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 (or two stacks in computer science applications) 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 $$\rho _1 =\lambda _1/\mu _1 \geqslant \rho _2 = \lambda _2/\mu _2$$ . Let $$\tau _n$$ be the first time X hits the line $$\partial A_n = \{x \in {\mathbb Z}^2:x(1)+x(2) = n \}$$ , i.e., when the sum of the components of X equals n for the first time. 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 (or the two stacks) represented by X (if the queues/stacks share a common buffer, then $$p_n$$ is the probability that this buffer overflows during the system’s first busy cycle). Stability of the process implies that $$p_n$$ decays exponentially in n when the process starts off the exit boundary $$\partial A_n.$$ We show that, for $$x_n= \lfloor nx \rfloor $$ , $$x \in {{\mathbb {R}}}_+^2$$ , $$x(1)+x(2) \leqslant 1$$ , $$x(1) > 0$$ , $$P_{(n-x_n(1),x_n(2))}( \tau < \infty )$$ 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 $$\rho _1 \ne \rho _2$$ , we construct a class of harmonic functions from single and conjugate points on a related characteristic surface for Y with which the probability $$P_y(\tau < \infty )$$ can be approximated with bounded relative error. For $$r^2 = \rho _1 \rho _2$$ , we obtain the exact formula $$P_y(\tau < \infty ) = r^{y(1)-y(2)} +\frac{r(1-r)}{r-\rho _2}\left( \rho _1^{y(1)} - r^{y(1)-y(2)} \rho _1^{y(2)}\right) .$$