Approximation of excessive backlog probabilities of two tandem queues

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.