Queuing with Alternating Priorities

Abstract

For a service facility serving more than one class of customers a problem arises as to the order in which customers should be served (i.e., the priority rule). A typical example is a manufacturing facility that produces two products to meet a random stream of incoming orders. In this paper the alternating priority rule (also known as the “zero switch rule”) is introduced and investigated. Two classes of customers are served by a single service facility. Customers of class i(i = 1, 2) have priority over customers of class j(j = 1, 2; j ≠ 1) whenever a class i customer is in service. When the service facility is idle, the first arriving customer enters service and acquires the priority right for customers of his class. Within classes the “first-come, first-served” rule is observed. Customers' arrivals are assumed to be Poisson and service times are assumed to be arbitrarily distributed independent random variables. The steady-state densities of queuing times are formulated by the use of a special mathematical procedure. The expectations of queuing times and sizes of queues as well as the first two moments of the busy periods are obtained in terms of the basic parameters of the arrival process and service time distributions. All the results are related to the case where switching from one type of customers to another is not penalized by setup time. The alternating priority rule is compared to the “head-of-the-line” and the “first-come, first-served” rules with respect to expected queuing times and queue sizes. The last part of the paper discusses the possibilities of extending the suggested rule to cases involving more than two classes of customers and nonzero setup times and setup costs.