Explicit Solution of a Quorum Queueing System

This paper deals with the queueing system M/Gr/1 in which customers are served in batches of fixed size r. The author considers the discrete time parameter process, an embedded Markov chain, and the continuous time parameter process, a semi-regenerative process. In both cases, explicit solutions are given for the system size steady-state probabilities in terms of the zeros, inside and/or outside the closed unit ball, of the characteristic equation. Moments are derived and illustrative examples are provided.