Exact asymptotics for the stationary distribution of a Markov chain: a production model
type: Item
instance of: scholarly article
title: Exact asymptotics for the stationary distribution of a Markov chain: a production model
zbMATH Open document ID: 1188.60046
author: Robert D. Foley
author: David R. Mcdonald
author: Ivo Adan
published in: Queueing Systems
publication date: 2009-11-06T00:00:00Z
review text: The authors are interested in estimating the probability of rare events related to the stationary distribution \(\pi \) of Markov chains that typically arise in modeling queueing networks. They develop an approach to deriving the exact asymptotics of \(\pi \) that allows them to analyze situations where the fluid limit of excursions to the (increasingly) rare event is nonlinear. This nonlinear behavior can arise in a pair of stable, \(M/M/1\) queues in tandem. To illustrate the power of the approach, they completely describe the exact asymptotics of \(\pi \) for a production model in all directions and for all stable parameter settings. The production model, described in the paper, has unbounded jumps; at every point in the state space, the boundaries influence the possible transitions. In addition, for certain regions of the parameters, the fluid limits of excursions to the rare events are nonlinear.
reviewed by: Oleg K. Zakusilo
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 60K20
Mathematics Subject Classification ID: 90B22
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 5627123
zbMATH Keywords: Rare events
zbMATH Keywords: large deviations
zbMATH Keywords: exact asymptotics
zbMATH Keywords: Markov chain
zbMATH Keywords: production model
MaRDI profile type: Publication
full work available at URL: s11134-009-9140-y
OpenAlex ID: W2111170904
cites work: Q4109064
cites work: Large deviations of multiclassM/G/1 queues
cites work: Q3059475
cites work: Q3721531
cites work: Join the shortest queue: Stability and exact asymptotics
cites work: Bridges and networks: exact asymptotics
cites work: Large deviations of a modified Jackson network: stability and rough asymptotics
cites work: A maximal coupling for Markov chains
cites work: Martin boundary of a killed random walk on a half-space
cites work: Renewal theory for functionals of a Markov chain with general state space
cites work: Geometric Decay in a QBD Process with Countable Background States with Applications to a Join-the-Shortest-Queue Model
cites work: Asymptotics of first passage times for random walk in an orthant
cites work: Markov chains and stochastic stability
cites work: Q4865050
DOI: 10.1007/S11134-009-9140-Y
description: scientific article; zbMATH DE number 5627123