Approximation of excessive backlog probabilities of two tandem queues

Author:

Sezer Ali Devin

Abstract

AbstractLetXbe the constrained random walk on ℤ+2having increments (1,0), (-1,1), and (0,-1) with respective probabilities λ, µ1, and µ2representing the lengths of two tandem queues. We assume thatXis stable and µ1≠µ2. Let τnbe the first time when the sum of the components ofXequalsn. LetYbe 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 ofYare equal to each other. We prove thatPn-xn(1),xn(2)(τ<∞) approximatespn(xn) with relative errorexponentially decayinginnforxn=⌊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 lettingn→∞ connect theXandYprocesses. We use a linear combination of basis functions constructed from single and conjugate points on a characteristic surface associated withXto derive a simple expression for ℙy(τ<∞) in terms of the utilization rates of the nodes. The proof that the relative error decays exponentially innuses a sequence of subsolutions of a related Hamilton‒Jacobi‒Bellman equation on a manifold consisting of three copies of ℝ+2glued 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.

Publisher

Cambridge University Press (CUP)

Subject

Statistics, Probability and Uncertainty,General Mathematics,Statistics and Probability

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