Rare event asymptotics for a random walk in the quarter plane

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...

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Bibliographic Details
Published in:Queueing systems 2011-01, Vol.67 (1), p.1-32
Main Authors: Guillemin, Fabrice, van Leeuwaarden, Johan S. H.
Format: Article
Language:English
Subjects:
Asymptotic methods
Boundary value problems
Business and Management
Computer Communication Networks
Control
Mathematical analysis
Mathematics
Operations Research/Decision Theory
Probability Theory and Stochastic Processes
Queuing theory
Random walk theory
Studies
Supply Chain Management
Systems Theory
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Summary: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.
ISSN:0257-0130
1572-9443
DOI:10.1007/s11134-010-9197-7