Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process

Kroese, D. P., Scheinhardt, W. R. W. and Taylor, P. J. (2004) Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process. Annals of Applied Probability, 14 4: 2057-2089. doi:10.1214/105051604000000477

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Author Kroese, D. P.
Scheinhardt, W. R. W.
Taylor, P. J.
Title Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process
Journal name Annals of Applied Probability   Check publisher's open access policy
ISSN 1050-5164
Publication date 2004-01-01
Sub-type Article (original research)
DOI 10.1214/105051604000000477
Open Access Status File (Publisher version)
Volume 14
Issue 4
Start page 2057
End page 2089
Total pages 33
Place of publication Beachwood, OH, United States
Publisher Institute of Mathematical Statistics
Language eng
Subject 230119 Systems Theory and Control
230117 Operations Research
Abstract Quasi-birth-and-death (QBD) processes with infinite “phase spaces” can exhibit unusual and interesting behavior. One of the simplest examples of such a process is the two-node tandem Jackson network, with the “phase” giving the state of the first queue and the “level” giving the state of the second queue. In this paper, we undertake an extensive analysis of the properties of this QBD. In particular, we investigate the spectral properties of Neuts’s R-matrix and show that the decay rate of the stationary distribution of the “level” process is not always equal to the convergence norm of R. In fact, we show that we can obtain any decay rate from a certain range by controlling only the transition structure at level zero, which is independent of R. We also consider the sequence of tandem queues that is constructed by restricting the waiting room of the first queue to some finite capacity, and then allowing this capacity to increase to infinity. We show that the decay rates for the finite truncations converge to a value, which is not necessarily the decay rate in the infinite waiting room case. Finally, we show that the probability that the process hits level n before level 0 given that it starts in level 1 decays at a rate which is not necessarily the same as the decay rate for the stationary distribution.
Keyword Quasi-birth-and-death processes
Tandem Jackson network
Decay Rate
Stationary distribution
Hitting probabilities
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status UQ

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Created: Wed, 09 May 2007, 12:02:34 EST by Mrs Leith Woodall on behalf of School of Mathematics & Physics