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Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory

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Abstract

Multi-dimensional asymptotically quasi-Toeplitz Markov chains with discrete and continuous time are introduced. Ergodicity and non-ergodicity conditions are proven. Numerically stable algorithm to calculate the stationary distribution is presented. An application of such chains in retrial queueing models with Batch Markovian Arrival Process is briefly illustrated.

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References

  1. S. Asmussen, Applied Probability and Queues. (Springer, New York, 2003).

    Google Scholar 

  2. J. Artalejo, D.S. Orlovsky, and A.N. Dudin, Multiserver retrial model with variable number of active servers. Computers and Industrial Engineering 48 (2005) 273–288.

    Article  Google Scholar 

  3. R. Bellman, Introduction to Matrix Analysis. (McGraw Hill Book Company, New York, 1960).

    Google Scholar 

  4. L. Breuer, A.N. Dudin, and V.I. Klimenok, A retrial BMAP/PH/N system. Queueing Systems. 40 (2002) 433–457.

    Article  Google Scholar 

  5. L. Breuer, V. Klimenok, A. Birukov, A. Dudin, and U. R. Krieger, Mobile Networks Modeling the access to a wireless network at hot spots. European Transactions on Telecommunications 16 (2005) 309–316.

    Article  Google Scholar 

  6. L. Bright and P.G. Taylor, Calculating the equilibrium distribution in Level Dependent Quasi-Birth-and-Death processes. Communications in Statistics-Stochastic Models 11 (1995) 497–525.

    Google Scholar 

  7. S. Chakravarthy, The batch Markovian arrival process: A review and future work. Advances in Probability Theory and Stochastic Processes. (Notable Publications, New Jersey, 2001), pp. 21–49.

    Google Scholar 

  8. A.N. Dudin and V.I. Klimenok, Multi-dimensional quasitoeplitz Markov chains. J. Appl. Math. and Stochast. Analysis 12 (1999) 393–415.

    Google Scholar 

  9. A.N. Dudin and V.I. Klimenok, BMAP/SM/1 model with Markov modulated retrials. Top 7 (1999) 267–278.

    Google Scholar 

  10. A.N. Dudin and V.I. Klimenok, Retrial BMAP/SM/1 system operating in a synchronous random environment. Probab. Analysis of Rare Events: Theory and Problems of Safety, (Insurance and Ruin, Riga, 1999), pp. 143–148.

    Google Scholar 

  11. A.N. Dudin and V.I. Klimenok, A retrial BMAP/SM/1 system with linear repeated requests. Queueing Systems 34 (2000) 47–66.

    Article  Google Scholar 

  12. A.N. Dudin, G.V. Tsarenkov, and V.I. Klimenok, Software “SIRIUS++” for performance evaluation of modern communication networks. Modelling and Simulation 2002. 16th European Simulation Multi-conference. Darmstadt (2002), pp. 489–493.

  13. A.N. Dudin, V.I. Klimenok, G.V. Tsarenkov, O.V. Semenova, and A.A. Birukov, Software “SIRIUS-C” for synthesis of optimal control by queues, In: Proceedings of 11-th International Conference on Analytical and Stochastic Modelling Techniques and Applications (ASMTA 2004), 13–16 June 2004. (Magdeburg, Germany, 2004), pp. 123–129.

  14. A.N. Dudin, A. Krishnamoorthy, V.C. Joshua, and G.V. Tsarenkov, Analysis of the BMAP/G/1 retrial system with search of customers from the orbit. European Journal of Operational Research 157 (2004) 169–179.

    Article  Google Scholar 

  15. H.R. Gail, S.L. Hantler, and B.A. Taylor, Spectral analysis of M/G/1 and G/M/1 type Markov chains. Adv. in Appl. Probab 28 (1996) 114–165.

    Article  Google Scholar 

  16. F.R. Gantmakher, The Matrix Theory. (Science, Moscow, 1967).

    Google Scholar 

  17. W. Grassmann and D. Heyman, Equilibrium equations of block-structured Markov chains with repeated rows. J. of Appl. Probab 27 (1990) 557–576.

    Article  Google Scholar 

  18. J. Hofmann, The BMAP/G/1 queue with level-dependent arrivals - an overview. Telecommunication Systems 16 (2001) 347–359.

    Article  Google Scholar 

  19. J.G. Kemeni, J.L. Snell, and A.W. Knapp, Denumerable Markov Chains (Van Nostrand, New York, 1966).

    Google Scholar 

  20. C.S. Kim, V. Klimenok, A. Birukov, and A. Dudin, Optimal multi-threshold control by the BMAP/SM/1 retrial system. Annals of Operations Research 141 (2006) 193–210.

