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Diffusion limits for the queue length of jobs in multi-server open queueing networks

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A mathematical model of a multiserver open queueing network in heavy traffic is developed. This model is that of a multiserver computer system network in heavy traffic. A limit theorem for the length of the queue has been presented.
Rocznik
Strony
71--84
Opis fizyczny
Bibliogr. 42 poz.
Twórcy
autor
  • Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, 08663 Vilnius, Lithuania
  • Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, 08663 Vilnius, Lithuania
Bibliografia
  • [1] BILLINGSLEY P., Convergence of Probability Measures, Wiley, 1968.
  • [2] BOROVKOV A.A., Asymptotic methods in queueing theory, Nauka, Moscow 1980 (in Russian).
  • [3] BOROVKOV A.A., Stochastic processes in queueing theory, Nauka, Moscow 1972 (in Russian).
  • [4] BOXMA O., On a tandem queueing model with identical service times at both counters, I, II, Adv. Appl.Prob., 1979, 11 (3), 616–643, 644–659.
  • [5] BRAMSON M., DAI J.G., Heavy traffic limits for some queueing networks, Ann. Appl. Prob., 2001, 11 (1), 49–90.
  • [6] BRAMSON M., State space collapse with application to heavy traffic limits for multiclass queuing networks, Queuing Sys., 1998, 30 (1, 2), 89–140.
  • [7] CHEN H., MALDELBAUM A., Stochastic discrete flow networks. Diffusion approximations and bottlenecks, Ann. Appl. Prob., 1991, 19 (4), 1463–1519.
  • [8] CHEN H., YAO D.D., Fundamentals of Queueing Networks. Performance, Asymptotics and Optimization, Springer-Verlag, New York 2001.
  • [9] CHEN H., ZHANG H., A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service disciplines, Queing Syst., Theory Appl., 2000, 34 (1–4), 237–268.
  • [10] CHEN H., YE H.Q., Existence condition for the diffusion approximations of multiclass priority queueing networks, Queueing Syst., Theory Appl., 2001, 38, 435–470.
  • [11] CHEN H., Fluid approximations and stability of multiclass queuing networks: work-conserving disciplines, Ann. Applied Probability., 1995, 5 (3), 637–665.
  • [12] DAI J.G., Stability of open multiclass queuing networks via fluid models, F. Kelly, R. Williams (Eds.), Proc. IMA Workshop on Stochastic Networks, Springer-Verlag, New York 2000, 100–121.
  • [13] DAI J.G., On positive Harris recurrence of multiclass queueing networks. A unified approach via unified fluid limit models, Ann. Appl. Prob., 1995, 5 (1), 49–77.
  • [14] FLORES C., Diffusion approximations for computer communications networks, [In:] B. Gopinath (Ed.), Comp. Comm., Proc. Syrup. Appl. Math., American Mathematical Society, 1985, 83–124.
  • [15] GLYNN E.M., Diffusion approximations, [In:] D.P. Heyman, M.J. Sobel (Eds.), Handbooks in Operations Research and Management Science, Vol. 2. Stochastic Models, Elsevier, 1990.
  • [16] HARRISON J.M., LEMOINE A.J., A note on networks of infinite-server queues, J. Appl. Prob., 1981, 18, 561–567.
  • [17] HARRISON J.M., Brownian Motion and Stochastic Flow Processes, Wiley, New York 1985.
  • [18] HARRISON J.M., A broader view of Brownian networks, Ann. Appl. Prob., 2003, 13 (3), 1119–1150.
  • [19] HARRISON J.M., NQUYEN V., Brownian models of multiclass queueing networks: current status and open problems, Queueing Syst., Theory Appl., 1993, 13, 5–40.
  • [20] IGLEHART D., Weak convergence in queueing theory, Adv. Appl. Prob., 1973, 5, 570–594.
  • [21] IGLEHART D., WHITT W., Multiple channel queues in heavy traffic, Adv. Appl. Prob., 1970, 2, 150–177.
  • [22] IGLEHART D., WHITT W., Multiple channel queues in heavy traffic. II. Sequences, networks and batches, Adv. Appl. Prob., 1970, 2, 355–369.
  • [23] JOHNSON D.P., Diffusion Approximations for Optimal Filtering of Jump Processes and for Queueing Networks, PhD Diss., University of Wisconsin, 1983.
  • [24] KANG W.N., KELLY F.P., LEE N.H., WILLIAMS R.J., State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy, Ann. Appl. Probab., 2009, 19 (5), 1719–1780.
  • [25] KARPELEVICH F., KREININ A., Heavy traffic limits for multiphase queues, Amer. Math. Soc., Providence, Rhode Island, 1994.
  • [26] KINGMAN J., On queues in heavy traffic, J. R. Statist. Soc., 1962, 24, 383–392.
  • [27] KINGMAN J., The single server queue in heavy traffic, Proc. Camb. Phil. Soc., 1962, 57, 902–904.
  • [28] KOBYASHI H., Application of the diffusion approximation to queueing networks, J. ACM, 1974, 21, 316–328.
  • [29] KOLMOGOROV A.N., Foundations of the Theory of Probability, 2nd Ed., Chelsea Publishing Company, New York 1956.
  • [30] LEMOINE A.J., Network of queues. A survey of weak convergence results, Manage. Sci., 1978, 24, 1175–1193.
  • [31] MANDELBAUM A., STOLYAR A.L., Scheduling flexible servers with convex delay costs: heavy-traffic optimality of the generalized c rule, Oper. Res., 2004, 52, 6, 836–855.
  • [32] MINKEVIČIUS S., Weak convergence in multiphase queues, Liet. Mat. Rink., 1986, 26, 717–722 (in Russian).
  • [33] MINKEVIČIUS S., Complex transient phenomena in multiphase queueing systems, Liet. Mat. Rink., 1999, 39 (3), 343–356 (in Russian).
  • [34] MINKEVIČIUS S., On the law of the iterated logarithm in multiserver open queueing networks, Stochastics, 2013, 86 (1), 46–59.
  • [35] PROHOROV Y., Transient phenomena in queues, Liet. Mat. Rink., 1963, 3, 199–206 (in Russian).
  • [36] REIMAN M., Open queueing networks in heavy traffic, Math. Oper. Res., 1984, 9, 441–459.
  • [37] RYBKO A.N., STOLYAR A.L., Ergodicity of stochastic processes describing the operation of open queuing networks, Probl. Peredachi Inf., 1992, 28 (3), 3–26, Probl. Inf. Trans., 1992, 28 (3), 199–220.
  • [38] SAKALAUSKAS L., MINKEVIČIUS S., On the law of the iterated logarithm in open queueing networks, Eur. J. Oper. Res., 2000, 120, 632–640.
  • [39] STRASSEN V., An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie Verwandte Gebiete, 1964, 3, 211–226.
  • [40] WHITT W., Heavy traffic limit theorems for queues: a survey, [In:] Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, 1974, 98, 307–350.
  • [41] WHITT W., On the heavy-traffic limit theorem for GIG/ queues, Adv. Appl. Prob., 1982, 14, 171–190.
  • [42] WHITT W., Large fluctuations in a deterministic multiclass network of queues, Manage. Sci., 1993, 39, 1020–1028.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-f1c2396f-b4da-4984-a17b-d2f1482a5e33
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