Optimal Policy for an Unreliable Service System

Services play a vital role in economies of nations and in world commerce. An economy cannot operate without the services provided by service firms. In its turn, a service firm cannot survive without a sound competitive strategy. Management of service systems is a continuous challenge, due to the dynamic and random nature of the service environment. The theory of waiting lines provides insight and identifies management options for improving customer service. A wide variety of queueing models have been developed and successfully exploited for very complex service situations. This chapter describes one such queueing model. For a comprehensive classification of various control policies applied in queueing systems, see the survey by Tadj and Choudhury (2005). An understanding of the queueing phenomenon is necessary before creative approaches to the management of service systems can be considered. Some of the main concern in dealing with waiting line problems are the customer arrival process, the service process, the procedure or priority rule to use in selecting the next product or customer to be served, etc. The service system we consider in this chapter is characterized by an unreliable server. Random breakdowns, when they occur, prevent the server from providing service to the customers. To model a realistic system, we assume that a repair may not be immediate. We also make assumptions regarding the server discipline in order to model a wide range of real-world systems. The first assumption is called a Bernoulli vacation schedule. It specifies that, at the end of a service, the server may either take a vacation or serve the next customer. The term server vacation is used when the server is away from the work station, attending some secondary job. The second assumption is called T-policy. It specifies that when the queue is empty, the server again takes vacations and scans the queue periodically, every T units of time, to check if some customer has arrived while he was away. For example in an inventory system, a T-policy would correspond to a periodic-review policy. The primary purpose of waiting lines theory is to obtain the performance measures of the service system. We thus obtain probability generating functions of the system state and the main performance measures such as the mean number of customers in the system, and the mean lengths of the idle period, busy period, and busy cycle. These measures are then used to design an optimal management policy of the system, by balancing various types of costs. We illustrate the results obtained by presenting a numerical example and perform some sensitivity analysis to show the effect of the system parameter on the optimal policy.