Fluid limits for the queue length of jobs in multiserver open queueing networks

Saulius Minkevičius

RAIRO - Operations Research - Recherche Opérationnelle (2014)

  • Volume: 48, Issue: 3, page 349-363
  • ISSN: 0399-0559

Abstract

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The object of this research in the queueing theory is a theorem about the Strong-Law-of-Large-Numbers (SLLN) under the conditions of heavy traffic in a multiserver open queueing network. SLLN is known as a fluid limit or fluid approximation. In this work, we prove that the long-term average rate of growth of the queue length process of a multiserver open queueing network under heavy traffic strongly converges to a particular vector of rates. SLLN is proved for the values of an important probabilistic characteristic of the multiserver open queueing network investigated as well as the queue length of jobs.

How to cite

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Minkevičius, Saulius. "Fluid limits for the queue length of jobs in multiserver open queueing networks." RAIRO - Operations Research - Recherche Opérationnelle 48.3 (2014): 349-363. <http://eudml.org/doc/275089>.

@article{Minkevičius2014,
abstract = {The object of this research in the queueing theory is a theorem about the Strong-Law-of-Large-Numbers (SLLN) under the conditions of heavy traffic in a multiserver open queueing network. SLLN is known as a fluid limit or fluid approximation. In this work, we prove that the long-term average rate of growth of the queue length process of a multiserver open queueing network under heavy traffic strongly converges to a particular vector of rates. SLLN is proved for the values of an important probabilistic characteristic of the multiserver open queueing network investigated as well as the queue length of jobs.},
author = {Minkevičius, Saulius},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {mathematical models of information systems; performance evaluation; queueing theory; multiserver open queueing network; heavy traffic; limit theorem; queue length of jobs; queue-length of jobs},
language = {eng},
number = {3},
pages = {349-363},
publisher = {EDP-Sciences},
title = {Fluid limits for the queue length of jobs in multiserver open queueing networks},
url = {http://eudml.org/doc/275089},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Minkevičius, Saulius
TI - Fluid limits for the queue length of jobs in multiserver open queueing networks
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 3
SP - 349
EP - 363
AB - The object of this research in the queueing theory is a theorem about the Strong-Law-of-Large-Numbers (SLLN) under the conditions of heavy traffic in a multiserver open queueing network. SLLN is known as a fluid limit or fluid approximation. In this work, we prove that the long-term average rate of growth of the queue length process of a multiserver open queueing network under heavy traffic strongly converges to a particular vector of rates. SLLN is proved for the values of an important probabilistic characteristic of the multiserver open queueing network investigated as well as the queue length of jobs.
LA - eng
KW - mathematical models of information systems; performance evaluation; queueing theory; multiserver open queueing network; heavy traffic; limit theorem; queue length of jobs; queue-length of jobs
UR - http://eudml.org/doc/275089
ER -

