On the multi-phase M/G/1 queueing system with random feedback

Abstract

In this paper we consider an M/G/1 queue with k phases of heterogeneous services and random feedback, where the arrival is Poisson and service times has general distribution. After the completion of the i-th phase, with probability θ i the (i + 1)-th phase starts, with probability p i the customer feedback to the tail of the queue and with probability 1 − θ i − p i = q i departs the system if service be successful, for i = 1, 2 , . . . , k. Finally in kth phase with probability p k feedback to the tail of the queue and with probability 1 − p k departs the system. We derive the steady-state equations, and PGF’s of the system is obtained. By using them the mean queue size at departure epoch is obtained.