On the asymptotic merging of the set of nodes in stochastic networks

Lebedev, E.; Livinska, G.

doi:10.1090/tpms/1119

On the asymptotic merging of the set of nodes in stochastic networks

Authors: E. O. Lebedev and G. V. Livinska
Translated by: S. V. Kvasko
Journal: Theor. Probability and Math. Statist. 101 (2020), 167-177
MSC (2020): Primary 60K25, 90B15
DOI: https://doi.org/10.1090/tpms/1119
Published electronically: January 5, 2021
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Abstract: We consider the problem of the asymptotic merging of the set of service nodes for multi-channel stochastic networks. A functional limit theorem on the convergence to a Gaussian diffusion process is proved for the multidimensional service process in the case of a critical overloaded mode. The dimension of the approximating process decreases under the asymptotic merging and its characteristics are expressed in terms of parameters of a network.

References

V. V. Anisimov, Limit Theorems for Stochastic Processes and Their Application for Discrete Schemes of Summation, “Vyshcha Shcola”, Kyiv, 1976. (in Russian)

V. V. Anisimov and E. A. Lebedev, Stochastic Queueing Networks. Markov Models, “Lybid”, Kyiv, 1992. (in Russian)

V. S. Korolyuk and A. F. Turbin, Polumarkovskie protsessy i ikh prilozheniya, Izdat. “Naukova Dumka”, Kiev, 1976 (Russian). MR 0420902

V. S. Korolyuk, Enlarging of complex systems, Kibernetika (1977), no. 1, 129–132; English transl. in Cybernetics (1977), no. 1, 131–134.

Ē. O. Lebēdēv, A steady-state mode and binomial moments for networks of the type $[SM|GI|\infty ]^r$, Ukraïn. Mat. Zh. 54 (2002), no. 10, 1371–1380 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 54 (2002), no. 10, 1656–1668. MR 2015488, DOI https://doi.org/10.1023/A%3A1023732303635

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 https://doi.org/10.1090/S0094-9000-04-00606-4

Ē. O. Lebēdēv, O. A. Chechel′nits′kiĭ, and G. V. Līvīns′ka, Multichannel networks with interdependent input flows in heavy traffic, Teor. Ĭmovīr. Mat. Stat. 97 (2017), 109–119 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 97 (2018), 113–125. MR 3746003, DOI https://doi.org/10.1090/tpms/1052

I. Sinyakova and S. Moiseeva, Mathematical model of insurance company as a queueing system $M|M|\infty$, Proc. Internat. Conf. “Modern Probabilistic Methods of Analysis, Design and Optimization of Networks of Information and Telecommunication”, Minsk, 2013, pp. 154–159.

Vladimir V. Anisimov, Switching processes in queueing models, Applied Stochastic Methods Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR 2437051

A. Dvurečenskij, L. A. Kulyukina, and G. A. Ososkov, Estimations of track ionization chambers, Transactions of United Nuclear Research Institute, Dubna (1981), no. 5-81-362.

Vladimir S. Korolyuk and Anatoly F. Turbin, Mathematical foundations of the state lumping of large systems, Mathematics and its Applications, vol. 264, Kluwer Academic Publishers Group, Dordrecht, 1993. Translated from the 1978 Russian original by V. V. Zayats and Y. A. Atanov and revised by the authors. MR 1281385

E. A. Lebedev and I. A. Makushenko, Profit maximization and risk minimization in semi-Markovian networks, Kibernet. Sistem. Anal. 43 (2007), no. 2, 65–79, 190 (Russian, with English and Ukrainian summaries); English transl., Cybernet. Systems Anal. 43 (2007), no. 2, 213–224. MR 2374415, DOI https://doi.org/10.1007/s10559-007-0040-z

E. Lebedev and G. Livinska, Gaussian approximation of multi-channel networks in heavy traffic, Modern Probabilistic Methods for Analysis of Telecommunication Networks 356 (2013), 122–130.

W. A. Massey and W. Whitt, A stochastic model to capture space and time dynamics in wire less communication systems, Prob. Eng. Inf. Sci. (1994), no. 8, 541–569.

J. H. Matis and T. E. Wehrly, Generalized stochastic compartmental models with Erlang transit times, J. Pharmacokinetics and Biopharmaceutics (1990), no. 18, 589–607.