ANNALS OF OPERATIONS RESEARCH, vol.293, no.1, pp.141-174, 2020 (Peer-Reviewed Journal)
Let X be the constrained random walk on Z2 + with increments (1, 0), (-1, 0), (0, 1) and (0,-1); X represents, at arrivals and service completions, the lengths of two queues (or two stacks in computer science applications) working in parallel whose service and interarrival times are exponentially distributed with arrival rates.i and service rates mu i, i = 1, 2; we assume.i < mu i, i = 1, 2, i.e., X is assumed stable. Without loss of generality we assume.1 =.1/mu 1 similar to.2 =.2/mu 2. Let tn be the first time X hits the line. An = {x. Z2 : x(1) + x(2) = n}, i.e., when the sum of the components of X equals n for the first time. Let Y be the same random walk as X but only constrained on {y. Z2 : y(2) = 0} and its jump probabilities for the first component reversed. Let. B = {y. Z2 : y(1) = y(2)} and let t be the first time Y hits. B. The probability pn = Px (tn < t0) is a key performance measure of the queueing system (or the two stacks) represented by X (if the queues/stacks share a common buffer, then pn is the probability that this buffer overflows during the system's first busy cycle). Stability of the process implies that pn decays exponentially in n when the process starts off the exit boundary. An. We show that, for xn = similar to nx similar to, x. R2+, x(1) + x(2) similar to 1, x(1) > 0, P( n- xn (1),xn (2))(t < 8) approximates Pxn (tn < t0) with exponentially vanishing relative error. Let r = (.1+.2)/(mu 1+mu 2); for r 2 <.2 and.1 similar to=.2, we construct a class of harmonic functions from single and conjugate points on a related characteristic surface for Y with which the probability Py(t < 8) can be approximated with bounded relative error. For r 2 =.1.2, we obtain the exact formula Py(t < 8) = r y(1)-y(2) + r (1-r) r-.2 similar to. y(1) 1 - r y(1)- y(2). y(2) 1 similar to.