Published online by Cambridge University Press: 16 November 2018
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.