Early birds versus last-minute arrivals: Empirical evidence and theoretical analysis of arrival time queueing game

<jats:p> In this study, we examine a scenario where each customer strategically selects their arrival time at a queue, aiming to minimize their total costs, which include waiting, tardiness, and earliness costs. Building on the mixed strategy framework studied by Moshe Haviv, which provides a closed-form arrival time distribution, we extend our focus to the transient behavior of this system in the presence of irrational customers, in which case it is not yet known whether an equilibrium exists or whether the system would converge to the equilibrium. We introduce a dynamic choice model to describe customers’ behavior in a repeated game setting. We empirically validate this model using data from a large endoscopy department of a major hospital in China, operating on a first-come first-served basis. The empirical results support the efficacy of this dynamic choice model in capturing the adaptive strategies of patients concerning their arrival times. Leveraging this model, we prove that there exists a unique equilibrium and the system converges to this equilibrium under both a fluid model and an <jats:inline-formula> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:mi>M</mml:mi> <mml:mo>/</mml:mo> <mml:mn>1</mml:mn> </mml:math> </jats:inline-formula> queueing model. By analyzing the arrival time distribution at equilibrium, we identify the optimal time for a patient to arrive at the department, which occurs either at the start or near the end of the morning session, depending on the model parameters. Furthermore, we formulate a dynamic control problem to characterize the optimal information sharing policy, suggesting that service providers may decrease social cost by deliberately modulating the communication of waiting time information to patients. Further examination of social cost unveils that increasing earliness cost can, counter-intuitively, detract from social cost at equilibrium. </jats:p>