Geodesic
On the approximation of the flow of events for a Poisson
Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 222-234.

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When modeling an extensive class of technical systems, the mathematical apparatus of queuing systems (QMS) is widely used. An example of such a system is the computer network, where computer applications are generated and executed. Applications are generated usually not regularly, but by accident, forming the so-called random order of applications (requirements). Service requests, it also continues some random time. One of the central issues in the organization of mass-service systems is the elucidation of the regularities that subordinate the moments when system requirements for service are submitted. The article explores the flow of events in technical systems of various purposes. On the basis of the fact that under the Poisson character of the flow mathematical modeling of the systems is greatly simplified, the problem of obtaining a simple criterion for determining the degree of approximation of the flow of events to a Poisson one is posed. Pearson's criterion, regression, correlation and parametric criteria were investigated. A criterion based on the calculation of the waiting function was obtained again. On the example of the study of the system with "competitions" it is shown that the flow of events generated by the system tends to Poisson with an infinite increase in the number of "competing" subjects. Bibliography: 14 titles.
Keywords: Event flow, Poisson flow, semi-Markov process, Pearson's criterion, correlation, regression, expectation function, uniform law. 
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E. V. Larkin; D. V. Gorbachev; A. N. Privalov. On the approximation of the flow of events for a Poisson. Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 222-234. http://geodesic.mathdoc.fr/item/CHEB_2017_18_2_a13/

[1] Sundarapandian V., Queueing Theory: Probability, Statistics and Queueing Theory, PHI Learning, New Delhi, 2009

[2] Gross D., Harris C. M., Fundamentals of Queue Theory, John Wiley Sons, N.Y., 1974 | MR

[3] Larkin E. V., Ivutin A. N., “Dispatching in Embedded Systems”, 5th Mediterranean Conference on Embedded Computing, MECO 2016 (12–16 June 2016, Bar, Montenegro), IEEE, 2016, 215–217

[4] Larkin E., Ivutin A., Esikov D., “Recursive Approach for Evaluation of Time Intervals between Transactions in Polling Procedure”, 8-th International Conference on Computer and Automation Engineering (ICCAE 2016) (March 3–4, 2016, Melbourne, Australia), MATEC Web of Conferences, 56, 2016, 01004 | DOI

[5] Larkin E. V., Ivutin A. N., Kotov V. V., Privalov A. N., “Interactive generator of commands”, Proceedings, 7-th International Conference ICSI-2016, LNCS Sublibrary: SL1 — Theoretical Computer Science and General Issues (Bali, Indonesia, June 25–30), v. 2, Lecture Notes in Computer Science, Springer, 2016, 601–608 | DOI | MR

[6] Larkin E. V., Privalov A. N., “Modeling of dialogue regimes of distance robot control”, Proceedings of 5th International Workshop on Mathematical Models and their Applications (Krasnoyarsk, Russia, November 7–9, 2016), 92–103

[7] Markov A. A., “Extension of the law of large numbers to dependent quantities”, Izvestiia Fiz.-Matem. Obsch. Kazan Univ., (2-nd Ser.), 1906, 135–156 | Zbl

[8] Boos D. D., Stefanski L. A., Essentioal Statistical Inference. Theory and methods, Springer Verlag, N.Y., 2013, xvii+568 pp. | MR

[9] Draper N. R., Smith H., Applied Regression Analysis, John Wiley Sons, Inc., 1998, 736 pp. | MR

[10] Rank M. K., Correlation Methods, Charles Griffin Company, London, 1955, 196 pp.

[11] Ventsel E. S., Probability theory, Mir Publisher, M., 1986, 86 pp.

[12] Ivutin A. N., Larkin E. V., “Simulation of Concurrent Games”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 8:2 (2015), 43–54 | Zbl

[13] Larkin E. V., Ivutin A. N., Kotov V. V., Privalov A. N., “Simulation of Relay-races”, Bulletin of the South Ural State University. Mathematical Modelling, Programming Computer Software, 9:4 (2016), 117–128

[14] Glushkov V. M, Amosov N. M., Artemenko I. A., Encyclopedia of Cybernetics, v. 2, Kiev, 1974

[15] Grigelionis B., “On the convergence of sums of random step processes to a Poisson process”, Theory Probab. Appl., 8:2 (1963), 177–182 | DOI | MR | Zbl