Acta Applicandae Mathematicae

, Volume 100, Issue 3, pp 201–226

Large Closed Queueing Networks in Semi-Markov Environment and Their Application

Article

Abstract

The paper studies closed queueing networks containing a server station and k client stations. The server station is an infinite server queueing system, and client stations are single-server queueing systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by a strictly stationary and ergodic sequence of random variables. The total number of units in the network is N. The expected times between departures in client stations are (Nμj)−1. After a service completion in the server station, a unit is transmitted to the jth client station with probability pj (j=1,2,…,k), and being processed in the jth client station, the unit returns to the server station. The network is assumed to be in a semi-Markov environment. A semi-Markov environment is defined by a finite or countable infinite Markov chain and by sequences of independent and identically distributed random variables. Then the routing probabilities pj (j=1,2,…,k) and transmission rates (which are expressed via parameters of the network) depend on a Markov state of the environment. The paper studies the queue-length processes in client stations of this network and is aimed to the analysis of performance measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex telecommunication networks, and the obtained results are expected to lead to the solutions to many practical problems of this area of research.

Keywords

Closed queueing network Random environment Martingales and semimartingales Skorokhod reflection principle 

Mathematics Subject Classification (2000)

60K25 60K30 60H30 60H35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abramov, V.M.: A large closed queueing network with autonomous service and bottleneck. Queueing Syst. 35, 23–54 (2000) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Abramov, V.M.: Some results for large closed queueing networks with and without bottleneck: Up- and down-crossings approach. Queueing Syst. 38, 149–184 (2001) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Abramov, V.M.: A large closed queueing network containing two types of node and multiple customers classes: One bottleneck station. Queueing Syst. 48, 45–73 (2004) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Abramov, V.M.: The stability of join-the-shortest-queue models with general input and output processes. arXiv: math/PR 0505040 (2005) Google Scholar
  5. 5.
    Abramov, V.M.: The effective bandwidth problem revisited. arXiv: math/PR 0604182 (2006) Google Scholar
  6. 6.
    Abramov, V.M.: Confidence intervals associated with performance analysis of symmetric large closed client/server computer networks. Reliab. Theory Appl. 2(2), 35–42 (2007) MathSciNetGoogle Scholar
  7. 7.
    Abramov, V.M.: Further analysis of confidence intervals for large client/server computer networks. Reliab. Theory Appl. (2007, to appear) Google Scholar
  8. 8.
    Anulova, S.V., Liptser, R.S.: Diffusion approximation for processes with normal reflection. Theory Probab. Appl. 35, 413–423 (1990) MathSciNetGoogle Scholar
  9. 9.
    Baccelli, F., Makovsky, A.M.: Stability and bounds for single-server queue in a random environment. Stoch. Models 2, 281–292 (1986) MATHCrossRefGoogle Scholar
  10. 10.
    Berger, A., Bregman, L., Kogan, Y.: Bottleneck analysis in multiclass closed queueing networks and its application. Queueing Syst. 31, 217–237 (1999) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Borovkov, A.A.: Stochastic Processes in Queueing Theory. Springer, Berlin (1976) MATHGoogle Scholar
  12. 12.
    Borovkov, A.A.: Asymptotic Methods in Queueing Theory. Wiley, New York (1984) MATHGoogle Scholar
  13. 13.
    Boxma, O.J., Kurkova, I.A.: The M/M/1 queue in heavy-tailed random environment. Stat. Neerl. 54, 221–236 (2000) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dellacherie, C.: Capacités et Processus Stochastiques. Springer, Berlin (1972) MATHGoogle Scholar
  15. 15.
    D’Auria, B.: M/M/∞ queues in quasi-Markovian random environment. arXiv: math/PR 0701842 (2007) Google Scholar
  16. 16.
    Fricker, C.: Etude d’une file GI/G/1 á service autonome (avec vacances du serveur). Adv. Appl. Probab. 18, 283–286 (1986) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fricker, C.: Note sur un modele de file GI/G/1 á service autonomé (avec vacances du serveur). Adv. Appl. Probab. 19, 289–291 (1987) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Gelenbe, E., Iasnogorodski, R.: A queue with server of walking type (autonomous service). Ann. Inst. H. Poincare 16, 63–73 (1980) MathSciNetMATHGoogle Scholar
  19. 19.
    Helm, W.E., Waldmann, K.-H.: Optimal control of arrivals to multiserver queues in a random environment. J. Appl. Probab. 21, 602–615 (1984) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kalmykov, G.I.: On the partial ordering of one-dimensional Markov processes. Theory Probab. Appl. 7, 456–459 (1962) CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Kogan, Y.: Another approach to asymptotic expansions for large closed queueing networks. Oper. Res. Lett. 11, 317–321 (1992) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kogan, Y., Liptser, R.S.: Limit non-stationary behavior of large closed queueing networks with bottlenecks. Queueing Syst. 14, 33–55 (1993) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kogan, Y., Liptser, R.S., Smorodinskii, A.V.: Gaussian diffusion approximation of a closed Markov model of computer networks. Prob. Inf. Transm. 22, 38–51 (1986) MATHMathSciNetGoogle Scholar
  24. 24.
    Krichagina, E.V., Liptser, R.S., Puhalskii, A.A.: Diffusion approximation for a system that an arrival stream depends on queue and with arbitrary service. Theory Probab. Appl. 33, 114–124 (1988) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Krichagina, E.V., Puhalskii, A.A.: A heavy-traffic analysis of closed queueing system with GI/∞ server. Queueing Syst. 25, 235–280 (1997) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Krieger, U., Klimenok, V.I., Kazmirsky, A.V., Breuer, L., Dudin, A.N.: A BMAP/PH/1 queue with feedback operating in a random environment. Math. Comput. Model. 41, 867–882 (2005) MATHCrossRefGoogle Scholar
  27. 27.
    Liptser, R.S., Shiryayev, A.N.: Theory of Martingales. Kluwer, Dordrecht (1989) MATHGoogle Scholar
  28. 28.
    McKenna, J., Mitra, D.: Integral representation and asymptotic expansions for closed Markovian queueing networks. Normal usage. Bell Syst. Tech. J. 61, 661–683 (1982) MATHMathSciNetGoogle Scholar
  29. 29.
    O’Cinneide, C., Purdue, P.: The M/M/∞ queue in a random environment. J. App. Probab. 23, 175–184 (1986) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Pittel, B.: Closed exponential networks of queues with saturation: The Jackson type stationary distribution and its asymptotic analysis. Math. Oper. Res. 6, 357–378 (1979) MathSciNetGoogle Scholar
  31. 31.
    Purdue, P.R.: The M/M/1 queue in a random environment. Oper. Res. 22, 562–569 (1974) MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Ramanan, K.: Reflected diffusions defined via extended Skorokhod map. Electron. J. Probab. 11, 934–992 (2006) MathSciNetGoogle Scholar
  33. 33.
    Skorokhod, A.V.: Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6, 264–274 (1961) CrossRefGoogle Scholar
  34. 34.
    Tanaka, H.: Stochastic differential equations with reflected boundary conditions in convex regions. Hiroshima Math. J. 9, 163–177 (1979) MATHMathSciNetGoogle Scholar
  35. 35.
    Whitt, W.: Open and closed models for networks of queues. AT&T Bell Lab. Tech. J. 63, 1911–1979 (1984) MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityClaytonAustralia

Personalised recommendations