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Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process
Kroese, DP, Scheinhardt, WRW and Taylor, PG (2004) Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process. Annals of Applied Probability, 14 4: 2057-2089.
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| Author(s) |
Kroese, DP
Scheinhardt, WRW
Taylor, PG
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| Title |
Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process
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| Journal name |
Annals of Applied Probability
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| Publication date |
2004
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| Volume number |
14
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| Issue number |
4
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| ISSN |
1050-5164
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| Start page |
2057
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| End page |
2089
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| Total pages |
33
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| Place of publication |
Beachwood
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| Publisher |
Inst Mathematical Statistics
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| 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 News'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.
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| Keyword(s) |
Statistics & Probability
decay rate
tandem Jackson network
QBD process
stationary distribution
hitting probabilities
Markov-chains
Stationary Distributions
Models
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