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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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