    Article  Google Scholar 

  21. V. Klimenok, A BMAP/SM/1 queueing system with hybrid operation mechanism. Automation and Remote Control 66 (2005) 779–790.

    Article  Google Scholar 

  22. V.I. Klimenok, S.R. Chakravarthy, and A.N. Dudin, Algorithmic analysis of a multiserver Markovian queue with primary and secondary services. Computers and Mathematics with Applications 50 (2005) 1251–1270.

    Article  Google Scholar 

  23. V.I. Klimenok, D.S. Orlovsky, and C.S. Kim, The BMAP/PH/N/N+R retrial queueing system with different disciplines of retrials. In Proceedings of 11-th International Conference on Analytical and Stochastic Modelling Techniques and Applications (ASMTA 2004), 13–16 June 2004, (Magdeburg, Germany, 2004) pp. 93–98.

  24. V. Klimenok, C.S. Kim, D. Orlovsky, and A. Dudin, Lack of invariant property of Erlang loss model in case of the MAP input. Queueing Systems 49 (2005) 187–213.

    Article  Google Scholar 

  25. V. Klimenok, D. Orlovsky, and A. Dudin, A BMAP/PH/N system with impatient repeated calls. Asia-Pacific Journal of Operational Research, accepted.

  26. Q.-L. Li, and J. Cao, Two types of RG -factorization of Quasi-Birth-and-Death processes and their applications to stochastic integral functionals. Communications in Statistics-Stochastic Models 20 (2004) 299–340.

    Article  Google Scholar 

  27. D. M. Lucantoni, K.S. Meier-Hellstern, and M.F. Neuts, A single-server queue with server vacations and a class of nonrenewal arrival processes. Adv. in Appl. Prob. 22 (1990) 676–705.

    Article  Google Scholar 

  28. D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process. Communications in Statistics-Stochastic Models 7 (1991) 1–46.

    Google Scholar 

  29. I. Mitrani, The spectral expansion solution method for Markov processes on lattice strips. Advances in Queueing: Theory, Methods and Open Problems. (CRC Press, Boca Raton, 1995), pp. 337–352.

    Google Scholar 

  30. M.D. Moustafa, Input-output Markov process. Proc. Koninkl. Net. Akad. Wetensch A60 (1957) 112–118.

    Google Scholar 

  31. V.V. Mushko, M.J. Jakob, K. O. Ramakrishnan, A. Krishnamoorthy, and A.N. Dudin, Multiserver queue with addressed retrials. Annals of Operations Research 141 (2006) 283–310.

    Article  Google Scholar 

  32. M.F. Neuts, A versatile Markovian point process. J. Appl. Prob. 16 (1979) 764–779

    Article  Google Scholar 

  33. M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models—An Algorithmic Approach. (Johns Hopkins University Press, 1981).

  34. M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and Their Applications. (Marcel Dekker, New York, 1989).

    Google Scholar 

  35. V. Ramaswami, A stable recursion for the steady-state vector in Markov chains of M/G/1 type. Communications in Statistics-Stochastic Models 4 (1988) 183–188.

    Google Scholar 

  36. V. Ramaswami, Matrix analytic methods: A tutorial overview with some extensions and new results. Matrix analytic methods in stochastic models. Lecture Notes in Pure and Applied Mathematics. (Marcel Dekker, New York, 1997), pp. 261–296.

    Google Scholar 

  37. L.I. Sennot, P.A. Humblet, and R.L. Tweedie, Mean drifts and non-ergodicity of Markov chains. Operations Research 31 (1983) 783–789.

    Article  Google Scholar 

  38. Q. Ye, High accuracy algorithms for solving nonlinear matrix equations in queueing models. Advances in Algorithmic Methods for Stochastic Models. (Notable Publications, New Jersey, 2000), pp. 401–415.

    Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Belarusian State University, 4 Independence Ave., 220030, Minsk, Belarus

    Valentina Klimenok & Alexander Dudin

Authors
  1. Valentina Klimenok
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  2. Alexander Dudin
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Correspondence to Alexander Dudin.

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AMS Subject Classifications Primary 60K25 · 60K20

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Klimenok, V., Dudin, A. Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory. Queueing Syst 54, 245–259 (2006). https://doi.org/10.1007/s11134-006-0300-z

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  • DOI: https://doi.org/10.1007/s11134-006-0300-z

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