nach oben

Queueing Systems

Erschienen in:

01.04.2015

A law of iterated logarithm for multiclass queues with preemptive priority service discipline

verfasst von: Yongjiang Guo, Yunan Liu

Erschienen in: Queueing Systems | Ausgabe 3-4/2015

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

A law of iterated logarithm (LIL) is established for a multiclass queueing model, having a preemptive priority service discipline, one server and \(K\) customer classes, with each class characterized by a renewal arrival process and i.i.d. service times. The LIL limits quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The LIL is established in three cases: underloaded, critically loaded, and overloaded, for five performance measures: queue length, workload, busy time, idle time, and number of departures. The proof of the LIL is based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. We conduct numerical examples to provide insights on these LIL results.

Vorheriger Artikel The stability of the deterministic Skorokhod problem is undecidable

Nächster Artikel Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

Bramson, M., Dai, J.G.: Heavy traffic limits for some queueing networks. Ann. Appl. Probab. 11(1), 49–90 (2001)CrossRef

Harrison, M.: A limit theorem for priority queues in heavy traffic. J. Appl. Probab. 10(4), 907–912 (1973)CrossRef

Whitt, W.: Weak convergence theorems for priority queues: preemptive-Resume discipline. J. Appl. Probab. 8(1), 74–94 (1971)CrossRef

Chen, H., Zhang, H.Q.: A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service disciplines. Queueing Syst. 34(1–4), 237–268 (2000)CrossRef

Peterson, W.P.: A heavy traffic limit theorem for networks of queues with multiple customer types. Math. Oper. Res. 16(1), 90–118 (1991)CrossRef

Chen, H.: Fluid approximations and stability of multiclass queueing networks I: work-conserving disciplines. Ann. Appl. Probab. 5(3), 637–665 (1995)CrossRef

Chen, H., Shen, X.: Strong approximations for multiclass feedforward queueing networks. Ann. Appl. Probab. 10(3), 828–876 (2000)CrossRef

Dai, J.G.: On the positive Harris recurrence for multiclass queueing networks: a unified approach via fluid limit models. Ann. Appl. Probab. 5(1), 49–77 (1995)CrossRef

Zhang, H.Q., Hsu, G.X.: Strong approximations for priority queues: head-of-the-line-first discipline. Queueing Syst. 10(3), 213–234 (1992)CrossRef

10.

Lévy, P.: Théorie de l’addition des variables aléatories. Gauthier-Villars, Paris (1937)

11.

Lévy, P.: Procesus stochastique et mouvement Brownien. Gauthier-Villars, Paris (1948)

12.

Csörgő, M., Révész, P.: How big are the increments of a Wiener process? Ann. Probab. 7(4), 731–737 (1979)CrossRef

13.

Csörgő, M., Révész, P.: Strong Approximations in Probability and Statistics. Academic Press, New York (1981)

14.

Iglehart, G.L.: Multiple channel queues in heavy traffic: IV. Law of the iterated logarithm. Z.Wahrscheinlichkeitstheorie verw. Geb. 17, 168–180 (1971)CrossRef

15.

Minkevičius, S.: On the law of the iterated logarithm in multiserver open queueing networks. Stoch. Int. J. Probab. Stoch. Process. 86(1), 46–59 (2014)

16.

Sakalauskas, L.L., Minkevičius, S.: On the law of the iterated logarithm in open queueing networks. Eur. J. Oper. Res. 120(3), 632–640 (2000)CrossRef

17.

Minkevičius, S., Steišūnas, S.: A law of the iterated logarithm for global values of waiting time in multiphase queues. Statist. Probab. Lett. 61(4), 359–371 (2003)CrossRef

18.

Chen, H., Yao, D.D.: Fundamentals of Queueing Networks. Springer-Verlag, New York (2001)CrossRef

19.

Chen, H., Mandelbaum, A.: Hierarchical modeling of stochastic network, part II: strong approximations. In: Yao, D.D. (Eds.) Stochastic Modeling and Analysis of Manufacturing Systems, 107–131 (1994)

20.

Strassen, V.: An invariance principle for the law of the iterated logarith. Z. Wahrscheinlichkeitstheorie verw. Geb. 3(3), 211–226 (1964)CrossRef

21.

Whitt, W.: Stochastic-Process Limits. Springer, Berlin (2002)

22.

Glynn, P.W., Whitt, W.: A central-limit-theorem version of \(L=\lambda W\). Queueing Syst. 1(2), 191–215 (1986)CrossRef

23.

Glynn, P.W., Whitt, W.: Sufficient conditions for functional limit theorem versions of \(L=\lambda W\). Queueing Syst. 1(3), 279–287 (1987)CrossRef

24.

Glynn, P.W., Whitt, W.: An LIL version of \(L=\lambda W\). Math. Oper. Res. 13(4), 693–710 (1988)CrossRef

25.

Horváth, L.: Strong approximation of renewal processes. Stoch. Process. Appl. 18(1), 127–138 (1984)CrossRef

26.

Horváth, L.: Strong approximation of extended renewal processes. Ann. Probab. 12(4), 1149–1166 (1984)CrossRef

27.

Csörgő, M., Deheuvels, P., Horváth, L.: An approximation of stopped sums with applications in queueing theory. Adv. Appl. Probab. 19(3), 674–690 (1987)CrossRef

28.

Csörgő, M., Horváth, L.: Weighted Approximations in Probability and Statistics. Wiley, New York (1993)

29.

Glynn, P.W., Whitt, W.: A new view of the heavy-traffic limit for infinite-server queues. Adv. Appl. Probab. 23(1), 188–209 (1991)CrossRef

30.

Zhang, H.Q., Hsu, G.X., Wang, R.X.: Strong approximations for multiple channels in heavy traffic. J. Appl. Probab. 27(3), 658–670 (1990)CrossRef

31.

Glynn, P.W., Whitt, W.: Departures from many queues in series. Ann. Appl. Probab. 1(4), 546–572 (1991)CrossRef

32.

Horváth, L.: Strong approximations of open oueueing networks. Math. Oper. Res. 17(2), 487–508 (1992)CrossRef

33.

Zhang, H.Q.: Strong approximations of irreducible closed queueing networks. Adv. Appl. Probab. 29(2), 498–522 (1997)CrossRef

34.

Mandelbaum, A., Massey, W.A.: Strong approximations for time-dependent queues. Math. Oper. Res. 20(1), 33–64 (1995)CrossRef

35.

Mandelbaum, A., Massey, W.A., Reiman, M.: Strong approximations for Markovian service networks. Queueing Syst. 30, 149–201 (1998)CrossRef

36.

Dai, J.G.: Stability of Fluid and Stochastic Processing Networks, vol. 9. MaPhySto Miscellanea Publication, Aarhus, Denmark (1999)

37.

Chen, H.: Rate of convergence of the fluid approximation for generalized Jackson networks. J. Appl. Probab. 33(3), 804–814 (1996)CrossRef

Titel: A law of iterated logarithm for multiclass queues with preemptive priority service discipline
verfasst von: Yongjiang Guo
Yunan Liu
Publikationsdatum: 01.04.2015
Verlag: Springer US
Erschienen in: Queueing Systems / Ausgabe 3-4/2015
Print ISSN: 0257-0130
Elektronische ISSN: 1572-9443
DOI: https://doi.org/10.1007/s11134-014-9419-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 3-4/2015

Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

Stochastic grey-box modeling of queueing systems: fitting birth-and-death processes to data

The stability of the deterministic Skorokhod problem is undecidable

Optimal design of measurements on queueing systems

Optimal service rate perturbations of many server queues in heavy traffic

Premium Partner