Steady-state GI/GI/n\mathit{GI}/\mathit{GI}/\mathit{n} queue in the Halfin-Whitt regime

We consider the FCFS $\mathit{GI}/\mathit{GI}/n$ queue in the so-called Halfin-Whitt heavy traffic regime. We prove that under minor technical conditions the associated sequence of steady-state queue length distributions, normalized by $n^{1/2}$, is tight. We derive an upper bound on the large deviation exponent of the limiting steady-state queue length matching that conjectured by Gamarnik and Momcilovic [Adv. in Appl. Probab. 40 (2008) 548-577]. We also prove a matching lower bound when the arrival process is Poisson. Our main proof technique is the derivation of new and simple bounds for the FCFS $\mathit{GI}/\mathit{GI}/n$ queue. Our bounds are of a structural nature, hold for all $n$ and all times $t\geq0$, and have intuitive closed-form representations as the suprema of certain natural processes which converge weakly to Gaussian processes. We further illustrate the utility of this methodology by deriving the first nontrivial bounds for the weak limit process studied in [Ann. Appl. Probab. 19 (2009) 2211-2269].Comment: Published in at http://dx.doi.org/10.1214/12-AAP905 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Performance Analysis of a Multi-Class, Preemptive Priority Call Center with Time-Varying Arrivals

We model a call center as a an $M_{t}/M/n$, preemptive-resume priority queue with time-varying arrival rates and two priority classes of customers. The low priority customers have a dynamic priority where they become high priority if their waiting time exceeds a given service-level time. The performance of the call center is estimated by the mean number in the system and mean virtual waiting time for both classes of customers. We discuss some analytical methods of measuring the performance of call center models, such as Laplace transforms. We also propose a more-robust fluid approximations method to model a call center. The accuracy of the performance measures from the fluid approximation method depend on an asymptotic scheme developed by Halfin and Whitt. Here, the offered load and number of servers are scaled by the same factor, which maintains a constant system utilization. The fluid approximations provide estimates for the mean number in system and mean virtual waiting time. The approximations are solutions of a system of nonlinear differential equations. We analyze the accuracy of the fluid approximations through a comparison with a discrete-event simulation of a call center. We show that for a large enough scale factor, the estimates of the performance measures derived from the fluid approximations method are relatively close to those from the discrete-event simulation. Finally, we demonstrate that these approximations remain relatively close to the simulation estimates as the system state varies between under-loaded and over-loaded status

Large scale queueing systems : asymptotics and insights

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 195-203).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.by David Alan Goldberg.Ph.D