M/G/1 Queue with Multiple Optional Services and Deterministic Repair Times

Abstract

We analyze the steady state behavior of an $M/G/1$ queue with Poisson arrivals subject to multiple optional services and random system breakdowns. Arriving customer has to undergo first essential service and there are $j$ optional services, where $j=1,2,\ldots, n$. As soon as the essential service of a customer is complete, then with probability $r_{j},j=1\ldots n$, he may opt for any one of the $j$ optional services, in which case his any one of the $j$ services will immediately commence or else with probability $1-\sum\limits_{j=1}^{n}r_{j}$, he may opt to leave the system, in which case another customer at the head of the queue is taken up for his essential service.The service times follow arbitrary (general) service distributions. The system is prone to random breakdowns and just after a breakdown the server undergoes repair of a fixed duration. We obtain time dependent as well as steady state probability generating functions for the number in the system. For steady state we obtain explicitly the mean number and the mean waiting time for the system and for the queue. Results for some special cases of interest are derived.