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Approximations of open queueing networks by reflection mappings

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Abstract

This paper presents a three-step procedure which allows to approximate the queue-length and the busy-time processes associated with open queueing networks. These three approximations are based on reflection mappings and are introduced with explicit estimates of their accuracy. The third one may be treated as approximation by accompanying reflection Brownian motions with rates of convergence.

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References

  1. A.A. Borovkov, A.A. Mogulskii and A.I. Sakhanenko, Limit Theorems for Random Processes. Enciclopaedia of Mathematics (Springer, New York, 1994).

    Google Scholar 

  2. H. Chen and A. Mandelbaum, Leontief systems, RBV's and RBM's, in: Proc.of Imperial College Workshop on Applied Stochastic Processes, eds. M.H.A. Davis and R.J. Elliott (Gordon and Breach, New York, 1991) pp. 1–43.

    Google Scholar 

  3. H. Chen and A. Mandelbaum, Discrete flow networks: Bottleneck analysis and fluid approximations, Math. Oper. Res. 16(1991) 408–446.

    Google Scholar 

  4. H. Chen and A. Mandelbaum, Stochastic discrete flow networks: Diffusion approximations and bottlenecks, Ann. Probab. 19(1991) 302–308.

    Google Scholar 

  5. H. Chen and A. Mandelbaum, Hierarchical modelling of stochastic networks, Part II: Strong approximations, in: Stochastic Modeling and Analysis of Manufacturing Systems, ed. D.D. Yao (Springer, New York, 1994) pp. 107–131.

    Google Scholar 

  6. M. Csörgő, L. Horvath and J. Steinebach, Invariance principle for renewal processes, Ann. Probab. 15(1987) 1441–1460.

    Google Scholar 

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

    Google Scholar 

  8. U. Einmahl, Extensions of results of Komlos, Major and Tusnady to the multivariate case, J. Multivariate Anal. 28(1989) 20–68.

    Article  Google Scholar 

  9. J.M. Harrison and M.I. Reiman, Reflected Brownian motion on an orthant, Ann. Probab. 9(1981) 302–308.

    Google Scholar 

  10. L. Horvath, Strong approximation of renewal process, Stochastic Process. Appl. 18(1984) 127–138.

    Article  Google Scholar 

  11. L. Horvath, Strong approximations of open queueing networks, Math. Oper. Res. 17(1992) 487–508.

    Article  Google Scholar 

  12. M.I. Reiman, Open queueing networks in heavy traffic, Math. Oper. Res. 9(1984) 441–458.

    Google Scholar 

  13. A.I. Sakhanenko, On the accuracy of normal approximation in the invariance principle, Siberian Adv. Math. 1(1991) 58–91.

    Google Scholar 

  14. A.V. Skorohod, On a representation of random variables (in Russian), Theory Probab. Appl. 21 (1976) 645–648.

    Google Scholar 

  15. V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36 (1965) 423–439.

    Google Scholar 

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  1. Surgut State University, 14, Energetikov, Surgut, Tumen region, 626400, Russia

    Alexander I. Sakhanenko

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  1. Alexander I. Sakhanenko
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Sakhanenko, A.I. Approximations of open queueing networks by reflection mappings. Queueing Systems 32, 41–64 (1999). https://doi.org/10.1023/A:1019130919230

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