Optimal and Self Selection of Service Type in a Queueing System where Long Service Postpones the Need for the Next Service

Abstract

We study a make-to-order system with a finite set of customers. Production is stochastic with a nonlinear dependence between the ordered quantity and the production rate. Customers may have to queue until their turn arrives, and therefore their order decisions interact. Specifically, while being served, customers are aware of the queue length and choose one of two order quantities (or service types). The time to the next replenishment (their activity time) is stochastic and depends on the order quantities. A customer is inactive during service and while waiting in the queue. We refer to the type of service with a greater ratio of expected activity to service time as ``more efficient''. In the centralized case, the system is interested in maximizing the steady-state average number of active customers, which is referred to as the efficiency of the system. We show that choosing the more efficient service is not always optimal, but the optimal strategy can be approximated well by selecting one of three threshold strategies which depend on the number of inactive customers. In the decentralized case, each customer acts to maximize the fraction of time she is active. We observe that individuals and the manager have opposite incentives: When the queue is long, individuals tend to choose the long service, while the manager prefers the short service in this case. This makes the system difficult to regulate. However, we show that simply removing the less efficient service significantly increases efficiency.