Queuing with reneging and multiple heterogeneous servers

C. J. Ancker and A. V. Cafarian

Naval Research Logistics Quarterly, 1963, vol. 10, issue 1, 125-149

Abstract: A Poisson stream of items arrive at a multiple parallel serving facility consisting of s heterogeneous servers. The service time density functions are all negative exponential and may all have different means. An arrival balks (refuses to enter) if the queue size is equal to N. If the queue size is less than N, the arrival enters the queue and is served on a first‐come, first‐served basis unless it reneges (leaves the queue). Each entering item has a definite length of time it will wait in queue before reneging. This time is a random variable with negative exponential density function. If on arrival more than one server is free, the server to be used is chosen at random. The results obtained are for steady‐state and include (1) state probabilities; (2) the probability of j or more in system, j=s, s+I,…, s+N; (3) mean number in queue and system and mean number of busy servers; (4) probability that an arrival acquires service; (5) mean rate of loss due to balking and reneging; and (6) density functions and mean values of waiting times in queue for those who are served, those who renege, and all who enter. Certain important special cases are derived from the preceding, namely, (1) unbounded queues, (2) homogeneous servers, (3) no reneging, (4) no reneging and unbounded queues, and (5) single server. Finally, a comparison is made of a heterogeneous system with an equivalent homogeneous system.

Date: 1963
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