Analysis of Single Server Markovian Queueing Model with Differentiated Working Vacation, Vacation Interruption, Soft Failure, Reneging of Customers

This article investigates the reneging of customers in M/M/1 model with differentiated working vacation, vacation interruption and soft failure. The customers come with rate λ and receives service during busy period with rate μ, where λ and μ obeys markovian distribution. In this model two distinct vacations are considered: one has been taken just after serving all customers in busy period with slow service rate θ as some soft failure occurs during working vacation (Vacation I). At an epoch of completion of working vacation, if any customers are present in the system, then the server moves to busy period for serving customers otherwise move to vacation II. During vacation II if customer comes then interruption is assumed to occur in the vacation and server returns to busy period otherwise remain in vacation .When an arriving customer finds server is on working vacation, it makes customer impatient and it start up an impatient timer T0 with an exponentially distributed rate α0 If service does not begin before T0 expires, the customer might renege with probability p without getting served or wait for their turn with probability 1-p=q. By using PGF technique we have derived different steady state probabilities and various system performances analytically. Effect of few parameters on different system performances have been shown numerically and illustrated graphically.