Abstract
In previous chapters, we were interested mainly in the limiting or pre-limiting the behaviour of the probabilities of the states of queueing models. However, there exists another class of problems associated with non-stationary behaviour. Namely, imagine that we are seeking the time of the first accessing of a queue-length Q(•) of some fixed level Q*. If we study a redundant system, we may want to know what is the first break-down time. When studying a storage problem, it is interesting to estimate the first time of zero stock or the first time of overflow. In all such situations, the first-occurrence time is actually an r.v. and the problem consists of finding either the d.f. of this r.v., or its reasonable bounds.
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Buslenko, N., Kalashnikov, V., and Kovalenko, I. (1973) Lectures on complex systems theory. Sov. Radio, Moscow (in Russian).
Buslenko, N., Kalashnikov, V., and Kovalenko, I. (1973) Lectures on complex systems theory. Sov. Radio, Moscow (in Russian).
Disney, R., and Kiessler, P. (1987) Traffic Processes in Queueing Networks. The Johns Hopkins Univ. Press, Baltimore and London.
Korol’uk, V. (1965) Occupation time of a fixed set of states by a semi-Markov process. Ukrainaian Math. J., No. 3 (in Russian).
Buslenko, N., Kalashnikov, V., and Kovalenko, I. (1973) Lectures on complex systems theory. Sov. Radio, Moscow (in Russian).
LaSalle, J., and Lefschetz, S. (1961) Stability by Lyapunov’s Direct Method with Applications. Academic Press, New York.
Kalashnikov, V. (1978) Qualitative Analysis of the Behaviour of Complex Systems by the Method of Test Functions. Nauka, Moscow (in Russian).
Kalashnikov, V. (1985) The rate of convergence of Erlang distribution to the degenerate one. LN in Math., 1155, 95–101, Springer-Verlag, Berlin.
Kalashnikov, V. (1989) Analytical and simulation estimates of reliability for regenerative models. Syst. Anal. Model. Simul., 6, 833–851.
Kalashnikov, V., and Vsekhsviatskii, S. (1985) Metric estimates of the first occurrence time in regenerative processes. LN in Math., 1155, 102–130. Springer-Verlag, Berlin.
Soloviev, A. (1971) Asymptotic behaviour of the time of first occurrence of a rare event. Engineering Cybernetics, 9, 1038–1048.
Kalashnikov, V. (1990a) Upper and lower bounds for geometric convolutions. Tech. Report No. 141, October 1990, Dept. of Statistics, UCSB.
Kalashnikov, V. (1993) Two-side estimates of geometric convolutions. LN in Math., 1546, 76–88, Springer-Verlag, Berlin.
Brown, M. (1990) Error bounds for exponential approximation of geometric convolutions. The Annals of Probabil., 18, 1388–1402.
Kalashnikov, V. (1990) Regenerative queueing processes and their qualitative and quantitative analysis. Queueing Systems, 6, 113–136.
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© 1994 Springer Science+Business Media Dordrecht
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Kalashnikov, V.V. (1994). First-Occurrence Events. In: Mathematical Methods in Queuing Theory. Mathematics and Its Applications, vol 271. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2197-4_11
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DOI: https://doi.org/10.1007/978-94-017-2197-4_11
Publisher Name: Springer, Dordrecht
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