FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1996, VOLUME 2, NUMBER 4, PAGES 1107-1115

Asymptotic of maxima in the infinite server queue with bounded batch sizes

A. V. Lebedev

Abstract

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This paper considers the infinite server queue with the batch input $MX|G|$¥. Let all servers be free at time zero and $M(t)$ denote the maximum number of customers simultaneously present in the queue during $[0,t]$. The following theorem is proved.

Theorem 1. If $L$ is the maximum number of customers in a batch, then almost sure

Some generalizations are discussed: nonstationary queues (with time-dependent parameters) and queues with heterogeneous customers. For these monotony theorems are proved. Conditions under which the asymptotic $($*) stays correct are obtained.

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