A System of Markov Chains with Random Life Times

Abstract

0. Introduction. The purpose of this paper is to investigate the limiting properties of random variables associated with a system of random processes. The system is described as follows. At each discrete integer time n > 0, Mn particles enter a denumerable set of states A at a given state denoted by (0,0). Assume {M, n e I} to be a sequence of independent Poisson variables with common mean A. (Here and throughout, I denotes the set of nonnegative integers.) Moreover, at each integer time n ? 1, each particle already in the system may undergo a transition independently of the other particles and independently of {Mn, n E I}. A particle which entered the system at time k < n, moves according to the probability law of Z(n k) where {Z(n), n e I} is a random process described below.