Delay analysis for the fixed-cycle traffic-light queue

We consider the fixed-cycle traffic-light (FCTL) queue, where vehicles arrive at an intersection controlled by a traffic light and form a queue. The traffic-light signal alternates between green and red periods, and delayed vehicles are assumed to depart during the green period at equal time intervals. Most of the research done on the FCTL queue assumes that the vehicles arrive at the intersection according to a Poisson process and focuses on deriving formulas for the mean queue length at the end of green periods and the mean delay. For a class of discrete arrival processes, including the Poisson process, we derive the probability generating function of both the queue length and delay, from which the whole queue length and delay distribution can be obtained. This allows for the evaluation of performance characteristics other than the mean, such as the variance and percentiles of the distribution.