THE EFFECT OF LEFT TURNS ON THE CAPACITY OF A TRAFFIC INTERSECTION.
A model is proposed for the traffic flow through a fixed cycle traffic signal on a narrow two lane highway. The average flows in the two opposing lanes are computed when the queues are arbitrarily long assuming that left turns are permitted and occur with fixed probabilities in the two lanes. It is found that the existence of left turns tends to favor short cycles of the light and, under certain conditions, the competition between this effect and the obvious advantages of the long cycle light gives rise to an optimal cycle time at which the average flows have a maximum value. Some simple models for multilane highways are also considered and show optimal cycle times. It is possible to analyse in considerable detail the queuing problem for a fixed cycle traffic light on a single lane highway at which cars are not allowed to make left or right turns [1]. The corresponding problem for a highway with two lanes of opposing traffic in which cars are allowed to make left and right turns (particularly the left turns) is much more difficult because cars that wish to turn left at a typical intersection with no special facilities for left turning cars interact with cars traveling in the opposite direction. The two queues of traffic in opposing lanes are therefore not statistically independent and one must consider the behavior of two queues simultaneously. One of the major complications in this problem arises from the fact that cars in lane 1 may turn left if there is a sufficiently large gap in the traffic of lane 2. To analyse this feature of the traffic flow, one must study rather detailed models which specify the size of the required gap and the statistical properties of 'the entering traffic. It is, however, relatively easy to determine from simple models the conditions under which the queues in both lanes will grow indefinitely. This defines the maximum traffic that the intersection can carry before both lanes have simultaneously reached capacity. If the traffic of lane 2 is below capacity, the capacity of lane 1 decreases with increasing flow in lane 2 and vice versa. The flows at which both lanes reach capacity is therefore also the minimum capacity of each lane as a function of the flow in the opposing lane. If the incoming traffic flows in both lanes equal or exceed the average rate at which cars can go through the intersection, the queues grow indefinitely and the behavior of cars at the intersection is independent of the queue lengths or the arrival times for the incoming flow. Furthermore, there are no gaps in the flow through which a car can turn left, except at the end of the green light period. The average rate at which cars can go through the intersection under such conditions therefore depends upon only a relatively few properties of the intersection. We consider here a model intended to simulate an intersection on a narrow two lane highway on which there is no room for passing at the intersection. Our purpose is to *Received Jan. 4, 1958. "Alfred P. Sloan Foundation Research Fellow. 68 GORDON F. NEWELL [Vol. XVII, No. 1 find the flow through the intersection when capacity is reached or exceeded in both lanes. The model is described by the following properties: 1. Each car in lane 1 will turn left with probability p and each car in lane 2 will turn left with probability p'. 2. During any cycle of the traffic light, cars will leave the intersection on a first come first served basis only at discrete and specified times tx < t2 < • • • < tn (the same for both lanes) with the most one car leaving the intersection in each lane at each time tj. For a less rigid interpolation of this, we could say that we observe the intersection only at discrete times U ,t2, • • • chosen so that tj — f,_i represents some average time interval between cars leaving the intersection under the ideal situation in which all cars move forward. We then approximate the exact leaving times defined in some suitable way by a neighboring value of t,-, either the preceding value of tf, the nearest t,-, or the subsequent value of tj . It makes little difference how one actually does this. 3. Forward moving cars or right turning cars in each lane will leave at consecutive times tj but a car that wishes to turn left must wait until a left turning car appears in the opposing lane or until the time tn , the end of the green cycle of the light. We neglect the possibility that a left turning car in lane 1 might be able to turn left if a car in lane 2 turns right. This we can eliminate either by saying that there are no right turning cars or that this practically never happens even if there are right turning cars. Actually one could extend the model to include such possibilities but at the expense of complicating the algebra. Although the same techniques described below could still be applied to such an extension, the capacities would depend upon two more parameters, the probabilities of right turns