Queueing for gaps in traffic

Garwood (1940) considered the time for which a single minor road vehicle had to wait to cross a major road at a junction controlled by a vehicle-actuated signal. The problem was essentially one of waiting for a gap of at least T in the major road flow, with gaps assumed to be exponentially distributed. Tanner (1951) used the same model for pedestrians crossing the road. With the arrival of pedestrians forming a Poisson process, he found the distribution of the number waiting at any instant. Mayne (1954) generalized this result for an arbitrary distribution of gaps. Weiss & Maradudin (1962) introduced a gap acceptance function az(t) which is the probability that the minor road vehicle will accept a gap of duration t. Using the general gap distribution they derived the distribution of delay to each of two vehicles arriving simultaneously. Oliver & Bisbee (1962) also assumed a general gap distribution and that a vehicle would not cross the main road if there was an arrival due on it within a time T. They further imposed the restriction that no two vehicles could cross in the same gap. Thus minor road vehicles, arriving at random, were forced to queue. By considering a Markov chain embedded at the times of crossing they found the distribution of queue size at these epochs. Gaver (1963) has used a model which differs from the others in not requiring the protection of a gap which is distinctly larger than the time it takes the minor road vehicle to cross the junction. This latter time is supposed to be a random service time in the queueing of minor road vehicles. On reaching the head of the queue a vehicle must wait for a gap which is greater than its service time. Gaps are distributed exponentially and alternate with blocks which have an arbitrary distribution, and during which no vehicle may cross. This is equivalent to pre-emptive-repeat-identical queueing with active interruptions which he studied in his 1962 paper. He derives the distribution of waiting time. When service time is constant this model is equivalent to that of ? 5 below with a = T. Tanner (1962) also considers this form of main road flow, and in particular the blocks are considered as being formed by the passage of bunches of vehicles which have been formed by queueing in a system with constant service time, and taking account of the usual constant T. Minor road vehicles arrive at random and cross the road with constant headway during a gap. He derives the mean waiting time. This is very similar to Gaver's model with constant service time except that at least one vehicle crosses in every gap, provided that a vehicle is waiting. In this paper we assume the same system of gaps and blocks, or greens and reds. We also assume that minor road vehicles arrive in bunches, but during a gap they cross one at a time with an interval of at least a between them. The equilibrium distribution of waiting time is obtained. It is shown how the system of greens and reds may be generated by a more general form of main road flow than Tanner's.