A combinatorial analysis of the M/M ^{[ m ]} /1 queue

Neuts’ Matrix Geometric Method makes use of the left-skip free characteristic of M/G/1-type Markov chains and determines the first passage distribution matrix G by solving a non-linear matrix equation. In this paper, we focus on the k-step first passage problem. In particular, we identify three associated matrices, namely the matrix G _k , the conditional first passage probability matrix P _k , and the first passage count matrix T _k . The reformulation allows for combinatorial techniques. Specifically, we refer to an extension of Takács’ ballot theorem. We note that the matrix P _k exhibits some ballot properties. In the case of the M/M ^{[ m ]} /1 queue, we establish the special structure of the count matrix T _k using lattice path arguments. Furthermore, we obtain a closed-form expression for the G matrix, where the first passage probabilities are expressed in terms of generalized hypergeometric functions.