Non-stationary distribution of customers number in markov queueing systems |

N. I. Golovko, V. V. Katrakhov |

2004, issue 2, P. 211–217 |

Abstract |

There are many investigation results devoted to the analysis of the customers distribution in markov non-stationary queueing systems. For the first time non-stationary distribution of customers number in markov non-stationary queueing system $M/M/1$ with constant intensity of input stream $\lambda$ and service $\mu$ it has been received in Clark's work. However it is impossible to apply a method offered by Clark to the analysis of markov non-stationary queueing systems from wide class, for example, queueing systems $M(t)/M(t)/1$ with variables intensities of input stream $\lambda (t)$ and service $\mu (t)$, or queueing systems with a various configuration, for example, queueing system with the final store and so on. In this work for calculation of customers number probabilities nonstationary distributions in markov non-stationary queueing systems various configuration with variables intensities of input stream $lambda (t)$ and service $\mu (t)$ the method of making functions is offered with a variation of the right part which is shown on examples of queueing systems $M(t)/M(t)/1$ and $M(t)/M(t)/1/N_0$ with infinite and final stores accordingly. |

queueing system theory, input a Poisson stream, exponential service, the infinite and final store, single server, non-stationary distribution of customers numberKeywords: |

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## References |

[1] N. I. Golovko, I. A. Korotaev, “Raschet xarakteristik nestacionarnyx sistem massovogo obsluzhivaniya”, Avtomatika i telemexanika, 1991, № 2, 97–102. [2] A. N. Dudin, “Ob obsluzhivayushhej sisteme s peremennym rezhimom raboty”, Avtomatika i vych. texnika, 1985, № 2, 27–29. [3] A. B. Clark, “A Waiting line process of Markov type”, Annals of Mathematical Statistics, 27:2 (1965), 452–459. [4] T. Rolski, “Queues with non-stationary input stream: Ross's conjecture”, Adv. Appl. Probab., 13:3 (1991), 603–618. [5] R. Syski, “Further comments on the solution of the M/M/1 queue”, Adv. Appl. Probab., 20:3 (1988), 693. [6] A. A. Borovkov, Veroyatnostnye processy v teorii massovogo obsluzhivaniya, Nauka, M., 1971, 368 s. [7] V. V. Katraxov, D. E. Ryzhkov, O funkcional'no-analiticheskom metode v teorii massovogo obsluzhivaniya, Preprint № 10 IPM DVO RAN, Dal'nauka, Vladivostok, 2004, 64 s. [8] V. V. Katraxov, D. E. Ryzhkov, O sisteme massovogo obsluzhivaniya s konechnym nakopitelem, Preprint № 11 IPM DVO RAN, Dal'nauka, Vladivostok, 2004, 12 s. [9] V. Feller, Vvedenie v teoriyu veroyatnostej i ee prilozheniya, t. 2, Mir, M., 1984, 751 s. [10] N. I. Golovko, V. V. Katpaxov, E. A. Svitelik, Stacionarnoe raspredelenie chisla zayavok v sistemax obsluzhivaniya s beskonechnym nakopitelem pri diffuzionnoj intensivnosti vxodnogo potoka, Preprint № 21 IPM DVO RAN, Dal'nauka, Vladivostok, 2004, 28 s. |