Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks

Abstract

Two variants of an M/G/1 queue with negative customers lead to the study of a random walkX _n+1=[X _n +ξ _n ]⁺ where the integer-valuedξ _n are not bounded from below or from above, and are distributed differently in the interior of the state-space and on the boundary. Their generating functions are assumed to be rational. We give a simple closed-form formula for$\mathbb{E}(s^{X_n } )$, corresponding to a representation of the data which is suitable for the queueing model. Alternative representations and derivations are discussed. With this formula, we calculate the queue length generating function of an M/G/1 queue with negative customers, in which the negative customers can remove ordinary customers only at the end of a service. If the service is exponential, the arbitrarytime queue length distribution is a mixture of two geometrical distributions.