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

A. V. Lebedev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 2 (1996) no. 4, pp. 1107-1115.

Résumé

This paper considers the infinite server queue with the batch input $M^X|G|\infty$. 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 $$ M(t)\frac{\ln\ln t}{\ln t}\to L\quadas $t\to\infty$.\eqno (*) $$ 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.