Time-Dependent Solution of a Priority Queue with Bulk Arrival

The time-dependent distribution of the number of customers in the queue and the lapsed service time is obtained, in terms of Laplace transforms, for a head-of-the-line priority queue in which both classes of customer arrive in bunches. From this we obtain the equilibrium distribution and the distribution of the number of customers left behind a departing customer. The latter is then used to find the equilibrium distribution of queuing times for both classes of customer. It is not easy to obtain a complete explicit solution, even for exponential service times, but the mean queuing times are readily obtained.