Asymptotic analysis of the effect of arrival model uncertainties in some optimal routing problems

  • B. Mohanty
  • C. G. Cassandras
Contributed Papers


We study the effect of arrival model uncertainties on the optimal routing in a system of parallel queues. For exponential service time distributions and Bernoulli routing, the optimal mean system delay generally depends on the interarrival time distribution. Any error in modeling the arriving process will cause a model-based optimal routing algorithm to produce a mean system delay higher than the true optimum. In this paper, we present an asymptotic analysis of the behavior of this error under heavy traffic conditions for a general renewal arrival process. An asymptotic analysis of the error in optimal mean delay due to uncertainties in the service time distribution for Poisson arrivals was reported in Ref. 6, where it was shown that, when the first moment of the service time distribution is known, this error in performance vanishes asymptotically as the traffic load approaches the system capacity. In contrast, this paper establishes the somewhat surprising result that, when only the first moment of the arrival distribution is known, the error in optimal mean delay due to uncertainties in the arrival model is unbounded as the traffic approaches the system capacity. However, when both first and second moments are known, the error vanishes asymptotically. Numerical examples corroborating the theoretical results are also presented.

Key Words

Routing optimization queueing systems robustness distributed algorithms 


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  1. 1.
    Kleinrock, L.,Queueing Systems, Vol. 1, John Wiley, New York, New York, 1975.Google Scholar
  2. 2.
    Kumar, A., andBonomi, F.,Adaptive Load Balancing in a Multiprocessor System with a Central Job Scheduler, Proceedings of the 2nd International Workshop on Applied Mathematics and Performance/Reliability Models of Computer Communication Systems, Rome, Italy, pp. 173–188, 1987.Google Scholar
  3. 3.
    Tsitsiklis, J. N., andBertsekas, D. P.,Distributed Asynchronous Optimal Routing in Data Networks, IEEE Transactions on Automatic Control, Vol. 31, pp. 325–332, 1986.Google Scholar
  4. 4.
    Gallager, R. G.,A Minimum Delay Routing Algorithm Using Distributed Computation, IEEE Transactions on Communications, Vol. 23, pp. 73–85, 1977.Google Scholar
  5. 5.
    Chang, F., andWu, L.,An Optimal Adaptive Routing Algorithm, IEEE Transactions on Automatic Control, Vol. 31, pp. 690–700, 1986.Google Scholar
  6. 6.
    Mohanty, B., andCassandras, C. G.,The Effect of Model Uncertainty on Some Optimal Routing Problems, Journal of Optimization Theory and Applications, Vol. 77, pp. 256–290, 1993.Google Scholar
  7. 7.
    Ho, Y. C., andCao, X. R.,Perturbation Analysis of Discrete-Event Dynamic Systems, Kluwer Academic Publishers, Boston, Massachusetts, 1991.Google Scholar
  8. 8.
    Glasserman, P.,Gradient Estimation via Perturbation Analysis, Kluwer Academic Publishers, Boston, Massachusetts, 1991.Google Scholar
  9. 9.
    Glynn, P.,Likelihood Ratio Gradient Estimation: An Overview, Proceedings of the 1987 Winter Simulation Conference, pp. 336–375, 1987.Google Scholar
  10. 10.
    Reimann, M., andWeiss, A.,Sensitivity Analysis for Simulations via Likelihood Ratios, Operations Research, Vol. 37, pp. 830–844, 1989.Google Scholar
  11. 11.
    Bertsekas, D., andGallager, R.,Data Networks, Prentice-Hall, Englewood Cliffs, New Jersey, 1987.Google Scholar
  12. 12.
    Cassandras, C. G., Abidi, M. V., andTowsley, D.,Distributed Routing with Online Marginal Delay Estimation, IEEE Transactions on Communications, Vol. 38, pp. 348–359, 1990.Google Scholar
  13. 13.
    Simha, R.,Optimization of Resource Control in Communication Systems, PhD Thesis, University of Massachusetts, Amherst, Massachusetts, 1990.Google Scholar
  14. 14.
    Ni, L. M., andHwang, H.,Optimal Load Balancing in a Mulitple Processor with Many Job Classes, IEEE Transactions on Software Engineering, Vol. 11, pp. 491–496, 1985.Google Scholar
  15. 15.
    Tantawi, A. N., andTowsley, D.,Optimal Static Load Balancing in Distributed Computer Systems, Journal of the ACM, Vol. 32, pp. 445–465, 1985.Google Scholar
  16. 16.
    Buzen, J. P., andChen, P. P. S.,Optimal Load Balancing in Memory Hierarchies, Information Processing, North Holland, Amsterdam, Holland, pp. 271–275, 1974.Google Scholar
  17. 17.
    Dowdy, L., andFoster, D.,Comparative Models of the File Assignment Problem, ACM Computing Surveys, Vol. 14, pp. 267–314, 1982.Google Scholar
  18. 18.
    Wah, B.,File Placement in Distributed Computer Systems, IEEE Computer, Vol. 17, pp. 23–33, 1984.Google Scholar
  19. 19.
    Buzacott, J. A., andShanthikumar, J. G.,Stochastic Models of Manufacturing Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1992.Google Scholar
  20. 20.
    Shaked, M., andShanthikumar, J. G.,Stochastic Convexity and Its Applications, Advances in Applied Probability, Vol. 20, pp. 427–446, 1988.Google Scholar
  21. 21.
    Suri, R.,Robustness of Queueing Network Formulas, Journal of the ACM, Vol. 30, pp. 564–594, 1983.Google Scholar
  22. 22.
    Cox, D. R., andIsham, V.,Point Processes, Chapman and Hall, London, England, 1980.Google Scholar
  23. 23.
    De Bruijn, N. G.,Asymptotic Methods in Analysis, North-Holland, Amsterdam, Holland, 1961.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • B. Mohanty
    • 1
    • 2
  • C. G. Cassandras
    • 1
  1. 1.Department of Electrical and Computer EngineeringUniversity of MassachusettsAmherst
  2. 2.San Diego

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