Overflow at a Traffic Light

Suppose vehicles of uniform length arrive at an intersection controlled by a traffic light, with arrival times which constitute a homogeneous Poisson process, with parameter A. The effect of acceleration will be ignored; once a vehicle is near the traffic light it travels with a uniform speed S unless stopped. When a queue of vehicles is stopped by the light, headway (distance separation between corresponding parts of adjacent vehicles) is constant, and when the cars move off the headway will be a larger constant, denoted by K. The letter , will be used to indicate the length of the red phase, so that the expected number of arrivals during a red phase will be A, for a fixed cycle light and AE(f8) for a variable cycle light. During the green phase, when there is a queue of cars waiting, they are discharged with time separation K/S = T. Once the queue present at the beginning of the green phase has been emptied, the Poisson arrivals continue through the intersection without delay for the remainder of the green phase. Instead of characterizing the green phase by its length, we will use instead an integer N, which is the largest multiple of T contained in its length: length green = a = NT + OT (O < 0 < 1). Thus, with a sufficiently long queue, at most N vehicles can be discharged during a green phase. N is, however, by no means the maximum number of vehicles which may pass through the intersection during a green phase, for once the queue is dissipated, the Poisson stream may send cars through with time separation < T. In some circumstances N will be a constant, and in others a stochastic variable governed by vehicle actuation on the side street. In speaking of input and output, we do not refer to input and output to the intersection, but only to the congestion at the intersection. Thus, when there is no queue, a car may come into the intersection and pass out of it, leaving both input and output zero. It is essential