Excessive Backlog Probabilities of Two Parallel Queues

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 $P_y(\tau < \infty) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-\rho_2}\left( \rho_1^{y(1)} - r^{y(1)-y(2)} \rho_1^{y(2)}\right).$