Multi-Stage Retrial Queueing System with Bernoulli Feedback

TLDR: An M/G/1 retrial queueing system with k stages of heterogeneous services and feedback is considered, and the condition under which the steady state exists is investigated.

Abstract: Abstract — An M/G/1 retrial queueing system with k stages of heterogeneous services and feedback is considered. Primary customers get into the system according to Poisson process. If the server is free, an arriving customer receives first stage service immediately otherwise, he enters a retrial orbit. After the completion of the first stage, the customer may proceed to second stage with probability θ1, or feedback to the retrial group with probability p 1 or depart the system with probability 1 – θ1 – p1 = q1. In general, after the completion of the i, (i = 1, 2, ...., k−1) stage, the customer may opt (i+1) stage with probability θ1, or feedback to the retrial group with probability p i or depart the system with probability 1 – θ1 – pi = qi. The customer in final stage will feedback to the retrial group with probability pk or depart the system with probability 1 – pk = qk. It is assumed that the service times and the retrial times are arbitrarily distributed. The condition under which the steady state exists is investigated. Performance measures are obtained. A stochastic decomposition is presented.