Analysis of a Single Server Queueing System Controlled by a Random Switch

Abstract

In this paper, a single server queueing system operating in a doubly stochastic environment is analysed. The random environment makes transitions among N levels controlled by a random switch which performs the task of assigning a job to the server. The queueing system resides in level k of the environment till the completion of the service of the last customer in the system and immediately reports to the random switch to get a new service job. The random switch generates a set up time and waits till a customer arrives and assigns a job to the server in any one of the N levels with a positive probability governed by a binomial distribution. In each level r of the environment, the queueing system behaves like M(λr)/M(μr)/1 queue subject to the condition that the server reports to the random switch immediately after performing exhaustive service for getting a new assignment. For this model, time-dependent state probabilities are explicitly found and the corresponding steady-state probabilities are deduced. Some key performance measures are also obtained. A numerical study is also made.