Stein's method for steady-state diffusion approximations

Diffusion approximations have been a popular tool for performance analysis in queueing theory, with the main reason being tractability and computational efficiency. This dissertation is concerned with establishing theoretical guarantees on the performance of steady-state diffusion approximations of queueing systems. We develop a modular framework based on Stein's method that allows us to establish error bounds, or convergence rates, for the approximations. We apply this framework three queueing systems: the Erlang-C, Erlang-A, and $M/Ph/n+M$ systems. The former two systems are simpler and allow us to showcase the full potential of the framework. Namely, we prove that both Wasserstein and Kolmogorov distances between the stationary distribution of a normalized customer count process, and that of an appropriately defined diffusion process decrease at a rate of $1/\sqrt{R}$, where $R$ is the offered load. Futhermore, these error bounds are \emph{universal}, valid in any load condition from lightly loaded to heavily loaded. For the Erlang-C model, we also show that a diffusion approximation with state-dependent diffusion coefficient can achieve a rate of convergence of $1/R$, which is an order of magnitude faster when compared to approximations with constant diffusion coefficients.Comment: PhD Thesi

An approximation algorithm for the queue length distributions of time-varying many-server queues

This paper presents a novel methodology for approximating the queue length (thenumber of customers in the system) distributions of time-varying non-Markovian manyserverqueues (e.g., Gt/Gt/nt queues), where the number of servers (nt) is large. Ourmethodology consists of two steps. The first step uses phase-type distributions toapproximate the general inter-arrival and service times, thus generating an approximatingP ht/P ht/nt queue. The second step develops strong approximation theory toapproximate the P ht/P ht/nt queue with fluid and diffusion limits. However, by naivelyrepresenting the P ht/P ht/nt queue as a Markov process by expanding the state space,we encounter the lingering phenomenon even when the queue is overloaded. Lingeringtypically occurs when the mean queue length is equal or near the number of servers,however, in this case it also happens when the queue is overloaded and this time is notof zero measure. As a result, we develop an alternative representation for the queuelength process that avoids the lingering problem in the overloaded case, thus allowingfor the derivation of a Gaussian diffusion limit. Finally, we compare the effectivenessof our proposed method with discrete event simulation in a variety parameter settingsand show that our approximations are very accurate

On the modelling and performance measurement of service networks with heterogeneous customers

Service networks are common throughout the modern world, yet understanding how their individual services effect each other and contribute to overall system performance can be difficult. An important metric in these systems is the quality of service. This is an often overlooked measure when modelling and relates to how customers are affected by a service. Presented is a novel perspective for evaluating the performance of multi-class queueing networks through a combination of operational performance and service quality—denoted the “flow of outcomes”. Here, quality is quantified by customers moving between or remaining in classes as a result of receiving service or lacking service. Importantly, each class may have different flow parameters, hence the positive/negative impact of service quality on the system’s operational performance is captured. A fluid–diffusion approximation for networks of stochastic queues is used since it allows for several complex flow dynamics: the sequential use of multiple services; abandonment and possible rejoin; reuse of the same service; multiple customers classes; and, class and time dependent parameters. The scalability of the approach is a significant benefit since, the modelled systems may be relatively large, and the included flow dynamics may render the system analytically intractable or computationally burdensome. Under the right conditions, this method provides a framework for quickly modelling large time-dependent systems. This combination of computational speed and the “flow of outcomes” provides new avenues for the analysis of multi-class service networks where both service quality and operational efficiency interact

Fluid flow models in performance analysis

We review several developments in fluid flow models: feedback fluid models, linear stochastic fluid networks and bandwidth sharing networks. We also mention some promising new research directions