Retrial Queueing System of MAP/PH/N Type with a Finite Buffer and Group Service. The Process Describing the System Dynamics

A queuing system with many identical servers, a finite buffer, and order retrials is under study. The order’s arrival is described by the Markov arrival process. Service is offered for orders in groups. The size of the group is bounded from above by the capacity of the buffer and from below by a fixed threshold. A group’s service time follows a phase-type distribution with the irreducible representation determined by group size. A classical retrial strategy is assumed. Retrying and waiting orders can renege from the service after a random time interval. The duration of this interval has an exponential distribution. We built a multidimensional continuous-time Markov chain that includes the number of retrying orders, the number of busy servers, the state of the underlying arrivals process, and an auxiliary multidimensional Markov chain that defines the number of servers providing service at all possible service phases. The infinitesimal generator of the constructed chain is written down and, and the explicit expressions for the matrix blocks of the generator are presented.