Strong Approximations for Queues with Customer Choice and Constant Delays

Many service systems provide queue length information to customers thereby allowing customers to choose among many options of service. However, queue length information is often delayed and is often not provided in real time. Recent work by Dong et al. [7] explores the impact of these delays in an empirical study in U.S. hospitals. Work by Pender et al. [27] uses a two-dimensional fluid model to study the impact of delayed information and determine the exact threshold under which delayed information can cause oscillations in the dynamics of the queue length. In this work, we confirm that the fluid model analyzed by Pender et al. [27] can be rigorously obtained as a functional law of large numbers limit of a stochastic queueing process and we generalize their threshold analysis to arbitrary dimensions. Moreover, we prove a functional central limit theorem for the queue length process and show that the scaled