Title: Perfect Simulation, Sample-path Large Deviations, and Multiscale Modeling for Some Fundamental Queueing Systems
Description: As a primary branch of Operations Research, Queueing Theory models and analyzes engineering systems with random fluctuations. With the development of internet and computation techniques, the engineering systems today are much bigger in scale and more complicated in structure than 20 years ago, which raises numerous new problems to researchers in the field of queueing theory. The aim of this thesis is to explore new methods and tools, from both algorithmic and analytical perspectives, that are useful to solve such problems. In Chapter 1 and 2, we introduce some techniques of asymptotic analysis that are relatively new to queueing applications in order to give more accurate probabilistic characterization of queueing models with large scale and complicated structure. In particular, Chapter 1 gives the first functional large deviation result for infinite-server system with general inter-arrival and service times. The functional approach we use enables a nice description of the whole system over the entire time horizon of interest, which is important in real problems.

In Chapter 2, we construct a queueing model for the so-called limit order book that is used in main financial markets worldwide. We use an asymptotic approach called multi-scale modeling to disentangle the complicated dependence among the elements in the trading system and to reduce the model dimensionality. The asymptotic regime we use is inspired by empirical observations and the resulting limit process explains and reproduces stylized features of real market data. Chapter 2 also provides a nice example of novel applications of queueing models in systems, such as the electronic trading system, that are traditionally outside the scope of queueing theory. Chapter 3 and 4 focus on stochastic simulation methods for performance evaluation of queueing models where analytic approaches fail. In Chapter 3, we develop a perfect sampling algorithm to generate exact samples from the stationary distribution of stochastic fluid networks in polynomial time. Our approach can be used for time-varying networks with general inter-arrival and service times, whose stationary distributions have no analytic expression.

In Chapter 4, we focus on the stochastic systems with continuous random fluctuations, for instance, the workload arrives to the system in continuous flow like a Levy process. We develop a general framework of simulation algorithms featuring a deterministic error bound and an almost square root convergence rate. As an application, we apply this framework to estimate the stationary distributions of reflected Brownian motions and the performance of our algorithm is better than existing prevalent numeric methods.