Strategic arrivals to a queue with service rate uncertainty
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This paper studies the problem of strategic choice of arrival time to a single-server queue with opening and closing times when there is uncertainty regarding service speed. A Poisson population of customers need to arrive during a specified acceptance period and are served on a first-come first-served basis. We assume there are two types of customers that differ in their beliefs regarding the service time distribution. The inconsistent beliefs may arise from randomness in the server state along with noisy signals that customers observe. Customers choose their arrival-times with the goal of minimizing their expected waiting times. Assuming that customers are aware of the two types of customers with differing beliefs, we characterize the Nash equilibrium arrival profiles for exponentially distributed service times and provide an explicit solution for a fluid approximation of this game. For general service time distributions we provide an algorithm for computing the equilibrium in a discrete time system. Furthermore, we present a dynamic learning model in which customers make joining decisions based on their signals and past experience. The learning model assumes customers do not know all of the system and uncertainty parameters, or are unable to correctly compute the respective posterior expected waiting times. We numerically compare the long-term average outcome of the learning process, which has an inherent estimation bias due to the unknown signal quality, with the equilibrium solution.
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