Asymptotic methods to understand the performance of large-scale service systems

Modern service systems tend to be large and require sophisticated operational strategies. Examples are customer contact centers with multi-class calls and multiple pools of agents, and hospitals with patient flows in many diagnosis and treatment centers. Large-scale service systems have the advantage of economy of scale, providing the possibility of achieving both high quality of service and high efficiency, if they are well managed. However, the complexity of such systems presents challenges for decision making in several ways, including routing, staffing, scheduling and revenue management. My primary work in this dissertation has concentrated on the following aspects of large-scale service systems. First, I try to understand the congestion (e.g., delays) caused by service interruptions, so that managers can plan ahead to reduce the consequences. Second, I seek ways to determine if we can effectively set staffing levels to meet performance targets by looking at extreme values of observed performance over time intervals. Third, I seek to obtain the performance measures for general non-exponential service-time and customer-patience-time distributions.;Stochastic models are often useful to provide supporting insights into these challenging problems. Analysis of these problems is facilitated by applying many-server queueing models and their variants with customer abandonment and service interruptions. Although many-server queueing models have traditionally been used to design and manage telecommunications, manufacturing systems and telephone contact centers, the modeling complexity makes it substantially harder to derive exact formulas for various performance measures and requires further extensive investigation of the classical Erlang models and their variants. The few available formulas turn out to be too complicated to provide many useful insights. Since these systems tend to be large, it is appealing to obtain insightful asymptotic approximations in heavy-traffic regimes in which the scale increases. The limits strip away unessential details while retaining key features. I have made contributions by establishing heavy-traffic stochastic-process limits for these complex systems, from which their transient and steady-state behavior can be approximately described.;I seek to use and develop simple but effective methods to solve practical problems. For many-sever queues with exponential service times, in Pang, Talreja and Whitt (2007), we review and advocate the approach using continuous mappings together with the martingale functional central limit theorem to prove the convergence in distribution of appropriately scaled queueing processes to diffusion processes (briefly outlined in § 1.2.4). This approach is further developed to account for convergence of processes with unmatched jumps in my other works on service interruptions; see §2. I have adopted both strong approximations for Poisson processes and extreme-value theory for diffusion processes to derive an insightful approximation for the maximum queue length and the maximum number of idle servers over certain time intervals in Erlang delay models; see §3.;One significant phenomenon in large-scale service systems is non-exponential service duration. A major focus of my research is to seek an effective mathematical framework to establish Markov stochastic-process limits for general non-Markov many-server queueing models with non-exponential service times. I have explored the two-parameter stochastic-process approach, where "two-parameter" means that the time domain is two dimensional with the second component keeping track of elapsed or residual service times. This two-parameter approach has been used to establish Markov limit processes for general infinite-server queues; see §4. That result involves Kiefer processes and stochastic integrals with respect to Brownian sheets, and I am generalizing this approach to study infinite-server queues with delayed feedbacks. I expect to explore this approach further to study many-server queues with abandonment and other variants in future work.