Complex-analytic and matrix-analytic solutions for a queueing system with group service controlled by arrivals (Q5932234)

The following is the general description of the queueing model considered by the authors: Customers arrive at a service station by a Poisson process and are served in groups. The service time has a general distribution corresponding to the current service mode. The service station has two available modes of work: in Mode 1, the service group size is \(n\), in Mode 2 it is \(m\), where \(m > n\). Mode 2 being more ``expensive'', the station seeks to restrict its usage to ``heavy arrival'' situations by imposing ``arrival thresholds''. At the moment before completion of a service act, a decision is made on the next service mode: it switches ``up'' from 1 to 2 if the number of arrivals over the last service period reached or exceeded \(m\); it switches ``down'' from 2 to 1 if the number of arrivals was less than a ``down threshold'' \(N\), where \(n \leq N \leq m\); otherwise the mode remains as it was. They analyze this queueing process via an imbedded, two-dimensional Markov chain, and establish transition equations for its steady-state probabilities and a necessary and sufficient condition of ergodicity. They also obtain a system of equations for the generating functions of the steady-state probabilities and represent its solution in terms of a finite number of undetermined constants. The authors use the novel technique of ``matrix unfolding'' which reduces the problem to a matrix iteration process with the block size much smaller than in the direct application of the matrix-analytic method.