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Probabilistic models of computer systems—Part I (exact results)

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This paper is a survey of certain known results concerning networks of queues. The choice of the topics presented has been made with special emphasis on mathematical results which can be applied to the analysis and synthesis of mathematical or simulation models of complex computer systems in which an ensemble of resources is shared among a set of user programs. The subjects covered include the Jackson, and Gordon and Newell theorems; the work-rate theorems of Chang, Lavenberg and Traiger; the Baskett, Chandy, Muntz, Palacios results. We also discuss in a unified manner problems related to Poisson processes in queueing networks. Companion papers (Parts II, III) will present results concerning approximations to queueing networks and some of their applications to computer system performance evaluation.

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References

  1. Jackson, J. R.: Jobshop-like queueing system. Management Science 10, 131–142 (1963)

    Google Scholar 

  2. Baskett, F., Chandy, M., Muntz, R., Palacios, J.: Open, closed and mixed networks of queues with different classes of customers. J. ACM 22, 248–260 (1975)

    Google Scholar 

  3. Muntz, R. R.: Poisson departure processes and queueing networks. IBM Research Report RC4145, December 1972

  4. Gaver, D. P.: Probability models for multiprogramming computer systems. J. ACM 14, 423–438 (1967)

    Google Scholar 

  5. Gaver, D. P., Shedler, G. S.: Processor utilization in multiprogramming systems via diffusion approximations. Operations Research 21, 569–576 (1963)

    Google Scholar 

  6. Kobayashi, H.: Application of the diffusion approximation to queueing networks: Part I—Equilibrium queue distribution. J. ACM 21, 316–328 (1974)

    Google Scholar 

  7. Gelenbe, E.: On approximate computer system models. J. ACM 22, 261–269 (1975)

    Google Scholar 

  8. Gelenbe, E.: A non-Markovian diffusion model and its application to the approximation of queueing and computer system behaviour. Research Report 74-1, Chaire de Systèmes Informatiques, University of Liege, Belgium, 1974

    Google Scholar 

  9. Reiser, M., Kobayashi, H.: Accuracy of a diffusion approximation for some queueing systems. IBM J. R and D. 18, 110–124 (1974)

    Google Scholar 

  10. Badel, M., Shum, A. V. Y.: Accuracy of a diffusion approximation method for computing work-rates in multiprogramming systems. IRIA Le Chesnay Research Report, 1975

  11. Chandy, M., Herzog, U., Woo, L.: Approximate analysis of general queueing networks. IBM Research Report RC4931, July 1974

  12. Courtois, P. J.: On the near complete decomposability of networks of queues and of stochastic models of multiprogramming systems. Carnegie-Mellon University, Computer Science Department, Research Report, November 1971

  13. Courtois, P. J., Georges, P.: An evaluation of the stationary behaviour of computations in multiprogramming computer systems. Proc. ACM-International Computing Symposium Bonn, 1970, p. 98–115

  14. Courtois, P. J.: Decomposability, instabilities, and saturation in multiprogramming systems. Comm. ACM 18, 371–377 (1975)

    Google Scholar 

  15. Avi-Itzhak, B., Heyman, P.: Approximate models of multiprogramming computer systems. Operations Research 21, 1212–1230 (1973)

    Google Scholar 

  16. Reiser, M., Kobayashi, H.: Blocking in a queueing network with two exponential servers. IBM Research Report RJ1360, 1974

  17. Chang, A., Lavenberg, S. S.: Work-rates in closed queueing networks with general independant servers. IBM Research Report RJ989, 1972

  18. Lavenberg, S. S., Traiger, I. L., Chang, A.: Bounds on work-rates for closed queueing networks with non-exponential stages. IBM Research Report RJ1182, March 1973

  19. Feller, W.: Diffusion processes in one dimension. Trans. American Mathematical Society 77, 1–31 (1954)

    Google Scholar 

  20. Cohen, J.: The single server queue. Amsterdam: North-Holland 1969

    Google Scholar 

  21. Gordon, W. J., Newell, G. F.: Closed queueing systems with exponential servers. Operations Research 15, 254–265 (1976)

    Google Scholar 

  22. Howard, R. A.: Dynamic probabilistic systems, Vol. I and II. New York: Wiley 1973

    Google Scholar 

  23. Buzen, J.: Computational algorithms for closed queueing networks with exponential servers. Comm. ACM 16, 527–531 (1973)

    Google Scholar 

  24. Cox, D. R.: A use of complex probabilities in the theory of stochastic processes. Proc. Cambridge Philosophical Society 51, 313–319 (1955)

    Google Scholar 

  25. Gaver, D. P.: Diffusion approximations and models for certain congestion problems. J. Applied Probability 5, 607–623 (1968)

    Google Scholar 

  26. Newell, G. F.: Applications of queueing theory. London: Chapman and Hall 1971

    Google Scholar 

  27. Buzen, J. P.: Queueing network models of multiprogramming. Ph. D. Thesis, Harvard University, 1971

  28. Gelenbe, E.: Modèles de Systèmes Informatiques. Thèse de Docotorat d'Etat, Université Paris VI, 1973

  29. Brandwajn, A., Buzen, J., Gelenbe, E., Potier, D.: A model of performance for virtual memory systems. Proc. ACM-Sigmetrics Symposium, Montreal, September 1974

  30. Muntz, R. R.: Analytical modelling of interactive systems. Proc. IEEE, 63, 946–953 (1975).

    Google Scholar 

  31. Badel, M., Gelenbe, E., Leroudier, J., Potier, D.: Adaptive optimization of a virtual memory system. Proc. IEEE, 63, 958–965 (1975).

    Google Scholar 

  32. Cochi, B.: Parameter analysis of some queueing models. Rapport de l'Ecole des Mines de Paris, February 1975

  33. Fishman, G.: Concepts and methods in discrete event digital simulation. New York: Wiley 1973

    Google Scholar 

  34. Leroudier, J., Parent, M.: Quelques aspects de la modélisation des systèmes informatiques par simulation à evènements discrets, (to appear in Revue d'Informatique de l'AFCET)

  35. Chandy, K. M.: The analysis and solutions for general queueing networks. Proc. Sixth Annual Princeton Conf. on Information Sciences and Systems, Princeton (N. J.), March 1972

  36. Williams, A. C., Bhandiwad, R. A.: A generating function approach to queueing network analysis of multiprogrammed computers, (to be published in Networks)

  37. Chandy, K. M., Howard, J., Keller, T. W., Towsley, D. J.: Local balance, robustness, poisson departures and the product form in queueing networks. (Submitted to J. ACM)

  38. Denning, P. J., Graham, S.: Multiprogrammed memory management. Proc. IEEE 63, 924–939 (1975)

    Google Scholar 

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Gelenbe, E., Muntz, R.R. Probabilistic models of computer systems—Part I (exact results). Acta Informatica 7, 35–60 (1976). https://doi.org/10.1007/BF00265220

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