Excessive Backlog Probabilities of Two Parallel Queues

TL;DR

This paper analyzes the probabilities of queue overflow in a two-parallel queue system modeled by constrained random walks, providing exponential decay estimates and constructing harmonic functions for approximation.

Contribution

It introduces a novel method to approximate overflow probabilities using harmonic functions derived from characteristic surfaces of related random walks.

Findings

Overflow probability decays exponentially with system size.

Harmonic functions effectively approximate key probabilities.

Explicit formulas are derived for specific parameter regimes.

Abstract

Let $X$ be the constrained random walk on ${\mathbb Z}_+^2$ with increments $(1,0)$, $(-1,0)$, $(0,1)$ and $(0,-1)$; $X$ represents, at arrivals and service completions, the lengths of two queues working in parallel whose service and interarrival times are exponentially distributed with arrival rates $\lambda_i$ and service rates $\mu_i$, $i=1,2$; we assume $\lambda_i < \mu_i$, $i=1,2$, i.e., $X$ is assumed stable. Without loss of generality we assume $\rho_1 =\lambda_1/\mu_1 \ge \rho_2 = \lambda_2/\mu_2$. Let $\tau_n$ be the first time $X$ hits the line $\partial A_n = \{x \in {\mathbb Z}^2:x(1)+x(2) = n \}$. Let $Y$ be the same random walk as $X$ but only constrained on $\{y \in {\mathbb Z}^2: y(2)=0\}$ and its jump probabilities for the first component reversed. Let $\partial B =\{y \in {\mathbb Z}^2: y(1) = y(2) \}$ and let $\tau$ be the first time $Y$ hits $\partial B$. The probability $p_n = P_x(\tau_n < \tau_0)$ is a key performance measure of the queueing system represented by $X$ (probability of overflow of a shared buffer during system's first busy cycle). Stability of $X$ implies $p_n$ decays exponentially in $n$ when the process starts off $\partial A_n.$ We show that, for $x_n= \lfloor nx \rfloor$, $x \in {\mathbb R}_+^2$, $x(1)+x(2) \le 1$, $x(1) > 0$, $P_{(n-x_n(1),x_n(2))}( \tau < \infty)$ approximates $P_{x_n}(\tau_n < \tau_0)$ with exponentially vanishing relative error. Let $r = (\lambda_1 + \lambda_2)/(\mu_1 + \mu_2)$; for $r^2 < \rho_2$ and $\rho_1 \neq \rho_2$, we construct a class of harmonic functions from single and conjugate points on a characteristic surface of $Y$ with which $P_y(\tau < \infty)$ can be approximated with bounded relative error. For $r^2 = \rho_1 \rho_2$, we obtain $P_y(\tau < \infty) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-\rho_2}\left( \rho_1^{y(1)} - r^{y(1)-y(2)} \rho_1^{y(2)}\right).$