Related papers: Stein's method for diffusive limit of queueing pro…
This paper uses the generator approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. Until now, the standard way to invoke Stein's method for this problem was to use the…
The original Donsker theorem says that a standard random walk converges in distribution to a Brownian motion in the space of continuous functions. It has recently been extended to enriched random walks and enriched Brownian motion. We use…
We extend the ideas of (Barbour 1990) and use Stein's method to obtain a bound on the distance between a scaled time-changed random walk and a time-changed Brownian Motion. We then apply this result to bound the distance between a…
Diffusion approximations are widely used in the analysis of service systems, providing tractable insights into complex models. While heavy-traffic limit theorems justify these approximations asymptotically, they do not quantify the error…
Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite…
Donsker's theorem shows that random walks behave like Brownian motion in an asymptotic sense. This result can be used to approximate expectations associated with the time and location of a random walk when it first crosses a nonlinear…
We provide a general steady-state diffusion approximation result which bounds the Wasserstein distance between the reversible measure $\mu$ of a diffusion process and the measure $\nu$ of an approximating Markov chain. Our result is…
Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term…
We investigate the steady-state diffusion-approximation error for continuous-time queueing systems with generally distributed primitives. Across four canonical systems -- the $G/G/1$ and $G/M/\infty$ queues, the join-the-shortest-queue…
Motivated by queues with many servers, we study Brownian steady-state approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steady-state, we prove,…
One of the key ingredients to successfully apply Stein's method for distributional approximation are solutions to the Stein equations and their derivatives. Using Barbour's generator approach, one can solve for the solutions to the Stein…
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…
We consider the process of partial sums of moving averages of finite order with a regular varying memory function, constructed from a stationary sequence, variance of the sum of which is a regularly varying function. We study the Gaussian…
Diffusion processes have been widely used for approximations in the queueing theory. There are different types of diffusion approximations. Among them, we are interested in those obtained through limits of a sequence of models which…
The paper is concerned with the equilibrium distributions of continuous-time density dependent Markov processes on the integers. These distributions are known typically to be approximately normal, and the approximation error, as measured in…
We develop a general approach to Stein's method for approximating a random process in the path space $D([0,T]\to R^d)$ by a real continuous Gaussian process. We then use the approach in the context of processes that have a representation as…
We propose a unified approach to establishing diffusion approximations for queues with impatient customers within a general framework of scaling customer patience time. The approach consists of two steps. The first step is to show that the…
This paper studies the diffusion limit for a network of infinite-server queues operating under Markov modulation (meaning that the system's parameters depend on an autonomously evolving background process). In previous papers on (primarily…
We derive and analyze new diffusion approximations of stationary distributions of Markov chains that are based on second- and higher-order terms in the expansion of the Markov chain generator. Our approximations achieve a higher degree of…
In this paper, we analyze a discrete-time queue that is motivated from studying hospital inpatient flow management, where the customer count process captures the midnight inpatient census. The stationary distribution of the customer count…