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

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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 (N μ j )−1. After a service completion in the server station, a unit is transmitted to the jth client station with probability p j (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 p j (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.

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References

  1. Abramov, V.M.: A large closed queueing network with autonomous service and bottleneck. Queueing Syst. 35, 23–54 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  4. Abramov, V.M.: The stability of join-the-shortest-queue models with general input and output processes. arXiv: math/PR 0505040 (2005)

  5. Abramov, V.M.: The effective bandwidth problem revisited. arXiv: math/PR 0604182 (2006)

  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)

    MathSciNet  Google Scholar 

  7. Abramov, V.M.: Further analysis of confidence intervals for large client/server computer networks. Reliab. Theory Appl. (2007, to appear)

  8. Anulova, S.V., Liptser, R.S.: Diffusion approximation for processes with normal reflection. Theory Probab. Appl. 35, 413–423 (1990)

    MathSciNet  Google Scholar 

  9. Baccelli, F., Makovsky, A.M.: Stability and bounds for single-server queue in a random environment. Stoch. Models 2, 281–292 (1986)

    Article  MATH  Google Scholar 

  10. Berger, A., Bregman, L., Kogan, Y.: Bottleneck analysis in multiclass closed queueing networks and its application. Queueing Syst. 31, 217–237 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Borovkov, A.A.: Stochastic Processes in Queueing Theory. Springer, Berlin (1976)

    MATH  Google Scholar 

  12. Borovkov, A.A.: Asymptotic Methods in Queueing Theory. Wiley, New York (1984)

    MATH  Google Scholar 

  13. Boxma, O.J., Kurkova, I.A.: The M/M/1 queue in heavy-tailed random environment. Stat. Neerl. 54, 221–236 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dellacherie, C.: Capacités et Processus Stochastiques. Springer, Berlin (1972)

    MATH  Google Scholar 

  15. D’Auria, B.: M/M/∞ queues in quasi-Markovian random environment. arXiv: math/PR 0701842 (2007)

  16. Fricker, C.: Etude d’une file GI/G/1 á service autonome (avec vacances du serveur). Adv. Appl. Probab. 18, 283–286 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gelenbe, E., Iasnogorodski, R.: A queue with server of walking type (autonomous service). Ann. Inst. H. Poincare 16, 63–73 (1980)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kalmykov, G.I.: On the partial ordering of one-dimensional Markov processes. Theory Probab. Appl. 7, 456–459 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kogan, Y.: Another approach to asymptotic expansions for large closed queueing networks. Oper. Res. Lett. 11, 317–321 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kogan, Y., Liptser, R.S.: Limit non-stationary behavior of large closed queueing networks with bottlenecks. Queueing Syst. 14, 33–55 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  25. Krichagina, E.V., Puhalskii, A.A.: A heavy-traffic analysis of closed queueing system with GI/∞ server. Queueing Syst. 25, 235–280 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  27. Liptser, R.S., Shiryayev, A.N.: Theory of Martingales. Kluwer, Dordrecht (1989)

    MATH  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  29. O’Cinneide, C., Purdue, P.: The M/M/∞ queue in a random environment. J. App. Probab. 23, 175–184 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  31. Purdue, P.R.: The M/M/1 queue in a random environment. Oper. Res. 22, 562–569 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  32. Ramanan, K.: Reflected diffusions defined via extended Skorokhod map. Electron. J. Probab. 11, 934–992 (2006)

    MathSciNet  Google Scholar 

  33. Skorokhod, A.V.: Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6, 264–274 (1961)

    Article  Google Scholar 

  34. Tanaka, H.: Stochastic differential equations with reflected boundary conditions in convex regions. Hiroshima Math. J. 9, 163–177 (1979)

    MATH  MathSciNet  Google Scholar 

  35. Whitt, W.: Open and closed models for networks of queues. AT&T Bell Lab. Tech. J. 63, 1911–1979 (1984)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Monash University, Building 28M, Wellington road, Clayton, VIC, 3800, Australia

    Vyacheslav M. Abramov

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  1. Vyacheslav M. Abramov
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Correspondence to Vyacheslav M. Abramov.

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Abramov, V.M. Large Closed Queueing Networks in Semi-Markov Environment and Their Application. Acta Appl Math 100, 201–226 (2008). https://doi.org/10.1007/s10440-007-9180-4

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