Large scale queueing systems : asymptotics and insights

Abstract

Parallel server queues are a family of stochastic models useful in a variety of applications, including service systems and telecommunication networks. A particular application that has received considerable attention in recent years is the analysis of call centers. A feature common to these models is the notion of the 'trade-off' between quality and efficiency. It is known that if the underlying system parameters scale together according to a certain 'square-root scaling law', then this trade-off can be precisely quantified, in which case the queue is said to be in the Halfin-Whitt regime. A common approach to understanding this trade-off involves restricting one's models to have exponentially distributed call lengths, and restricting one's analysis to the steady-state behavior of the system. However, these are considered shortcomings of much work in the area. Although several recent works have moved beyond these assumptions, many open questions remain, especially w.r.t. the interplay between the transient and steady-state properties of the relevant models. These questions are the primary focus of this thesis. In the first part of this thesis, we prove several results about the rate of convergence to steady-state for the A/M/rn queue, i.e. n-server queue with exponentially distributed inter-arrival and processing times, in the Halfini-Whitt regime. We identify the limiting rate of convergence to steady-state, discover an asymptotic phase transition that occurs w.r.t. this rate, and prove explicit bounds on the distance to stationarity. The results of the first part of this thesis represent an important step towards understanding how to incorporate transient effects into the analysis of parallel server queues. In the second part of this thesis, we prove several results regarding the steadystate G/G/n queue, i.e. n-server queue with generally distributed inter-arrival and processing times, in the Halfin-Whitt regime. We first prove that under minor technical conditions, the steady-state number of jobs waiting in queue scales like the square root of the number of servers. We then establish bounds for the large deviations behavior of this model, partially resolving a conjecture made by Gamarnik and Momcilovic in [431. We also derive bounds for a related process studied by Reed in [91]. We then derive the first qualitative insights into the steady-state probability that an arriving job must wait for service in the Halfin-Whitt regime, for generally distributed processing times. We partially characterize the behavior of this probability when a certain excess parameter B approaches either 0 or oo. We conclude by studying the large deviations of the number of idle servers, proving that this random variable has a Gaussian-like tail. We prove our main results by combining tools from the theory of stochastic comparison [99] with the theory of heavy-traffic approximations [113]. We compare the system of interest to a 'modified' queue, in which all servers are kept busy at all times by adding artificial arrivals whenever a server would otherwise go idle, and certain servers can permanently break down. We then analyze the modified system using heavy-traffic approximations. The proven bounds hold for all n, have representations as the suprema of certain natural processes, and may prove useful in a variety of settings. The results of the second part of this thesis enhance our understanding of how parallel server queues behave in heavy traffic, when processing times are generally distributed.