Mean waiting times of the alternating traffic with starting delays

We analyze alternating traffic crossing a narrow one-lane bridge on a two-lane road. Once a car begins to cross the bridge in one direction, arriving cars from the other direction must wait, forming a queue, until all the arrivals in the first direction finish crossing the bridge. Such a situation can often be observed when road-maintenance work is carried out. Cars are assumed to arrive at the queues according to independent Poisson processes and to cross the bridge in a constant time. In addition, once cars join the queue, each car needs a starting delay, a constant, to start crossing the bridge. We model the situation where a signal controls the traffic so that the signal gives a priority to one direction at least for a fixed time. Under an assumption, the first two moments of a period during which the signal keeps giving a priority to one direction are obtained. Using a stochastic decomposition property the mean waiting times are obtained of cars to start crossing the bridge from each direction.