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The tail asymptotic behavior of the stationary distribution of a double M/G/1 process and their applications to a batch arrival Jackson network

Authors:

Masahiro Kobayashi,

Masakiyo MiyazawaAuthors Info & Claims

QTNA '09: Proceedings of the 4th International Conference on Queueing Theory and Network Applications

Article No.: 8, Pages 1 - 4

https://doi.org/10.1145/1626553.1626561

Published: 29 July 2009 Publication History

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QTNA '09: Proceedings of the 4th International Conference on Queueing Theory and Network Applications

The tail asymptotic behavior of the stationary distribution of a double M/G/1 process and their applications to a batch arrival Jackson network

Pages 1 - 4

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Abstract

We consider a two dimensional reflected random walk on the nonnegative integer quadrant. This random walk is assumed to be skip free in the direction to the boundary, and referred to as a double M/G/1-type process. We are interested in tail asymptotic behavior of its stationary distribution, provided it exists. Assuming the arriving batch size distributions have light tails, we derive supremum for rough decay rates of the marginal stationary distributions in the coordinate directions. We then apply these results to a batch arrival Jackson network with two nodes. It is shown that the stochastic upper bound of Miyazawa and Taylor [4] is not tight except for a special case.

References

[1]

A. A. Borovkov and A. A. Mogul'skii (2001) Large deviations for Markov chains in the positive quadrant, Russian Math. Surveys 56, 803--916.

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[2]

W. L. Feller (1971) An Introduction to Probability Theory and Its Applications, 2nd edition, John Wiley&Sons, New York.

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[3]

M. Miyazawa (2009) Tail decay rates in double QBD processes and related reflected random walks, to appear in Mathematics of Operations Research.

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[4]

M. Miyazawa and P. G. Taylor (1997) A geometric product-form distribution for a queueing network with non-standard batch arrivals and batch transfers. Adv. Appl. Prob. 29, 523--544

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Kobayashi MMiyazawa MZhao Y(2010) Tail Asymptotics of the Occupation Measure for a Markov Additive Process with an M / G /1-Type Background Process Stochastic Models10.1080/15326349.2010.49831926:3(463-486)Online publication date: 4-Aug-2010
https://doi.org/10.1080/15326349.2010.498319

The tail asymptotic behavior of the stationary distribution of a double M/G/1 process and their applications to a batch arrival Jackson network
1. Mathematics of computing

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QTNA '09: Proceedings of the 4th International Conference on Queueing Theory and Network Applications

July 2009

143 pages

ISBN:9781605585628

DOI:10.1145/1626553

General Chair:
Winston Seah
Singapore

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Association for Computing Machinery

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Published: 29 July 2009

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Japan Society for the Promotion of Science

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QTNA '09: International Conference on Queueing Theory and Network Applications

July 29 - 31, 2009

Fusionopolis, Singapore

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Kobayashi MMiyazawa MZhao Y(2010) Tail Asymptotics of the Occupation Measure for a Markov Additive Process with an M / G /1-Type Background Process Stochastic Models10.1080/15326349.2010.49831926:3(463-486)Online publication date: 4-Aug-2010
https://doi.org/10.1080/15326349.2010.498319

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The M/G/1 Queue with Finite Workload Capacity

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Tail asymptotics for the $M_{1}, M_{2} / G_{1}, G_{2} / 1$ retrial queue with non-preemptive priority