On Distributions of One Class of Random Sums and their Applications

  • Ivan Matsak Department of Operation Research, Taras Shevchenko National University of Kyiv, Ukraine
  • Mikhail Moklyachuk Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv,Ukraine
DOI: https://doi.org/10.19139/soic-2310-5070-698
Keywords: Random sums, queueing, reliability, redundant system, renewal

Abstract

We propose results of the investigation of properties of the random sums of random variables. We consider the case, where the number of summands is the first moment of an event occurrence. An integral equation is presented that determines distributions of random sums. With the help of the obtained results we analyse the distribution function of the time during which the Geiger-Muller counter will not lose any particles, the distribution function of the busy period of a redundant system with renewal, and the distribution function of the sojourn times of a single-server queueing system.

Author Biographies

Ivan Matsak, Department of Operation Research, Taras Shevchenko National University of Kyiv, Ukraine
Department of Operation Research, Taras Shevchenko National University of Kyiv, Ukraine
Mikhail Moklyachuk, Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv,Ukraine
Department of Probability Theory, Statistics and Actuarial Mathematics, Professor

References

A. A. Borovkov, Probability theory, London: Universitext, Springer, 2013.

Y. S. Chow, H. Robbins, and D. Siegmund, Great expectations: The theory of optimal stopping, Boston etc.: Houghton Mifflin Company. XII, 1971.

D. R. Cox, Renewal theory, Methuen & Co. Ltd., London; John Wiley & Sons, Inc., New York, 1962.

B.V. Dovgai, and I.K. Matsak, On a redundant system with renewals, Theory of Probability and Mathematical Statistics, Vol. 94, p.63–76, 2017.

W. Feller, An introduction to probability theory and its applications. Vol. 1. 2nd ed., New York etc.: JohnWiley and Sons, Inc., 1968.

W. Feller, An introduction to probability theory and its applications. Vol. 2. 2nd ed., New York etc.: JohnWiley and Sons, Inc., 1971.

V.K. Gedam, and S. B. Pathare, Estimation approaches for mean response time of a two stage open queueing network model, Statistics,Optimization & Information Computing, Vol. 3, pp. 249–258, 2015.

V. K. Gedam, and S. B. Pathare, Some estimation approaches of intensities for a two stage open queueing network, StatisticsOptimization and Information Computing. Vol. 2, No.1. pp. 33-46, 2014.

B. V. Gnedenko, On a two-unit redundant system, Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 4, p. 3–12, 1964 (In Russian).

B. V. Gnedenko, Doubling with renewal, Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 5, p. 111–118. 1964 (In Russian).

B. V. Gnedenko, Yu. K. Belyayev, and A. D. Solovyev, Mathematical methods of reliability theory, New York – London: Academic Press, 1969.

B. V. Gnedenko, and I. N. Kovalenko, Introduction to queuing theory, Moskva: Nauka, 1966 (In Russian).

A. Gut, Stopped random walks. Limit theorems and applications. 2nd ed., New York, NY: Springer, 2009.

V. Kalashnikov, Geometric sums: bounds for rare events with applications, Amsterdam: Springer, 1997.

L. Kleinrock, Queueing systems.Computer application. Vol.2, New York: Wiley, 1976.

A. N. Kolmogorov, Selected works by A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics. Ed. by A. N. Shiryayev, Mathematics and Its Applications. Soviet Series. 26. Dordrecht etc.: Kluwer Academic Publishers, 1992.

V. S. Korolyuk, Stochastic models of systems, Kyiv: Naukova Dumka, 1989 (In Russian).

I. N. Kovalenko, Studies in the reliability analysis of complex systems, Kyiv: Naukova Dumka, 1975 (In Russian).

V. M. Kruglov, and V. Yu. Korolev, Limit theorems for random sums, Moskva: Izdatelstvo Moskovskogo Universiteta, 1990 (In Russian).

I. K. Matsak, On a single-server queueing system with refusal, Theory of Probability and Mathematical Statistics, Vol. 90, P. 153-160, 2015.

S. I. Resnick, Adventures in stochastic processes, Boston etc.: Birkh¨auser, 2002.

W. L. Smith, Renewal theory and its ramifications, J. Roy. Statist. Soc. Ser. B 20, pp. 243–302, 1958.

A. Wald, Sequential analysis, New York: Wiley & Sons, 1947.

O. K. Zakusylo, and I. K. Matsak, On extreme values of some regenerative processes, Theory of Probability and Mathematical Statistics, Vol. 97, pp. 57–71, 2018.

Published
2020-02-18
How to Cite
Matsak, I., & Moklyachuk, M. (2020). On Distributions of One Class of Random Sums and their Applications. Statistics, Optimization & Information Computing, 8(1), 153-164. https://doi.org/10.19139/soic-2310-5070-698
Issue
Vol 8 No 1 (2020)
Section
Research Articles
Authors who publish with this journal agree to the following terms:

    1. Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
    2. Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
    3. Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).