On Extreme Values of Birth and Death Processes

Abstract

We establish the convergence rate to exponential distribution in a limit theorem for extreme values of birth and death processes. Some applications of this result are given to processes specifying queue length.). We establish uniform estimates for the convergence rate in the exponential distribution in a limit theorem for extreme values of birth and death processes. This topic is closely related to the problem on the time of first intersection of some level u by a regenerating process. Of course, we assume that both time t and level u grow infinitely. The proof of our main result is based on an important estimate for general regenerating processes. Investigations of the kind are needed in different fields: mathematical theory of reliability, queueing theory, some statistical problems in physics. We also provide with examples of applications of our results to extremal queueing problems M/M/s. In particular case of queueing M/M/1, we show that the obtained estimates have the right order with respect to the probability q(u) of the exceeding of a level u at one regeneration cycle, that is, only improvement of the corresponding constants is possible.