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Sufficient conditions for a geometric tail in a QBD process with many countable levels and phases. (English) Zbl 1065.60133
Summary: We present sufficient conditions, under which the stationary probability vector of a QBD process with both infinite levels and phases decays geometrically, characterized by the convergence norm \(\eta\) and the \(1/\eta\)-left-invariant vector \(x\) of the rate matrix \(R\). We also present a method to compute \(\eta\) and \(x\) based on spectral properties of the censored matrix of a matrix function constructed with the repeating blocks of the transition matrix of the QBD process. What makes this method attractive is its simplicity; finding \(\eta\) reduces to determining the zeros of a polynomial. We demonstrate the application of our method through a few interesting examples.

MSC:
60K25 Queueing theory (aspects of probability theory)
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References:
[1] Adan I., A Compensation Approach for Queueing Problems (1991) · Zbl 0746.60091
[2] DOI: 10.1023/A:1011040100856 · Zbl 0974.60079 · doi:10.1023/A:1011040100856
[3] DOI: 10.1137/0144074 · Zbl 0554.90041 · doi:10.1137/0144074
[4] Foley R. D., Ann. App. Probab. 11 pp 569– (2001)
[5] Kemeny J. G., Denumerable Markov Chains (1976) · Zbl 0178.53306
[6] DOI: 10.1287/moor.18.2.423 · Zbl 0771.60080 · doi:10.1287/moor.18.2.423
[7] Li Q.-L., Matrix-Analytic Methods: Theory and Applications pp 237– (2002)
[8] DOI: 10.1081/STM-120020387 · Zbl 1020.60088 · doi:10.1081/STM-120020387
[9] Meyn S. P., Markov Chains and Stochastic Stability (1993) · Zbl 0925.60001
[10] DOI: 10.1287/opre.29.5.945 · Zbl 0468.60089 · doi:10.1287/opre.29.5.945
[11] Neuts M. F., Matrix-Geometric Solutions in Stochastic Models (1981) · Zbl 0469.60002
[12] DOI: 10.1080/15326349608807377 · Zbl 0846.60086 · doi:10.1080/15326349608807377
[13] Seneta E., Non-Negative Matrices and Markov Chains (1981) · Zbl 1099.60004 · doi:10.1007/0-387-32792-4
[14] DOI: 10.1081/STM-100001397 · Zbl 0985.60074 · doi:10.1081/STM-100001397
[15] Zhao Y. Q., Advances in Algorithmic Methods for Stochastic Models pp 417– (2000)
[16] DOI: 10.1023/A:1024125320911 · Zbl 1022.60073 · doi:10.1023/A:1024125320911
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.