Queue analysis at signalized intersections

An involved discussion on analysis of queues at signalized intersections is not possible in this text as it requires substantial knowledge of stochastic queuing processes. However, certain characteristics of the arrival and departure processes from the point of view of queueing analysis are presented here so that the interested reader may pursue this topic further.

The primary purpose of doing a queueing analysis at a signalized intersection is to be able to determine the probability distribution of queues that form. That is, the aim should be to answer questions like, what is the probability that there will be vehicles in a particular queue at any time. This is important since not only values such as average queue length are important, what is more important is an idea of the queue length distribution so that one can determine lengths of auxiliary lanes  (see next lecture for details) for pre-designated values of probability of overflow (where the queue of vehicles is larger than the space provided by the auxiliary lane) and probability of blockage (where the queue of vehicles on the lane adjacent to the auxiliary lane is so large that it blocks the entrance to the auxiliary lane).

However, the theory of queues available so far is not able to model the queueing process at a signalized intersection in a simple manner. The primary reason for this is that the departure process is not a Poisson process. If one assumes (and correctly so) the time a vehicle waits at the top of the queue to be its service time, then it can be easily seen that the service time duration (to which the departure process is integrally connected) is not negative exponential. In fact the service time is either zero (if one reaches the top of the queue when the light is green) or equal to the duration of the red period (if one is the next to the last vehicle to have crossed the intersection); the probability of either of these service times being the service time of a particular vehicle is not easy to calculate. Matters can be further complicated if there is a permitted phase in the signal (as is sometimes the case for turning movements) where a vehicle can cross the intersection (during the non-green time) if a sufficient gap in the opposing stream exists.

Some researchers (see for example Kikuchi et al. ), however, have attempted to determine the probability distribution of queues through a Markov chain analysis of the queueing process. The interested reader may refer to Kikuchi et al.'s work cited above for a good understanding of the analysis procedure.