close together. The delays suffered at one light then depend only on the synchronization of the nearest previous light but are independent of lights further away. Thus the optimal synchronization of a long series of lights can be determined by a successive synchronization of con*Received April 8, 1963. This research was supported by a grant from General Motors Corporation' 316 G. F. NEWELL [Vol. XXI, No. 4 secutive pairs. This is particularly convenient when lights are not equally spaced along the highway since otherwise the optimal synchronization of one pair of lights would depend not only upon the distance between this pair of lights but also upon the distance and synchronization between the other lights. In the next section we shall describe some qualitative properties of one-directional traffic through a sequence of lights as predicted by existing car-following theories and identify the conditions necessary to guarantee that the delays suffered at any light depend only upon the synchronization of the previous light. We next consider what synchronization will minimize the total delay under these conditions. If the platoons approaching a light have spread so much in traveling from one light to the next that the time interval between the first and last cars exceeds the green time of the light then some car is certain to be delayed. We conclude that the total delay to all cars is least if the last car arrives at the light just before the light turns red and suffers no delay but the lead car arrives too early and is stopped. This has less delay, for example, than the commonly used synchronization in which the lead car arrives just as the light turns green but the last car consequently arrives after the light turns red. No attempt is made to evaluate the optimal phase of lights in terms of the velocities of the drivers, spacings, etc. In practice one can observe directly some average arrival times of last cars in platoons and then synchronize the lights so they turn red immediately afterwards. From an experimental point of view it would require less effort to determine the synchronization directly from observed arrival times than to record the velocities of the drivers, distances between lights, etc. which one would need to know in order to compute the arrival times. Finally we consider the corresponding problem for two directional traffic heavy in both opposing directions. The conditions for optimal synchronization in this case is complicated, however, and is sensitive to certain unrealistic details of the model. The model suggests, however, that in most cases the optimal synchronization will be nearly that for the direction of traffic with the lower flow, contrary to what one might expect. This is true only if the traffic is sufficiently heavy in both directions, however. The model. Suppose we have a source of traffic approaching the first of a sequence of traffic lights. This is represented at the bottom of Fig. 1 by the series of solid line space-time trajectories of cars. The traffic light at x — 0 and the red periods are indicated by the heavy horizontal line segments. For simplicity, we assume that the flow q, the average number of arrivals per unit time, approaching the first light is less than the capacity of the intersection and if T is the cycle time of the light, that qT — n is integer. The latter assumption is not essential but if it were not true, the flow would be more complicated in detail because the number of departures in each light cycle would have to oscillate between integer values on either side of qT in such a way as to give the average qT over many cycles. Present theories of traffic flow at high density predict the following behavior. As cars approach the first intersection during a red period, a deceleration shock moves back into the stream indicating the rear of the queue. When the light turns green a fan of acceleration waves shown in Fig. 1 by the broken lines is formed. As these waves move back to the shock, the shock gains forward speed, eventually changes direction, and passes through the intersection. If the flow approaching the intersection is less than the capacity of the intersection, this shock will pass the intersection during the green period as shown in Fig. 1. Cars which arrive after the shock has passed the intersection will 1964] SYNCHRONIZATION OF TRAFFIC LIGHTS 317