Internet and mobile services often provide waiting time or queue length information to customers. This information allows a customer to determine whether to remain in line or, in the case of multiple lines, better decide which line to join. Unfortunately, there is usually a delay associated with waiting time information. Either the information itself is stale, or it takes time for the customers to travel to the service location after having received the information. Recent empirical and theoretical work uses functional dynamical systems as limiting models for stochastic queueing systems. This work has shown that if information is delayed long enough, a Hopf bifurcation can occur and cause unwanted oscillations in the queues. However, it is not known how large the oscillations are when a Hopf bifurcation occurs. To answer this question, we model queues with functional differential equations and implement two methods for approximating the amplitude of these oscillations. The first approximation is analytic and yields a closed-form approximation in terms of the model parameters. The second approximation uses a statistical technique, and delivers highly accurate approximations over a wider range of parameters.

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Many service systems use technology to notify customers about their expected waiting times or queue lengths via delay announcements. However, in many cases, either the information might be delayed or customers might require time to travel to the queue 1