skip to main content
10.1145/223587.223613acmconferencesArticle/Chapter ViewAbstractPublication PagesmetricsConference Proceedingsconference-collections
Article
Free Access

Regenerative randomization: theory and application examples

Authors Info & Claims
Published:01 May 1995Publication History

ABSTRACT

Randomization is a popular method for the transient solution of continuous-time Markov models. Its primary advantages over other methods (i.e., ODE solvers) are robustness and ease of implementation. It is however well-known that the performance of the method deteriorates with the "stiffness" of the model: the number of required steps to solve the model up to time t tends to Λt for Λt → ∞. In this paper we present a new method called regenerative randomization and apply it to the computation of two transient measures for rewarded irreducible Markov models. Regarding the number of steps required in regenerative randomization we prove that: 1) it is smaller than the number of steps required in standard randomization when the initial distribution is concentrated in a single state, 2) for Λt → ∞, it is upper bounded by a function O(log(Λt/ε)), where ε is the desired relative approximation error bound. Using dependability and performability examples we analyze the performance of the method.

References

  1. Car94.J. A. Carrasco and A. CMder6n, "Regenerative Randomization: Theory an~l applicaton exanlples," Technical Report, UPC, 1994.Google ScholarGoogle Scholar
  2. Cin75.E. (Tinlar, Introduction to Stochastic Processes, Prentice- Hall, 1975, pp. 371-378.Google ScholarGoogle Scholar
  3. Cru76.K. S. Crump, "Numerical Inversion of Laplace Transforms Using a Fourier Series Approximation," Journal of the A CM, vol. 23, pp. 89-96, 1976. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Dij90.N. M. Dijk, "On a Simple Proof of Uniformization for Continuous and Discrete-State Continuous-Time Markov Chains," Adv. AppI. Prob., vol. 22, pp. 749-750, 1990.Google ScholarGoogle ScholarCross RefCross Ref
  5. Gra77.W. K. Grassmann, "Transient solutions in Markovian queuing systems," Comp~ft. Operatzons Res., vol. 4, pp. 47-53, 1977.Google ScholarGoogle ScholarCross RefCross Ref
  6. Gro84.D. Gross and D. R. Miller, "The randomization technique as a modelling tool and solution procedtu'e for transient Markov processes," Operations Re,., vol. 32, pp. 343-361, 1984.Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Jen53.A. Jensen, "Markoff chains as an aid in the study of Markoff processes," Skand. Akuarietidskr(ft, vol. 36, pp. 87-91, 1953.Google ScholarGoogle Scholar
  8. Koh82.J. Kohlas, Stoc~astlc }lfe~hod~ of Ot~~ratioT~ Re, earth, Cambridge University Press, Cambridge, 1982.Google ScholarGoogle Scholar
  9. Mal94.M. Malhotra, J. K. Muppala and I'~. 5. Trivedi, "'Stiffness- Tolerant Methods for Transient Analysis of Stiff Markov Chains," lllicroelectron. Rehab., vol. 34, no. 11, pp. 1825- 1841, 1994.Google ScholarGoogle Scholar
  10. Moo93.A. P. Moorsel and W. H. Sanders, "Adaptive Uniformization", Technical report, University of Arizona, 1993.Google ScholarGoogle Scholar
  11. Qur93.M. h. Qureshi and W. H. Sa,~ders, "Rewa,'d Model Solution Methods with Impulse and Rate Rewards: An Algorithm and Numerical Results," Technical Report, University of Arizona, 1993.Google ScholarGoogle Scholar
  12. Rei88.A. Reibman and K. S. Trivedi, "Numerical Transient Analysis of Markov Models," Comput. Operations Res., vol. 15, pp. 19-36, 1988. Google ScholarGoogle ScholarDigital LibraryDigital Library
  13. Rub93.G. Rubino and B. Sericola, "Interval Availability Distribution Computation," in Proc. 23th Int. Symp. on Fault. Tolerant Computing FTCS-~3, Toulouse, pp. 48- 55, June 1993.Google ScholarGoogle ScholarCross RefCross Ref
  14. Sou86.E. de Souza e Silva and H. R. Gall, "Calculating Cumulative Operational Time Distributions of Repairable Computer Systems," IEEE Trans. on Computers, vol. 35, pp. 322-332, 1986. Google ScholarGoogle ScholarDigital LibraryDigital Library
  15. Sou89.E. de Souza e Silva and H. R. Gall, "Calculating Availability and Performability Measures of Repairable Computer Systems using Randomization," Journal of the A CI~i, vol. 34, pp. 179-199, 1989. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Regenerative randomization: theory and application examples

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in
      • Published in

        cover image ACM Conferences
        SIGMETRICS '95/PERFORMANCE '95: Proceedings of the 1995 ACM SIGMETRICS joint international conference on Measurement and modeling of computer systems
        May 1995
        340 pages
        ISBN:0897916956
        DOI:10.1145/223587

        Copyright © 1995 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 May 1995

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • Article

        Acceptance Rates

        Overall Acceptance Rate459of2,691submissions,17%

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader
      About Cookies On This Site

      We use cookies to ensure that we give you the best experience on our website.

      Learn more

      Got it!