References

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  1. [1] P. Billingsley, Convergence of Probability Measures. Wiley, New York (1968). Zbl0944.60003MR233396
  2. [2] A.A. Borovkov, Weak convergence of functionals of random sequences and processes defined on the whole axis. Proc. Stecklov Math. Inst.128 (1972) 41–65. Zbl0287.60013MR319237
  3. [3] A.A. Borovkov, Stochastic Processes in Queueing Theory. Springer, Berlin (1976). Zbl0319.60057MR391297
  4. [4] A.A. Borovkov, Asymptotic Methods in Queueing Theory. Wiley, New York (1984). Zbl0544.60085MR745620
  5. [5] A.A. Borovkov, Limit theorems for queueing networks. Theory Prob. Appl.31 (1986) 413–427. Zbl0617.60089MR866868
  6. [6] H. Chen and A. Mandelbaum, Stochastic discrete flow networks: Diffusion approximations and bottlenecks. The Annals of Probability19 (1991) 1463–1519. Zbl0757.60094MR1127712
  7. [7] C. Flores, Diffusion approximations for computer communications networks. in Computer Communications, Proc. Syrup. Appl. Math., edited by B. Gopinath. American Mathematical Society (1985) 83–124. Zbl0581.90027MR807815
  8. [8] P.W. Glynn, Diffusion approximations. in Handbooks in Operations Research and Management Science, edited by D.P. Heyman and M.J. Sobel, Vol. 2 of Stochastic Models. North-Holland (1990). Zbl0703.60072MR1100747
  9. [9] P.W. Glynn and W. Whitt, A new view of the heavy-traffic limit theorems for infinite-server queues. Adv. Appl. Probab.23 (1991) 188–209. Zbl0716.60105MR1091098
  10. [10] B. Grigelionis and R. Mikulevičius, Diffusion approximation in queueing theory. Fundamentals of Teletraffic Theory. Proc. Third Int. Seminar on Teletraffic Theory (1984) 147–158. 
  11. [11] J.M. Harrison, The heavy traffic approximation for single server queues in series. Adv. Appl. Probab.10 (1973) 613–629. Zbl0287.60102MR359066
  12. [12] J.M. Harrison, The diffusion approximation for tandem queues in heavy traffic. Adv. Appl. Probab.10 (1978) 886–905. Zbl0387.60090MR509222
  13. [13] J.M. Harrison and A.J. Lemoine, A note on networks of infinite-server queues. J. Appl. Probab.18 (1981) 561–567. Zbl0459.60081MR611802
  14. [14] J.M. Harrison and M.I. Reiman, On the distribution of multidimensional reflected Brownian motion. SIAM J. Appl. Math.41 (1981) 345–361. Zbl0464.60081MR628959
  15. [15] J.M. Harrison and R.J. Williams, Brownian models of open queueing networks with homogeneous customer populations. Stochastics22 (1987) 77–115. Zbl0632.60095MR912049
  16. [16] D.L. Iglehart, Multiple channel queues in heavy traffic. IV. Law of the iterated logarithm. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 17 (1971) 168–180. Zbl0203.50402MR312604
  17. [17] D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic I. Adv. Appl. Probab.2 (1970) 150–175. Zbl0218.60098MR266331
  18. [18] D.L. Iglehart and W. Whitt, Multiple channel queues in heavy traffic II: Sequences, networks and batches. Adv. Appl. Probab.2 (1970) 355–364. Zbl0206.22503MR282443
  19. [19] D.P. Johnson, Diffusion Approximations for Optimal Filtering of Jump Processes and for Queueing Networks. Ph.D. dissertation, University of Wisconsin (1983). MR2632848
  20. [20] F.I. Karpelevitch and A.Ya. Kreinin, Joint distributions in Poissonian tandem queues. Queueing Systems12 (1992) 274–286. Zbl0811.60079MR1200867
  21. [21] F.P. Kelly, An asymptotic analysis of blocking. Modelling and Performance Evaluation Methodology. Springer, Berlin (1984) 3–20. Zbl0599.60082MR893651
  22. [22] G.P. Klimov, Several solved and unsolved problems of the service by queues in series (in Russian). Izv. AN USSR, Ser. Tech. Kibern. 6 (1970) 88–92. 
  23. [23] E.V. Krichagina, R.Sh. Liptzer and A.A. Pukhalsky, The diffusion approximation for queues with input flow, depending on a queue state and general service. Theory Prob. Appl.33 (1988) 124–135. Zbl0637.60101
  24. [24] Ya.A. Kogan and A.A. Pukhalsky, Tandem queues with finite intermediate waiting room and blocking in heavy traffic. Prob. Control Int. Theory17 (1988) 3–13. Zbl0643.60080MR935696
  25. [25] A.J. Lemoine, Network of queues – A survey of weak convergence results. Management Science24 (1978) 1175–1193. Zbl0396.60088MR652285
  26. [26] R.Sh. Liptzer and A.N. Shiryaev, Theory of Martingales. Kluwer, Boston (1989). 
  27. [27] S. Minkevičius, On the global values of the queue length in open queueing networks, Int. J. Comput. Math. (2009) 1029–0265. Zbl1191.60103
  28. [28] S. Minkevičius, On the law of the iterated logarithm in multiserver open queueing networks, Stochastics, 2013 (accepted). Zbl1306.60143
  29. [29] S. Minkevičius and G. Kulvietis, Application of the law of the iterated logarithm in open queueing networks. WSEAS Transactions on Systems6 (2007) 643–651. Zbl0981.60093
  30. [30] Yu.V. Prohorov, Convergence of random processes and limit theorems in probability theory. Theory Prob. Appl.1 (1956) 157–214. Zbl0075.29001MR84896
  31. [31] M.I. Reiman, Open queueing networks in heavy traffic. Math. Oper. Res.9 (1984) 441–458. Zbl0549.90043MR757317
  32. [32] M.I. Reiman, A multiclass feedback queue in heavy traffic. Adv. Appl. Probab.20 (1988) 179–207. Zbl0647.60100MR932539
  33. [33] M.I. Reiman and B. Simon, A network of priority queues in heavy traffic: one bottleneck station. Queueing Systems6 (1990) 33–58. Zbl0818.60081MR1053667
  34. [34] L. Sakalauskas and S. Minkevičius, On the law of the iterated logarithm in open queueing networks. Eur. J. Oper. Res.120 (2000) 632–640. Zbl0981.60093MR1781057
  35. [35] A.V. Skorohod, Studies in the Theory of Random Processes. Addison-Wesley, New York (1965). Zbl0146.37701MR185620
  36. [36] V. Strassen, An invariance principle for the law of the iterated logarithm. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete3 (1964) 211–226. Zbl0132.12903MR175194
  37. [37] W. Szczotka and F.P. Kelly, Asymptotic stationarity of queues in series and the heavy traffic approximation. The Annals of Probability18 (1990) 1232–1248. Zbl0726.60092MR1062067
  38. [38] W. Whitt, Weak convergence theorems for priority queues: preemptive resume discipline. J. Appl. Probab.8 (1971) 79–94. Zbl0215.53801MR307389
  39. [39] W. Whitt, Heavy traffic limit theorems for queues: a survey. in Lecture Notes in Economics and Mathematical Systems, Vol. 98. Springer-Verlag, Berlin, Heidelberg, New York (1971) 307–350. Zbl0295.60081MR394935
  40. [40] W. Whitt, On the heavy-traffic limit theorem for GI/G/∞ queues. Adv. Appl. Probab.14 (1982) 171–190. Zbl0479.60090MR644013

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