Analysis of M/G/1 feedback queue with two types of services, Bernoulli vacations and random breakdowns

Abstract

We analyse an M/G/1 feedback queueing system providing two types of services with vacations and breakdowns. Upon arrival to the system a customer can either choose type 1 service with probability p1 or type 2 service with probability p2 such that p1 + p2 = 1. After completion of each service he may join the tail of the queue with probability p, until he wishes for another service or leave the system with probability q = 1 – p. On each service completion, the server is allowed to take a vacation with probability θ or may continue service of a customer, if any, with probability 1 – θ. If no customer is found, it remains in the system until a customer arrives. The vacation times are exponentially distributed with parameter β. The system may breakdown at random and repair time follows exponential distribution with parameter η. Using supplementary variable technique, the Laplace transforms of time dependent probabilities of system state are derived. From this, we deduce the steady state results. We also obtain the average system size and average waiting time.