A Model of Rational Retrials in Queues

Customers often wait in queues before being served. Because waiting is undesirable, customers may come back later (i.e., retry) when the queue is too long. However, retrial attempts can be costly as a result of transportation fees and service delays. This paper introduces a framework for rational retrial decisions in stationary queues. Our approach accommodates retrials in queues by replicating the Naor's model [Naor P (1969) The regulation of queue size by levying tolls. Econometrica 37(1):15-24.] repeatedly over time periods. Within each period, we study an observable queue in which customers make rational state-dependent decisions to join, balk, or retry in a future period. We focus on a stationary environment where all arrivals, including new and retrying customers, will face the steady-state distribution of the system in equilibrium. Equilibrium analysis on customers' decision making is necessary, as they choose optimal strategies corresponding to the stationary queueing dynamics that are in turn determined by their decisions. We characterize the equilibria in both stable and overloaded systems. We find the following: (1) Compared with a system without retrials, the additional option to retry can hurt consumer welfare. (2) Compared with the socially optimal decisions, surprisingly, self-interested customers retry insufficiently (they join overly long queues) when the retrial cost is low and retry too often when the retrial cost is high. (3) Self-interested (retrial) customers can generate positive externalities by smoothing workload over time.