in the two lanes. We prefer to sacrifice generality for simplicity, in this case. Although we expect this model to yield a qualitatively correct description of the traffic flow under the appropriate conditions, there are certain quantitative weaknesses in this model. Assumption 1 is probably correct although there does not appear to be any experimental confirmation. Assumption 2 is a reasonable approximation for forward moving cars although it might be better to make the <,• random variables. The main inaccuracy of this assumption, however, arises in conjunction with assumption 3 where we postulate that right turning cars leave at the same rate (measured in terms of the time intervals tj — £,•_i) as forward moving cars and left turning cars when opposed by left turning cars also leave at this same rate. In both cases, we would expect that that the time intervals should be increased somewhat for the right and left turning cars. This model will therefore tend to underestimate slightly the influence of turns. The error is expected to be relatively small, however, in most cases of interest because the main effect of the turning cars arises from the left turning cars waiting for several time intervals, a time which is probably large compared with the excess time required to execute the left turn once the opportunity arises. The mathematical problem of calculating the flow through the intersection can be solved by first observing that a "state" of the intersection can be defined at any time tj by specifying the desires of the first car in each of the two queues. From this one can calculate the probabilities of various states at any subsequent times. For all practical purposes, however, the right turn states and the forward moving states are equivalent in this model and will therefore be considered as the same states. States in which both cars move forward (or right) or both turn left are also equivalent since they both lead to the result that the two cars leave the intersection immediately and new cars appear 1959] EFFECT OF LEFT TURNS ON INTERSECTION CAPACITY 69 in both lanes for the next value of t,-. We shall consider these states collectively as state 1. There are two other states. We denote by state 2 that state for which the car in lane 1 wishes to turn left and the car in lane 2 wishes to go forward. State 3 will then be that state for which the car in lane 1 wishes to go forward but the car in lane 2 wishes to go left. State 2 leads to the result that the car in lane 1 remains at the intersection but the car in lane 2 leaves and a new car appears in its place at the next value of tj . We can define a 3 X 3 stochastic matrix P that gives the transition probabilities from any of the three states at time t,to any of the three states at time ti+x. This matrix is P = pp' + (1 p)(l p') p' p v( 1 v') ' (1 v') 0 (1 v)v' 0 (1 v) (1) The second column, for example, describes the transitions from the state with a left turning car in lane 1 and a forward moving car in lane 2. The car in lane 1 remains as a left turning car but a new car appears in lane 2. The final states are therefore states with two left turning cars (probability v') or a left turning car in lane 1 and a forward moving car in lane 2 (probability 1 — p'). The third column describes a symmetric situation whereas column 1 gives the result of new cars appearing in both lanes. The first column of P also gives the state probabilities at time U but, for uniformity of notation, it is convenient to represent the probabilities for the states at time tx as given by Pei with ei the vector (1, 0, 0). The state probabilities at time tj are given by the three components of the vector P'ex . For j < n — 1, the three state probabilities at time tt also describe the probabilities that cars leave the intersection at time t,in both lanes, in the second lane only, and in the first lane only. They also represent the expectations for these events. The expectations of the number of times these three events occur during the first j values of tk are given by the components of the vector i Y. Pke1 for j < n. k-\ Since at time tn , the cars in both lanes leave the intersection regardless of which way they go, the expectations of the total number of times per cycle that cars leave in both lanes, in the second lane only, and in the first lane only are given by the components of the vector N = e, + 2 P\ = £ P\ . (2) k=l k"0 The sum of the first two components of N gives the average number of cars that leave in lane 2 per cycle, thus the capacity C2 of lane 2 in cars per cycle (provided the queue of lane 1 never vanishes). Similarly, the sum of the first and third components of N gives the capacity C\ of lane 1 in cars per cycle. The determination of Ci and C2 is thus reduced to evaluating the vector N. N can be evaluated most conveniently in terms of the eigenvectors and eigenvalues of the matrix P. If we let Xi , X2 , and X3 be the eigenvalues of P; fi , /2 and f3 the right eigenvectors; and gx ,