Rare event asymptotics for a random walk in the quarter plane

Abstract

This paper presents a novel technique for deriving asymptotic expressions for the occurrence of rare events for a random walk in the quarter plane. In particular, we study a tandem queue with Poisson arrivals, exponential service times and coupled processors. The service rate for one queue is only a fraction of the global service rate when the other queue is non-empty; when one queue is empty, the other queue has full service rate. The bivariate generating function of the queue lengths gives rise to a functional equation. In order to derive asymptotic expressions for large queue lengths, we combine the kernel method for functional equations with boundary value problems and singularity analysis.

Erratum to: Wireless three-hop networks with stealing II: exact solutions through boundary value problems

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Contrary to what we claimed in [5], the solution to the Riemann-Hilbert problem (4) considered in [5] for some domain D x is in general not the restriction to D x of the solution to the modified Riemann-Hilbert problem (6) in [5]. This occurs only when D x is a circle, which is not the case considered in that paper.

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“…To compute the function P(x, 0), by using arguments similar to those in [5], we consider the function P x (x) given by Eq. (20) in [5], namely P x (x) = P(x, 0) − R x x − x * + P(0, 0).…”

“…By using similar arguments as those for the conformal mapping γ x (x), we can prove that the conformal mapping γ y (y) is analytic in D(0, y 3 ) with continuous limits on the boundary. Equation (51) in [5] should read…”

On the structure and solutions of functional equations arising from queueing models

“…Now we describe the equation that arises [37] from two-stage tandem queues, illustrated in Fig. 21, where jobs arrive at queue 1 according to a Poisson process with rate λ, demanding service at both queues before leaving the system.…”

A note on solutions of a functional equation arising in a queuing model for a LAN gateway

“…There are many special cases of (1.1) arising in various models of applications (see [6][7][8][9][10][11][12]). In this paper we investigate a particular example of Eq.…”