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.
