The Armchair Decision: On Queue-Length Information When Customers Travel to a Queue

Abstract

Problem definition: We consider a service system in which customers must travel to the queue to be served. In our base model, customers observe the queue length and then decide whether to travel. We also consider alternative information models and investigate how the availability of queue-length information affects customer equilibrium strategies, throughput, and social welfare. Academic/Practical Relevance: A common assumption in queueing models is that once a customer decides to join the queue, joining is instantaneous. This assumption does not fit real-life settings, where customers possess online information about the current wait time at the service, but while traveling to the service, its queue length may change. Motivated by this realistic setting, we study how queue-length information prior to traveling affects customers' decision to travel. Methodology: We prove a symmetric equilibrium exists in our base model. We perform the calculation numerically as a result of the model complexity, which is due to the fact that the arrival rate to the traveling queue depends on the current state of the service queue, and vice versa. The alternative models are tractable, and we present their analytical solution. Results: When customers can observe the service-queue length prior to traveling, their probability of traveling is monotonically non-increasing with the observed queue length. We find customers may adopt a generalized mixed-threshold equilibrium strategy: travel when observing short queue lengths, avoid traveling when observing long queue lengths, and mix between traveling and not traveling when observing intermediate queue lengths, with a decreasing probability of traveling. Managerial Implications: Our results imply that when system congestion is high, the provider can increase throughput by disclosing the queue-length information, whereas at low congestion, the provider benefits from concealing the information. With respect to social welfare, queue-length information prior to departure is beneficial when congestion is at intermediate to high levels, and yields the same social welfare otherwise.