Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption

Abstract

We consider a single server renewal input queueing system under multiple vacation policy. When the system becomes empty, the server commences a vacation of random length, and either begins an ordinary vacation with probability q (0 ≤ q ≤ 1) or takes a working vacation with probability 1 - q. During a working vacation period, customers can be served at a rate lower than the service rate during a normal busy period. If there are customers in the system at a service completion instant, the working vacation can be interrupted and the server will come back to a normal busy period with probability p (0 ≤ p ≤ 1) or continue the working vacation with probability 1 - p. The server leaves for repeated vacations as soon as the system becomes empty. Upon arrival, customers decide for themselves whether to join or to balk, based on the observation of the system-length and/or state of the server. The equilibrium threshold balking strategies of customers under four cases: fully observable, almost observable, almost unobservable and fully unobservable have been studied using embedded Markov chain approach and linear reward-cost structure. The probability distribution of the system-length at pre-arrival epoch is derived using the roots method and then the system-length at an arbitrary epoch is derived with the help of the Markov renewal theory and semi-Markov processes. Various performance measures such as mean system-length, sojourn times, net benefit are derived. Finally, we present several numerical results to demonstrate the effect of the system parameters on the performance measures.