Dynamic Rate Queues: Estimation, Stabilization, and Control

Queueing models are used in a variety of application settings. Some of the applications of queueing models include but are not limited to the study of telecommunication networks, computer networks, call centers, energy systems, transportation networks, and healthcare systems. In most of these systems, there is a service manager whose goal it is to achieve and maintain a high quality of service for all customers. To this end, there are three types of analysis that are needed in order to achieve the service managers’ goals. The first type of analysis is estimation. A service manager must be able to estimate or compute accurate performance measures of the service quality at any given time. The second fundamental type of analysis is stability . This can be described by a service manager providing probabilistic uniformity of the customer experience. The last type of analysis is control . A service manager must be able to control and operate the system effectively within the given constraints. In this thesis entitled Dynamic Rate Queues: Estimation, Stability, and Control, we provide novel solutions to address these three fundamental issues in the context of a multitude of queueing models with dynamic rates. Much of the current queueing literature relies on either steady state analysis or stationary results to provide insights about these stochastic networks. However, the steady state analysis is not amenable to many of the time varying queueing models that we consider in this thesis. In fact, tools such as Laplace transforms, stationary distributions, and generating functions are often very difficult if not intractable to use for queues with time varying rates. In addition, to the complexity of the models with time varying rates, we also include the possibility of customer abandonment , which has a large effect on customers who are impatient or jobs that are delay sensitive. All of the queueing models that we consider in this thesis include these salient features of time varying rates or abandonment as they are both phenomena exhibited in realtime service networks. As a result, in this thesis we develop and create new tools to iii analyze queueing networks with time varying rates and abandonment. In addition to addressing the three fundamental types of analysis needed by service managers, another main goal of this thesis is to provide the stochastic analysis community with effective computational methods to compute important performance measures of complex queueing models. In the words of Professor Avishai Mandelbaum, our goal is to develop Simple Models for Complex Realities . Thus, we find it very important to be able to not only describe the queueing behavior, but also compute it in a meaningful, simple, and fast way. Our main contribution of this thesis is to combine dynamic queueing models with novel closure approximations for stochastic processes. In the quest for understanding the dynamics of time varying queues, we use Poisson random measures, the functional Kolmogorov forward equations, and orthogonal polynomials (Hermite polynomials) as the main ingredients for constructing our approximation methods. Lastly, in order to validate the applicability of our methods, numerous simulation studies are performed to demonstrate that our approximations are accurate in a large number of parameter settings.