Asymptotic Behavior of Parallel Queueing Systems with Discrete-Choice Driven Arrivals
Abstract: Asymptotic Behavior of Parallel Queueing Systems with Discrete-Choice Driven Arrivals
We study a parallel-queue system where each queue is served by a dedicated server at different locations or facilities. Upon arrival, customers observe real-time queue lengths at each facility and choose to join one or balk, whichever that maximizes their expected utility, defined as the service value minus the waiting cost. Our model acknowledges heterogeneity in customer preferences for services at different facilities and their varying waiting time tolerances. We derive fluid and diffusion limit processes to approximate the asymptotic behaviour of the queueing system, exploiting the distinctive features of the arrival rate functions dictated by the discrete choice model and sidestepping the traditional reliance on the Lipschitz-continuity assumption. We prove the uniqueness of the fluid limit process and its converges to a unique equilibrium. At the equilibrium, our analysis under the conditional logit assumption indicates that the social welfare is maximized as long as all service providers are operating at their capacity. Furthermore, we characterize the diffusion limit for the centered process as a reflected multi-dimensional Ornstein-Uhlenbeck process. Analysis of the diffusion model reveals that publicizing real-time wait times does not change the social welfare. Applications of our theoretical results are illustrated through a case study of vehicle queues at U.S.-Canada border-crossing ports.