Abstract
By using a purely probabilistic argumentation, two theorems are proved. They simplify the existing methods of analysis for the
Dedicated to the 90th birthday of Professor Ilya Mikadze (1928–2010)
Acknowledgements
The author is grateful to Prof. V. Tarieladze for his permanent support and to the referee for the positive report and useful remarks.
References
[1] R. E. Barlow and F. Proschan, Mathematical Theory of Reliability, Class. Appl. Math. 17, Society for Industrial and Applied Mathematics, Philadelphia, 1996. 10.1137/1.9781611971194Search in Google Scholar
[2] M. Ben-Daya, S. O. Duffuaa, A. Raouf, J. Knezevic and D. Ait-Kadi, Handbook of Maintenance Management and Engineering, Springer, London, 2009. 10.1007/978-1-84882-472-0Search in Google Scholar
[3] D. R. Cox, The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Proc. Cambridge Philos. Soc. 51 (1955), 433–441. 10.1017/S0305004100030437Search in Google Scholar
[4] W. Feller, An Introduction to Probability Theory and its Applications. Vol. I, 3rd ed., John Wiley & Sons, New York, 1969. Search in Google Scholar
[5] W. Feller, An Introduction to Probability Theory and its Applications. Vol. II, 2nd ed., John Wiley & Sons, New York-London, 1971. Search in Google Scholar
[6] B. V. Gnedenko, Y. K. Belyayev and A. D. Solovyev, Mathematical Methods of Reliability Theory, Probab. Math. Statist. 6, Academic Press, New York, 1969. 10.1016/B978-1-4832-3053-5.50007-7Search in Google Scholar
[7] B. V. Gnedenko and I. N. Kovalenko, Introduction to Queueing Theory, 2nd ed., Math. Model. 5, Birkhäuser, Boston, 1989. 10.1007/978-1-4615-9826-8Search in Google Scholar
[8]
W. Henderson,
Alternative approaches to the analysis of the
[9] N. K. Jaiswal, Priority Queues, Math. Sci. Eng. 50, Academic Press, New York, 1968. Search in Google Scholar
[10] J. Keilson and A. Kooharian, On time dependent queuing processes, Ann. Math. Statist. 31 (1960), 104–112. 10.1214/aoms/1177705991Search in Google Scholar
[11] D. G. Kendall, Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain, Ann. Math. Statist. 24 (1953), 338–354. 10.1214/aoms/1177728975Search in Google Scholar
[12] A. Y. Khinchin, Mathematical theory of stationary queues, Mat. Sbornik 39 (1932), 73–84. Search in Google Scholar
[13] L. Kleinrock, Queueing Systems. Vol. 1: Theory, John Wiley & Sons, New York, 1975. Search in Google Scholar
[14] V. S. Korolyuk and V. V. Korolyuk, Stochastic Models of Systems, Math. Appl. 469, Kluwer Academic, Dordrecht, 1999. 10.1007/978-94-011-4625-8Search in Google Scholar
[15] N. Limnios, A transient solution method for semi-Markov systems, Statist. Probab. Lett. 17 (1993), no. 3, 211–220. 10.1016/0167-7152(93)90169-JSearch in Google Scholar
[16] N. Limnios and G. Oprişan, Semi-Markov Processes and Reliability, Stat. Ind. Technol., Birkhäuser, Boston, 2001. 10.1007/978-1-4612-0161-8Search in Google Scholar
[17] F. Pollaczek, Über eine Aufgabe der Wahrscheinlichkeitstheorie. I, Math. Z. 32 (1930), no. 1, 64–100. 10.1007/BF01194620Search in Google Scholar
[18] T. L. Saaty, Elements of Queueing Theory, with Applications, McGraw-Hill, New York, 1961. Search in Google Scholar
[19] J. Sztrik, Finite source queueing systems and their applications, Technical Report, University of Debrecen, 2001. Search in Google Scholar
[20] L. Takács, A single-server queue with Poisson input, Oper. Res. 10 (1962), 388–394. 10.1287/opre.10.3.388Search in Google Scholar
[21] L. Takács, Introduction to the Theory of Queues, Univ. Texts Math. Sci., Oxford University, New York, 1962. Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
