Geodesic
Stability conditions for queueing systems with regenerative flow of interruptions
Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 4, pp. 623-653 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper is focused on the multichannel queueing system with heterogeneous servers, regenerative input flow, and a regenerative process of interruptions. Two service disciplines are studied: preemptive-repeat-different service discipline and preemptive resume service discipline. We consider discrete as well as continuous-time cases. We introduce an auxiliary service flow, which does not depend on the input flow, and construct the common points of regeneration for these two flows. Using such a synchronization method, we establish necessary and sufficient conditions for stability of the system under some additional assumptions. Additionally, under weaker assumptions, we also find the conditions needed for the queue length process to be stochastically bounded.
Keywords: multichannel queueing system, stability, priority, regeneration, synchronization.
Mots-clés : interruption 
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L. G. Afanas'eva; A. W. Tkachenko. Stability conditions for queueing systems with regenerative flow of interruptions. Teoriâ veroâtnostej i ee primeneniâ, Tome 63 (2018) no. 4, pp. 623-653. http://geodesic.mathdoc.fr/item/TVP_2018_63_4_a0/

[1] H. White, L. S. Christie, “Queuing with preemptive priorities or with breakdown”, Operations Res., 6:1 (1958), 79–95 | DOI | MR

[2] B. Avi-Itzhak, P. Naor, “Some queuing problems with the service station subject to breakdown”, Operations Res., 11:3 (1963), 303–320 | DOI | MR | Zbl

[3] K. Thiruvengadam, “Queuing with breakdowns”, Operations Res., 11:1 (1963), 62–71 | DOI | MR | Zbl

[4] D. P. Gaver Jr., “A waiting line with interrupted service, including priorities”, J. Roy. Statist. Soc. Ser. B, 24 (1962), 73–90 | MR | Zbl

[5] D. Fiems, H. Bruneel, “Discrete-time queueing systems with {Markovian} preemptive vacations”, Math. Comput. Modelling, 57:3-4 (2013), 782–792 | DOI | MR | Zbl

[6] A. Krishnamoorthy, P. K. Pramod, S. R. Chakravarthy, “Queues with interruptions: a survey”, TOP, 22:1 (2014), 290–320 | DOI | MR | Zbl

[7] A. V. Pechinkin, I. A. Sokolov, V. V. Chaplygin, “Mnogolineinaya sistema massovogo obsluzhivaniya s gruppovym otkazom priborov”, Inform. i ee primen., 3:3 (2009), 4–15

[8] L. G. Afanasyeva, E. E. Bashtova, “Coupling method for asymptotic analysis of queues with regenerative input and unreliable server”, Queueing Syst., 76:2 (2014), 125–147 | DOI | MR | Zbl

[9] A. V. Tkachenko, “Multichannel queueing system in a random environment”, Moscow Univ. Math. Bull., 69:1 (2014), 37–40 | DOI | MR | Zbl

[10] E. Morozov, D. Fiems, H. Bruneel, “Stability analysis of multiserver discrete-time queueing systems with renewal-type server interruptions”, Performance Evaluation, 68:12 (2011), 1261–1275 | DOI

[11] J. G. Dai, “On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models”, Ann. Appl. Probab., 5:1 (1995), 49–77 | DOI | MR | Zbl

[12] Hong Chen, “Fluid approximations and stability of multiclass queueing networks: work-conserving disciplines”, Ann. Appl. Probab., 5:3 (1995), 637–665 | DOI | MR | Zbl

[13] Hong Chen, D. D. Yao, Fundamentals of queueing networks. Performance, asymptotics, and optimization, Appl. Math. (N. Y.), Springer-Verlag, New York, 2001, xviii+405 pp. | DOI | MR | Zbl

[14] S. Foss, N. Konstantopoulos, “An overview of some stochastic stability methods”, J. Oper. Res. Soc. Japan, 47:4 (2004), 275–303 | DOI | MR | Zbl

[15] H. Thorisson, “A complete coupling proof of Blackwell's renewal theorem”, Stochastic Process. Appl., 26:1 (1987), 87–97 | DOI | MR | Zbl

[16] A. A. Borovkov, Stochastic processes in queueing theory, Appl. Math., 4, Springer-Verlag, New York–Berlin, 1976, xi+280 pp. | MR | MR | Zbl | Zbl

[17] S. Asmussen, Applied probability and queues, Appl. Math. (N. Y.), 51, 2nd ed., Springer-Verlag, New York, 2003, xii+438 pp. | MR | Zbl

[18] R. M. Loynes, “The stability of a queue with non-independent inter-arrival and service times”, Proc. Cambridge Philos. Soc., 58:3 (1962), 497–520 | DOI | MR | Zbl

[19] W. Feller, An introduction to probability theory and its applications, v. 1, 2, John Wiley Sons, Inc., New York–London–Sydney, 1968, 1971, xviii+509 pp., xxiv+669 pp. | MR | MR | Zbl

[20] T. N. Belorusov, “Ergodicity of a multichannel queueing system with balking”, Theory Probab. Appl., 56:1 (2012), 120–126 | DOI | DOI | MR | Zbl

[21] L. G. Afanasyeva, A. V. Tkachenko, “Multichannel queueing systems with regenerative input flow”, Theory Probab. Appl., 58:2 (2014), 174–192 | DOI | DOI | MR | Zbl

[22] W. L. Smith, “Regenerative stochastic processes”, Proc. Roy. Soc. London. Ser. A, 232:1188 (1955), 6–31 | DOI | MR | Zbl

[23] S. Meyn, R. L. Tweedie, Markov chains and stochastic stability, 2nd ed., Cambridge Univ. Press, Cambridge, 2009, xxviii+594 pp. | DOI | MR | Zbl