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 $λ_i$ and service rates $μ_i$, $i=1,2$; we assume $λ_i < μ_i$, $i=1,2$, i.e., $X$ is assumed stable. Without loss of generality we assume $ρ_1 =λ_1/μ_1 \ge ρ_2 = λ_2/μ_2$. Let $τ_n$ 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 $τ$ be the first time $Y$ hits $\partial B$. The probability $p_n = P_x(τ_n < τ_0)$ 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 $x_n= \lfloor nx \rfloor$, $x \in {\mathbb R}_+^2$, $x(1)+x(2) \le 1$, $x(1) > 0$, $P_{(n-x_n(1),x_n(2))}( τ< \infty)$ approximates $P_{x_n}(τ_n < τ_0)$ with exponentially vanishing relative error. Let $r = (λ_1 + λ_2)/(μ_1 + μ_2)$; for $r^2 < ρ_2$ and $ρ_1 \neq ρ_2$, we construct a class of harmonic functions from single and conjugate points on a characteristic surface of $Y$ with which $P_y(τ< \infty)$ can be approximated with bounded relative error. For $r^2 = ρ_1 ρ_2$, we obtain $P_y(τ< \infty) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-ρ_2}\left( ρ_1^{y(1)} - r^{y(1)-y(2)} ρ_1^{y(2)}\right).$