Acceptance-Rejection Methods for Generating Random Variates from Matrix Exponential Distributions and Rational Arrival Processes

  • Gábor Horváth
  • Miklós Telek
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 27)


Stochastic models based on matrix-exponential structures, like matrix-exponential distributions and rational arrival processes (RAPs), have gained popularity in analytical models recently. However, the application of these models in simulation-based evaluations is not as widespread yet. One of the possible reasons is the lack of efficient random-variate-generation methods. In this chapter we propose methods for efficient random-variate generation for matrix-exponential stochastic models based on appropriate representations of the models.


  1. 1.
    Asmussen, S., Bladt, M.: Point processes with finite-dimensional conditional probabilities. Stoch. Process. Appl. 82, 127–142 (1999)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bean, N.G., Nielsen, B.F.: Quasi-birth-and-death processes with rational arrival process components. Stoch. Models 26(3), 309–334 (2010)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brown, E., Place, J., de Liefvoort, A.V.: Generating matrix exponential random variates. Simulation 70, 224–230 (1998)MATHCrossRefGoogle Scholar
  4. 4.
    Buchholz, P., Telek, M.: Stochastic Petri nets with matrix exponentially distributed firing times. Perform. Eval. 67(12), 1373–1385 (2010)CrossRefGoogle Scholar
  5. 5.
    Buchholz, P., Telek, M.: On minimal representations of rational arrival processes. Ann. Oper. Res. (2011). doi:10.1007/s10479-011-1001-5 (to appear)Google Scholar
  6. 6.
    Cumani, A.: On the canonical representation of homogeneous Markov processes modelling failure-time distributions. Microelectron. Reliab. 22, 583–602 (1982)CrossRefGoogle Scholar
  7. 7.
    He, Q.M., Neuts, M.: Markov arrival processes with marked transitions. Stoch. Process. Appl. 74, 37–52 (1998)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Kronmal, R., Peterson, A.: On the alias method for generating random variables from a discrete distribution. Am. Stat. 33(4), 214–218 (1979)MathSciNetMATHGoogle Scholar
  9. 9.
    Latouche, G., Ramaswami, V.: Introduction to Matrix-Analytic Methods in Stochastic Model- ing. Society for Industrial and Applied Mathematics (1999)Google Scholar
  10. 10.
    van de Liefvoort, A.: The moment problem for continuous distributions. Technical report, WP-CM-1990-02, University of Missouri, Kansas City (1990)Google Scholar
  11. 11.
    Lipsky, L.: Queueing Theory: A Linear Algebraic Approach. MacMillan, New York (1992)MATHGoogle Scholar
  12. 12.
    Mitchell, K.: Constructing a correlated sequence of matrix exponentials with invariant first order properties. Oper. Res. Lett. 28, 27–34 (2001)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Mocanu, S., Commault, C.: Sparse representations of phase-type distributions. Comm. Stat. Stoch. Model 15(4), 759–778 (1999)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach. Dover, New York (1981)MATHGoogle Scholar
  15. 15.
    Neuts, M.F., Pagano, M.E.: Generating random variates from a distribution of phase type. In: WSC ’81: Proceedings of the 13th Conference on Winter Simulation, pp. 381–387. IEEE Press, Piscataway (1981)Google Scholar
  16. 16.
    Reinecke, P., Wolter, K., Bodrog, L., Telek, M.: On the cost of generating PH-distributed random numbers. In: International Workshop on Performability Modeling of Computer and Communication Systems (PMCCS), pp. 1–5. Eger, Hungary (2009)Google Scholar
  17. 17.
    Reinecke, P., Telek, M., Wolter, K.: Reducing the cost of generating APH-distributed random numbers. In: 15th International Conference on Measurement, Modelling and Evaluation of Computing Systems (MMB). Lecture Notes in Computer Science, vol. 5987, pp. 274–286. Springer, Essen (2010)Google Scholar
  18. 18.
    Robert, C., Casella, G.: Monte Carlo Statistical Methods. Springer, New York (2004)MATHGoogle Scholar
  19. 19.
    Telek, M., Horváth, G.: A minimal representation of Markov arrival processes and a moments matching method. Perform. Eval. 64(9–12), 1153–1168 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of TelecommunicationsTechnical University of BudapestBudapestHungary

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