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

Abstract

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 [Formula: see text] for all t in [0, t), in which Zt 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.

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 18 citations exist upstream.

Papers that cite the same references — the relational fallback while embeddings are still backfilling.