Far Eastern Mathematical Journal

To content of the issue


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.

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

Download the article (PDF-file)

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.

To content of the issue