The Distribution of the Maximum Length of a Poisson Queue During a Busy Period

In the design of waiting facilities for the customers in a queue, it is of interest to know probability distributions of extremal values of the queue length. In this paper we propose to calculate explicitly the probability that

for all t in [0, t), in which Z_t denotes the queue length at time t. This is the probability that in a Poisson queue the server remains busy throughout the time interval [0, t) without ever having as many as b customers in line. We express this probability in terms of the taboo-probabilities with taboo-states 0 and b for a finite continuous time random walk. As an incidental result we obtain new expressions for these taboo-probabilities. It is routine to determine the distribution of the maximum queue length before the next emptiness from our results.