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

TL;DR

This paper analyzes a queueing system where customers choose service types based on queue length, revealing that optimal strategies involve threshold policies and that removing less efficient services improves overall efficiency.

Contribution

It introduces a model of customer decision-making in a queue with multiple service types and demonstrates how threshold strategies can optimize system efficiency.

Findings

Optimal strategies can be approximated by three threshold policies.

Customers and managers have opposing incentives affecting service choices.

Removing less efficient service types enhances overall system efficiency.

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.