M/M/1 Retrial Queueing System with Vacation Interruptions Under Erlang-K Service

Abstract

a single server retrial queueing system in which customers arrive in a Poisson process with arrival rate λ. Let k be the number of phases in the service station. Let the service time follows an Erlang K-type distribution with service rate kμ for each phase. The server goes for vacation after exhaustively completing the service to the customers. This vacation rate follows an exponential distribution with parameter α. The concept of vacation interruption is introduced in this paper that is, the server comes from the vacation into normal working condition without completing his vacation period subject to some conditions. We assume that the services in all phases are independent and identical 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 sources 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, while the source which produced this repeated call disappears. Otherwise, the system state does not change. We assume that the access from orbit to the service facility is governed by the classical retrial policy. This model is solved using Matrix geometric Technique. Numerical study have been done for Analysis of Mean number of customers in the orbit (MNCO), Probability of server free ,busy and in vacation for various values of λ , μ ,k , α, N0 and σ in elaborate manner and also various particular cases of this model have been discussed.