Seminar: “Approximation of a Buffer Overflow Probability of a Stable Queueing System,” Ali Devin Sezer (METU), EA-409, 1:40PM April 14 (EN)


April 14, Friday 13:40


Ali Devin Sezer, Institute of Applied Mathematics, METU

Approximation of a Buffer Overflow Probability of a Stable Queueing System

For a stable constrained random walk $X$ on ${\mathbb Z}_+^d$ let $\tau_n$ denote the first time when the sum of the components of $X$ equals $n$. We will explain the derivation of a simple approximating formula for the probability $P_x(\tau_n < \tau_0)$ for the case of the random walk representing two tandem queues with exponentially distributed inter-arrival and service times. This will involve the following ideas: 1) derivation of a limit partially constrained and unstable random walk $Y$, 2) construction of explicit harmonic functions of $Y$ using what we call the characteristic surface of $Y$ and harmonic graphs. The harmonic functions will be used to compute $P_y(\tau < \infty)$ explicitly where $\tau$ is the first hitting time of $Y$ to a limit boundary determined by the hitting boundary used in the definition of $\tau_n.$ We will go over the proof of the convergence result that the relative approximation error $|P_{y_n}(\tau < \infty) -P_{x_n}(\tau_n < \tau_0)|/P_{x_n}(\tau_n < \tau_0)$ (where $x_n = \lfloor nx \rfloor$ and $y_n$ is the initial point for the $Y$ process corresponding to $x_n$) decays exponentially in $n$ for most points $x \in \{ 0 < x(1) + x(2) < 1\}.$

Ali Devin Sezer was born in Ankara, Turkey in 1977. He was graduated from Ankara Science High School in 1995, Middle East Technical University (METU), Departments of Computer Engineering and of Mathematics in 2000. He obtained his Ph.D. from Brown University, Division of Applied Mathematics in 2006. Since then he has worked at METU, Institute of Applied Mathematics. Between 2012 and 2014 he was on sabbatical leave from METU to spend two years as a Marie Sklodowska-Curie fellow in an EU-research-project at L'Université d’Evry, Le Laboratoire Analyse et Probabilités. His research interests include asymptotic analysis in (applied) probability, optimal control and simulation.