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Consider a single server retrial queueing system in which customers arrive in a Poisson process with arrival rate λ that which follows a Poisson process. Let k be the number of phases in the service station. The service time has Erlang k-type distribution with service rate kμ for each phase. Two types of vacation policies are discussed in this research paper that is Bernoulli type vacation and exhaustive type vacation. The vacation rate follows an exponential distribution with parameter α. 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 or on vacation, 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 by using Matrix geometric technique. Numerical study have been done for Analysis of Mean number of customers in the orbit (MNCO),Truncation level (OCUT), Probabilities of server free, busy and in vacation for various values of λ , μ ,k , p , α and σ in elaborate manner and also various particular cases of this model have been discussed.

M/M/1 Retrial Queueing System with two Types of Vacation Policies under Erlang – K Type Service