Functional Limit Theorems for Service Systems with Dependent Service and Patience Times

Abstract: We consider many-server queueing systems, assuming customers’ service requirements depend on their patience for waiting in queue. In this setting, establishing heavy-traffic limiting approximations is hard because the queue process does not admit a finite-dimensional Markov representation, and an infinite-dimensional measure-valued process representation lacks a martingale property that is key in proving weak limit theorems. In this presentation, I will discuss two of my recent works with my former PhD student Lun Yu: The first considers service systems with perfectly correlated service and patience times that are marginally exponentially distributed. Under the well-known square-root staffing rule, we prove that the sequence of diffusion-scaled queue processes converges to the Halfin-Whitt diffusion approximation for the Erlang-C model, which has no abandonment. In particular, when the traffic intensity converges to 1 from above, the limit process is transient, despite the stochastic systems in the pre-limit being ergodic due to abandonment. A lower-order fluid limit, combined with an interchange of limits result, proves that the steady-state queues in the transient-diffusion case are of order O(n^{3/4}) as n increases without bound. The second work considers general joint service-patience distributions with exponential marginals. In this case, the sequence of the diffusion-scaled queue processes converges weakly to the unique strong solution to a stochastic integral equation, which can be represented as a nonautonomous Stochastic Functional Differential Equation with Infinite Delay (SFDE-ID). We employ the SFDE-ID representation to establish a necessary and sufficient condition for the limit process to be ergodic, and prove that its stationary distribution is the limit of the sequence of stationary distributions of the diffusion-scaled queues. Interestingly, whether the SFDE-ID is ergodic depends on the joint distribution of the service and patience times only via a single parameter, which can therefore be considered as quantifying the strength of the dependence in our queueing setting.