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Approximation of excessive backlog probabilities of two tandem queues

Part of: Limit theorems Markov processes Stochastic processes

Published online by Cambridge University Press:  16 November 2018

Ali Devin Sezer
Ali Devin Sezer*
Affiliation:
Middle East Technical University
*
* Postal address: Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey. Email address: devin.sezer@gmail.com
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Abstract

Let X be the constrained random walk on ℤ+2 having increments (1,0), (-1,1), and (0,-1) with respective probabilities λ, µ1, and µ2 representing the lengths of two tandem queues. We assume that X is stable and µ1≠µ2. Let τn be the first time when the sum of the components of X equals n. Let Y be the constrained random walk on ℤ×ℤ+ having increments (-1,0), (1,1), and (0,-1) with probabilities λ, µ1, and µ2. Let τ be the first time that the components of Y are equal to each other. We prove that Pn-xn(1),xn(2)(τ<∞) approximates pn(xn) with relative error exponentially decaying in n for xn=⌊nx⌋, x ∈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→∞ 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 ℙy(τ<∞) 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 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.

Keywords

Large deviationconstrained random walkbuffer overflowqueueing systemexit timeharmonic system

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60F10: Large deviations 60J45: Probabilistic potential theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 968 - 997
Copyright
Copyright © Applied Probability Trust 2018 

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