Multichannel queueing systems with regenerative input flow

L. G. Afanasyeva   ; A. V. Tkachenko

Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 2, pp. 210-234

Résumé

We consider the multichannel queueing system with nonidentical servers and regenerative input flow. The necessary and sufficient condition for ergodicity is established, and functional limit theorems for high and ultra-high load are proved. As a corollary, the ergodicity condition for queues with unreliable servers is obtained. Suggested approaches are used to prove the ergodic theorem for systems with limitations. We also consider the hierarchical networks of queueing systems.

Détail

Keywords: multichannel queueing system; regenerative flow; ergodicity; stochastic boundedness.

Bibliographie

[1] Afanaseva L. G., Martynov A. V., “Ob ergodicheskikh svoistvakh sistem massovogo obsluzhivaniya s ogranichennym vremenem prebyvaniya”, Teoriya veroyatn. i ee primen., 14:1 (1969), 102–112 | MR | Zbl

[2] Afanaseva L. G., “Sistemy massovogo obsluzhivaniya s tsiklicheskimi upravlyayuschimi protsessami”, Kibernetika i sistemnyi analiz, 2005, no. 1, 54–69 | MR

[3] Afanaseva L. G., Bashtova E. E., “Predelnye teoremy dlya sistem massovogo obsluzhivaniya v usloviyakh vysokoi zagruzki”, Sovremennye problemy matematiki i mekhaniki, IV (2009), 40–54

[4] Afanaseva L. G., Bulinskaya E. V., Sluchainye protsessy v teorii massovogo obsluzhivaniya i upravleniya zapasami, Izd-vo Mosk. un-ta, M., 1980

[5] Afanaseva L. G., Rudenko I. V., “Sistemy obsluzhivaniya $ G|G|\infty$ i ikh prilozheniya k analizu transportnykh modelei”, Teoriya veroyatn. i ee primen., 57:3 (2012), 427–452 | DOI | MR

[6] Belorusov T. N., “Ergodichnost mnogokanalnoi sistemy obsluzhivaniya s vozmozhnostyu neprisoedineniya k ocheredi”, Teoriya veroyatn. i ee primen., 56:1 (2011), 145–152 | DOI | MR | Zbl

[7] Borovkov A. A., Veroyatnostnye protsessy v teorii massovogo obsluzhivaniya, Nauka, M., 1972, 367 pp.

[8] Borovkov A. A., Asimptoticheskie metody v teorii massovogo obsluzhivaniya, Nauka, M., 1980, 381 pp.

[9] Borovkov A. A., Ergodichnost i ustoichivost sluchainykh protsessov, Editorial URSS, M., 1999, 440 pp.

[10] Borovkov A. A., “Nekotorye predelnye teoremy teorii massovogo obsluzhivaniya. II: Mnogokanalnye sistemy”, Teoriya veroyatn. i ee primen., 10:3 (1965), 409–436

[11] Gnedenko B. V., Kovalenko I. N., Vvedenie v teoriyu massovogo obsluzhivaniya, Nauka, M., 1966, 431 pp.

[12] Koks D. R., Smit V. L., Teoriya vosstanovleniya, Sovetskoe radio, M., 1967, 299 pp.

[13] Maryanovich T. P., “Odnolineinaya sistema massovogo obsluzhivaniya s nenadezhnym priborom”, Ukr. matem. zhurn., 14:4 (1962), 417–422 | MR

[14] Rudenko I. V., “Dvukhfazovaya sistema obsluzhivaniya s nenadezhnymi priborami”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2012, no. 4, 4–12

[15] Afanasyeva L. G., Bashtova E. E., Bulinskaya E. V., “Limit theorems for semi-Markov queues and their applications”, Comm. Statist. Simulation Comput., 41:6 (2012), 688–709 | DOI | MR | Zbl

[16] Asmussen S., “Semi-Markov queues with heavy tails”, Semi-Markov Models and Application, eds. J. Janssen et al., Kluwer, Dordrecht, 1999, 269–284 | DOI | MR | Zbl

[17] Asmussen S., Applied Probability and Queues, Wiley, Chichester, 1987, 318 pp. | MR | Zbl

[18] Falin G., “An $ M|G|1$ retrial queue with an unreliable server and general repair times”, Perfomance Evaluation, 67:7 (2010), 569–582 | DOI

[19] Foss S., Kalashnikov V., “Regeneration and Renovation in Queues”, Queueing Syst., 8:3 (1991), 211–223 | DOI | MR

[20] Gaver D. P., Jr., “A waiting line with interrupted service, including priorities”, J. Roy. Statist. Soc., Ser. B, 24:1962 (1962), 73–96 | MR

[21] Grandell D. P., “Doubly stochastic poisson process”, Lecture Notes in Math., 529, 1976, 1–234 | DOI | MR

[22] Iglehart D., Whitt W., “Multiple channel queues in heavy traffic, I”, Adv. Appl. Probab., 2 (1970), 150–177 | DOI | MR | Zbl

[23] Kiefer J., Wolfowitz J., “On the theory of queues with many servers”, Trans. Amer. Math. Soc., 78 (1955), 1–18 | DOI | MR | Zbl

[24] Loynes R., “The stability of a queue with nonindependent inter-arrival and service times”, Proc. Cambridge Philos. Soc., 58 (1962), 497–520 | DOI | MR | Zbl

[25] Neuts M. F., Lucantoni D. M., “A Markovian queue with $ N$ servers subject to breakdowns and repairs”, Management Sci., 25:9 (1979), 849–861 | DOI | MR

[26] Morozov E., Stochastic boundness of some queueing processes, Preprint No R-95-2022, Dept. Math. and Computer Sci., Aalborg Univ., Aalborg, 1995, 28 pp.

[27] Morozov E., “The stability of a non-homogeneous queueing system with regenerative input”, J. Math. Sci. (N.Y.), 83:3 (1997), 407–421 | DOI | MR | Zbl

[28] Morozov E., “The tightness in the ergodic analysis of regenerative queueing processes”, Queueing Systems Theory Appl., 27:1–2 (1997), 179–203 | DOI | MR | Zbl

[29] Morozov E., “Weak regeneration in modeling of queueing processes”, Queueing Syst., 46:3–4 (2004), 295–315 | DOI | MR | Zbl

[30] Sadowsky J. S., Szpankowski W., “The probability of large queue lengths and waiting times in a heterogeneous multiserver queue. I: Tight limits”, Adv. Appl. Probab., 27:2 (1995), 532–566 | DOI | MR | Zbl

[31] Smith W. L., “Regenerative Stochastic Processes”, Proc. Roy. Soc. Lond., 232:1188 (1955), 6–31 | DOI | MR | Zbl

[32] Thorisson H., “A complete coupling proof of Blackwell's renewal theorem”, Stochasic Process. Appl., 26 (1987), 87–97 | DOI | MR | Zbl