On Distributions of One Class of Random Sums and their Applications
AbstractWe 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.
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.
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