A limit theorem for non-Markovian multi-channel networks under heavy traffic conditions
Author:
H. V. Livinska
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 93 (2016), 113-122
MSC (2010):
Primary 60K25, 90B15
DOI:
https://doi.org/10.1090/tpms/997
Published electronically:
February 7, 2017
MathSciNet review:
3553444
Full-text PDF Free Access
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Abstract: Open multi-channel stochastic networks are considered in the paper. The inputs are assumed to be non-homogeneous Poisson flows whose rates depend on time. Service times are random variables whose distribution functions are of the GI-type. A limit theorem for the service process is proved for such a network under heavy traffic conditions. Characteristics of the limit Gaussian process are expressed in an explicit form in terms of the network parameters.
References
- V. V. Anisimov and E. A. Lebedev, Stochastic Queueing Networks. Markov Models, “Lybid”, Kiev, 1992. (Russian)
- A. Yu. Veretennikov and A. M. Kulik, Diffusion approximation of systems with weakly ergodic Markov perturbations. I, Theory Probab. Math. Statist. 87 (2013), 13–29. Translation of Teor. Ǐmovīr. Mat. Stat. No. 87 (2012), 12–27. MR 3241443, DOI 10.1090/S0094-9000-2014-00901-1
- Ĭ. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, vol. 1, “Nauka”, Moscow, 1971; English transl., Springer-Verlag, New York–Heidelberg, 1974.
- Ē. O. Lebēdēv, A limit theorem for stochastic networks and its applications, Teor. Ĭmovīr. Mat. Stat. 68 (2003), 74–85 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 68 (2004), 81–92. MR 2000397, DOI 10.1090/S0094-9000-04-00606-4
- A. V. Livinskaya and E. A. Lebedev, Limit theorem for multichannel networks in heavy traffic, Cybernet. Systems Anal. 48 (2012), no. 6, 899–905. Translation of Kibernet. Sistem. Anal. 2012, no. 6, 106–113. MR 3228375, DOI 10.1007/s10559-012-9470-3
- G. V. Livins’ka, Approximation Gaussian process for networks of type $[M_t|M|\infty ]^r$ and its properties, Visnyk Kyiv Univ. Cibernet. (2012), no. 12, 32–37. (Ukrainian)
- Vladimir V. Anisimov, Switching processes in queueing models, Applied Stochastic Methods Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR 2437051, DOI 10.1002/9780470611340
- E. Lebedev and G. Livinska, Gaussian approximation of multi-channel networks in heavy traffic, Comm. Comp. Information Sci. (2013), no. 356, 122–130.
- H. V. Livinska and E. O. Lebedev, Conditions of Gaussian non-Markov approximation for multi-channel networks, Proc. 29th European Conf. Modelling and Simulation ECMS-2015, Albena (Varna), Bulgaria, 2015, pp. 642–649.
References
- V. V. Anisimov and E. A. Lebedev, Stochastic Queueing Networks. Markov Models, “Lybid”, Kiev, 1992. (Russian)
- A. Yu. Veretennikov and O. M. Kulik, Diffusion approximation of systems with weakly ergodic Markov perturbations. I, Teor. Ĭmovir. Mat. Stat. 87 (2012), 12–27; English transl. in Theory Probab. Math. Statist. 87 (2013), 13–29. MR 3241443
- Ĭ. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, vol. 1, “Nauka”, Moscow, 1971; English transl., Springer-Verlag, New York–Heidelberg, 1974.
- E. A. Lebedev, A limit theorem for stochastic networks and its applications, Teor. Ĭmovir. Mat. Stat. 68 (2003), 81–92; English transl. in Theory Probab. Math. Statist. 68 (2004), 74–85. MR 2000397
- A. V. Livinskaya and E. A. Lebedev, Limit theorem for multi-channel networks in heavy traffic, Kibernet. Sistem. Anal. (2012), no. 6, 106–113; English transl. in Cybernet. Systems Anal. 48 (2012), no. 6, 899–905. MR 3228375
- G. V. Livins’ka, Approximation Gaussian process for networks of type $[M_t|M|\infty ]^r$ and its properties, Visnyk Kyiv Univ. Cibernet. (2012), no. 12, 32–37. (Ukrainian)
- V. V. Anisimov, Switching Processes in Queueing Models, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR 2437051
- E. Lebedev and G. Livinska, Gaussian approximation of multi-channel networks in heavy traffic, Comm. Comp. Information Sci. (2013), no. 356, 122–130.
- H. V. Livinska and E. O. Lebedev, Conditions of Gaussian non-Markov approximation for multi-channel networks, Proc. 29th European Conf. Modelling and Simulation ECMS-2015, Albena (Varna), Bulgaria, 2015, pp. 642–649.
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Additional Information
H. V. Livinska
Affiliation:
Department of Applied Statistics, Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue, 4D, Kyiv 03127, Ukraine
Email:
livinskaav@gmail.com
Keywords:
Multi-channel queueing networks,
diffusion approximation,
heavy traffic regime
Received by editor(s):
June 16, 2015
Published electronically:
February 7, 2017
Additional Notes:
This paper was prepared following the talk at the International conference “Probability, Reliability and Stochastic Optimization (PRESTO-2015)” held in Kyiv, Ukraine, April 7–10, 2015
Article copyright:
© Copyright 2017
American Mathematical Society
