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Stability analysis of single server retrial queueing system with Erlang service

The objective of this paper is to study the stability analysis of single server retrial queueing system with Erlang-k service under various constraints over the behavior of server and customers, in which customers arrive in a Poisson process with arrival rate λ. These customers are identified as primary calls. Let k be the number of phases in the service station. Further assume that the service time has Erlang-k distribution with service rate kμ for each phase. The services in all phases are independent and identically exponentially distributed and only one customer at a time is in the service mechanism. If the server is free at the time of a primary call arrival, the arriving call begins to be served in phase 1 immediately by the server then progresses through the remaining phases and must complete the last phase and leave the system before the next customer enters the first phase. If the server is busy, then the arriving customer goes to orbit and becomes a source of repeated calls. This pool of source of repeated calls may be viewed as a sort of queue. Every such source produces a Poisson process of repeated calls with intensity σ. If an incoming repeated call finds the server free, it is served in the same manner and leaves the system after service completion, while the source which produced this repeated call disappears. Otherwise, the system state does not change. In this paper, we analyse the stability of this system by introducing various concepts namely vacation policies, unreliable server and negative arrival. We assume that the access from orbit to the service facility is governed by the classical retrial policy. Stability conditions are derived by using Matrix geometric method in detailed.