Advertisement

Acta Informatica

, Volume 23, Issue 6, pp 657–678 | Cite as

Information theoretic analysis for a general queueing system at equilibrium with application to queues in tandem

  • J. Cantor
  • A. Ephremides
  • D. Horton
Article

Summary

In this paper, information theoretic inference methology for system modeling is applied to estimate the probability distribution for the number of customers in a general, single server queueing system with infinite capacity utilized by an infinite customer population. Limited to knowledge of only the mean number of customers and system equilibrium, entropy maximization is used to obtain an approximation for the number of customers in the G¦ G¦1 queue. This maximum entropy approximation is exact for the case of G=M, i.e., the M¦M¦1 queue. Subject to both independent and dependent information, an estimate for the joint customer distribution for queueing systems in tandem is presented. Based on the simulation of two queues in tandem, numerical comparisons of the joint maximum entropy distribution is given. These results serve to establish the validity of the inference technique and as an introduction to information theoretic approximation to queueing networks.

Keywords

Maximum Entropy Single Server Queueing System System Equilibrium Numerical Comparison 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cohen, J.W.: The Single Server Queue. Amsterdam: North Holland 1982Google Scholar
  2. 2.
    Kleinrock, L.: Queueing Systems. Volume 1: Theory. New York: John Wiley 1975Google Scholar
  3. 3.
    Cooper, R.B.: Introduction to Queueing Theory. New York: North Holland 1981Google Scholar
  4. 4.
    Syski, R.: Introduction to Congestion Theory in Telephone Systems. London: Oliver and Boyd 1960Google Scholar
  5. 5.
    Kleinrock, L.: Communication Nets. New York: McGraw-Hill 1964Google Scholar
  6. 6.
    Tanenbaum, A.S.: Computer Networks. New Jersey: Prentice-Hall 1981Google Scholar
  7. 7.
    Trivedi, K.S.: Probability and Statistics with Reliability, Queueing and Computer Science Applications. New Jersey: Prentice-Hall 1982Google Scholar
  8. 8.
    Shore, J.E.: Derivation of equilibrium and time-dependent solutions of queueing systems using entropy maximization. Proceedings 1978 National Computer Conference, pp. 483–487. AFIPS 1978Google Scholar
  9. 9.
    Shore, J.E., Johnson, R.W.: Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy, IEEE Trans. Inf. Theory, 26, 26–37 (1980) (See also comments and corrections in IEEE Trans. Inf. Theory, 29, 942–943 (1983))Google Scholar
  10. 10.
    Beneš, V.E.: Mathematical Theory of Connecting Networks and Telephone Traffic. New York: Academic Press 1965Google Scholar
  11. 11.
    Ferdinand, A.E.: A Statistical Mechanics Approach to Systems Analysis. IBM J. Res. Dev. 14, 539–547 (1970)Google Scholar
  12. 12.
    Shore, J.E.: Information Theoretic Approximations for M¦G¦1 and G¦G¦1 Queueing Systems. Acta Inf. 17, 43–61 (1982)Google Scholar
  13. 13.
    El-Affendi, M.A., Kouvatos, D.D.: A Maximum Entropy Analysis of the M¦G¦1 and G¦M¦1 Queueing Systems at Equilibrium. Acta Inf. 19, 339–355 (1983)Google Scholar
  14. 14.
    Jaynes, E.T.: Information theory and statistical mechanics I. Phys. Rev. 106, 620–630 (1957)Google Scholar
  15. 15.
    Kullback, S.: Information Theory and Statistics. New York: Dover 1969; New York: Wiley 1959Google Scholar
  16. 16.
    Csiszár, I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3, 146–158 (1975)Google Scholar
  17. 17.
    Shore, J.E., Johnson, R.W.: Properties of cross-entropy minimization. IEEE Trans. Inf. Theory 27, 472–482 (1981)Google Scholar
  18. 18.
    Johnson, R.W.: Determining probability distributions by maximum entropy and minimum cross-entropy. APL79 Conference Proceedings, pp. 24–29, ACM0-89791-005, May 1979Google Scholar
  19. 19.
    Shore, J.E.: Cross-Entropy Minimization Given Fully Decomposable Subset and Aggregate Constraints, IEEE Trans. Inf. Theory 28, 956–961 (1982)Google Scholar
  20. 20.
    Mendenhall, W., Schaeffer, R.L.: Mathematical Statistics with Applications, North Scituate, MH: Duxbury Press 1974Google Scholar
  21. 21.
    Boxma, O.J.: On a Tandem Queueing Model with Identical Service Times at Both Counters I. Adv. Appl. Probab. 11, 616–643 (1979)Google Scholar
  22. 22.
    Pinedo, M, Wolff, R.W.: A Comparison Between Tandem Queues with Dependent and Independent Service Times. Oper. Res. 30, 464–479 (1982)Google Scholar
  23. 23.
    Cantor, J.L.: Information-Theoretic Analyses for a Multi-Server Queueing System at Equilibrium with Application to Queues in Tandem, M.SC. Thesis, Electrical Engineering Dept., Univ. of Maryland, College Park, Maryland 1984Google Scholar
  24. 24.
    Denning, P.J., Buzen, F.P.: The Operational Analysis of Queueing Network Models. Comput. Surv. 10, 225–261 (1978)Google Scholar
  25. 25.
    Zahorjan, J.: The Approximate Analysis of Large Queueing Network Models. Tech. Rep. CSR6-122 (Ph.D. Thesis), Univ. of Toronto, Toronto, Ontario 1980Google Scholar
  26. 26.
    Whitt, W.: The Queueing Network Analyzer. Bell. Syst. Tech. J. 62, 2279–2815 (1983)Google Scholar
  27. 27.
    Tembe, S.V., Wolff, R.W.: The Optimal Order of Service in Tandem Queues. Oper. Res. 22, 824–832 (1974)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • J. Cantor
    • 1
  • A. Ephremides
    • 2
  • D. Horton
    • 3
  1. 1.Business and Technological SystemsSeabrookUSA
  2. 2.Electrical Engineering DepartmentUniversity of MarylandCollege ParkUSA
  3. 3.Martin Marietta Aerospace OrlandoFloridaUSA

Personalised recommendations