On extreme values of the queue length in some queuing systems

Abstract

In this paper, the rate of convergence to the exponential distribution in the limit theorem for extreme values of the queue length in queuing systems M / G / 1 {M/G/1} and G / M / 1 {G/M/1} are studied. In proving these results, we first prove one general boundary theorem for extremal values of regenerative processes, which is of independent interest. This topic is closely related to the task of the first passage times to rare sets by the regenerative process. The need for such research arises in many areas: the mathematical theory of reliability, queuing theory, some statistical problems of physics.