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Approximation of excessive backlog probabilities of two tandem queues. (English) Zbl 1401.60081
Summary: Let $$X$$ be the constrained random walk on $$\mathbb Z_+^2$$ having increments $$(1,0)$$, $$(-1,1)$$, and $$(0,-1)$$ with respective probabilities $$\lambda$$, $$\mu_1$$, and $$\mu_2$$ representing the lengths of two tandem queues. We assume that $$X$$ is stable and $$\mu_1\neq\mu_2$$. Let $$\tau_n$$ be the first time when the sum of the components of $$X$$ equals $$n$$. Let $$Y$$ be the constrained random walk on $$\mathbb Z\times\mathbb Z_+$$ having increments $$(-1,0)$$, $$(1,1)$$, and $$(0,-1)$$ with probabilities $$\lambda$$, $$\mu_1$$, and $$\mu_2$$. Let $$\tau$$ be the first time that the components of $$Y$$ are equal to each other. We prove that $$P_{n-x_n(1),x_n(2)}(\tau<\infty)$$ approximates $$p_n(x_n)$$ with relative error exponentially decaying in $$n$$ for $$x_n=\lfloor nx\rfloor$$, $$x \in\mathbb R_+^2$$, $$0<x(1)+x(2)<1$$, $$x(1)>0$$. An affine transformation moving the origin to the point ($$n,0$$) and letting $$n\rightarrow\infty$$ connect the $$X$$ and $$Y$$ processes. We use a linear combination of basis functions constructed from single and conjugate points on a characteristic surface associated with $$X$$ to derive a simple expression for $$\mathbb P_y(\tau<\infty)$$ in terms of the utilization rates of the nodes. The proof that the relative error decays exponentially in $$n$$ uses a sequence of subsolutions of a related Hamilton-Jacobi-Bellman equation on a manifold consisting of three copies of $$\mathbb R_+^2$$ glued to each other along the constraining boundaries. We indicate how the ideas of the paper can be generalized to more general processes and other exit boundaries.

MSC:
 60G50 Sums of independent random variables; random walks 60K25 Queueing theory (aspects of probability theory) 60G40 Stopping times; optimal stopping problems; gambling theory 60F10 Large deviations 60J45 Probabilistic potential theory
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