Analysis of a multi-server retrial queue with a varying finite number of sources

A multi-server retrial queue with a finite number of sources of requests was considered. In contrast to similar models studied in the literature, we assumed this number is not constant but changes its value in a finite range. During the stay in the system, each source generates the service requests. These requests are processed in a finite pool of servers. After service completion of a request, the source is granted the possibility to generate another request. If the source does not use this possibility during an exponentially distributed time, it is deleted from the system. If the request finds all servers busy, it can make repeated attempts to enter the service. If all servers are busy, the request may depart from the system without service. In this case, with a fixed probability, the source that generated this request is deleted from the system. Sources arrive according to a Markov arrival process. If the number of sources in the system at the arrival epoch has the maximum allowed number, the arriving source is lost. This system is a more adequate model of many real-world systems than the standard finite source queue. Analysis of the considered system required a four-dimensional continuous-time Markov chain. The generator of the chain was obtained as a block matrix with four levels of nesting. The stationary distribution of this Markov chain was found numerically as well as the values of the system's performance measures. The dependence of these measures on the maximum allowed number of sources and the number of servers was numerically clarified. An example of solving an optimization problem was presented.