Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)
On asymptotic merging of nodes' set in stochastic networks
E. A. Lebedev, H. V. Livinska
Abstract: For multichannel stochastic networks the problem of asymptotic merging of the queueing nodes' set is considered. Under heavy traffic conditions in the network, a functional limit theorem for a multidimensional service process is proved. The statement of the theorem concerns to the convergence of the service process to a Gaussian diffusion process. Under the asymptotic merging condition, the dimension of the approximating process is reduced, and its characteristics can be written via network parameters in explicit form.
1. V. V. Anisimov, Limit Theorems for Stochastic Processes and Their Application for Discrete Schemes of Summation, Vyshcha Shcola, Kyiv, 1976.
2. V. V. Anisimov, E. A. Lebedev, Stochastic Queueing Networks. Markov Models, Lybid, Kyiv, 1992.
3. V. S. Korolyuk, A. F. Turbin, Semi-Markov Processes and Their Applications, Naukova Dumka, Kyiv, 1976.
4. V. S. Korolyuk, Enlarging of Complex Systems, Cybernetics (1977), no. 1, 129-132.
5. E. O. Lebedev, Stationary Regime and Binomial Moments for networks of [SM|GI|$\infty$]^r-Type, Ukrainian Math. Journal, 54 (2002), no. 10, 1371-1380.
6. E. O. Lebedev, A Limit Theorem for Stochastic Networks and its Application, Theory of Probability and Mathematical Statistics, 68 (2003), 81-92.
7. E. O. Lebedev, O. A. Chechelnitskyi, H. V. Livinska, Multichannel Networks with Interdependent Input Flows in Heavy Trac, Theory of Probability and Mathematical Statistics, 97 (2017), 109-119.
8. I. Sinyakova, S. Moiseeva, Mathematical Model of Insurance Company as a Queueing System M|M|$\infty$, Proceedings of International Conference "Modern Probabilistic Methods of Analysis, Design and Optimization of Networks of Information and Telecommunication", Minsk, 2013, 154-159.
9. V. V. Anisimov, Switching processes in Queueing Models, ISTE Ltd, 2008.
10. A. Dvurecenskij, L. A. Kulyukina, G. A. Ososkov, Estimations of track ionization chambers, Transactions of United Nuclear Research Institute, Dubna, (1981), no. 5-81-362.
11. V. Korolyuk, A. Turbin, Mathematical Foundation of the State Lumping of Large Systems, Springer, Kluver, Dordrecht, 1993.
12. E. Lebedev, I. Makushenko, Prot Maximization and Risk Minimization in Semi-Markovian Networks, Cybernetics and Systems Analysis, 43 (2007), no. 2, 213-224.
13. E. Lebedev, G. Livinska, Gaussian approximation of multi-channel networks in heavy trac, Communications in Computer and Information Science, 356 (2013), 122-130.
14. W. A. Massey, 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.
15. J. H. Matis, T. E. Wehrly, Generalized stochastic compartmental models with Erlang transit times, Journal Pharmacokin. Bioharm. (1990), no. 18, 589-607.