Retrial Queueing System of MAP/PH/N Type with a Finite Buffer and Group Service. Stationary Analysis of the System

In this paper, we is consider a multi-server queueing system with a finite buffer and customer (order) retrials. Arrival flow is defined by the Markov Arrival Process. Service is provided to groups of orders. The size of the group is bounded from below by a fixed threshold. The service time of a group has a phase-type distribution with irreducible representation depending on the size of a group. The linear dependence of the total retrial rate on the number of retrying orders is assumed. Retrying and waiting orders are assumed impatient. The behavior of the system is described by the multidimensional continuous-time Markov chain, including, as the components, the number of retrying orders, the number of orders waiting in the buffer, the number of busy servers, the state of the underlying process of arrivals, and the auxiliary multidimensional Markov chain, which defines the number of servers providing service at all possible phases of service. Ergodicity conditions of the analysed chain in cases of patient and impatient orders in the orbit are obtained. Algorithms for the computation of the stationary distribution and the main performance measures of the system are briefly outlined. The feasibility of the presented algorithms is confirmed by presenting the numerical results. The impact of the buffer capacity and the minimum group size on the main performance characteristics of the system is highlighted. An example of solving an optimization problem is presented.