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Numerical Methods in Markov Chain Modeling

Author

Listed:
  • Bernard Philippe

    (IRISA, Rennes, France)

  • Youcef Saad

    (University of Minnesota, Minneapolis, Minnesota)

  • William J. Stewart

    (North Carolina State University, Raleigh, North Carolina)

Abstract

This paper describes and compares several methods for computing stationary probability distributions of Markov chains. The main linear algebra problem consists of computing an eigenvector of a sparse, nonsymmetric matrix associated with a known eigenvalue. It can also be cast as a problem of solving a homogeneous, singular linear system. We present several methods based on combinations of Krylov subspace techniques, single vector power iteration/relaxation procedures and acceleration techniques. We compare the performance of these methods on some realistic problems.

Suggested Citation

  • Bernard Philippe & Youcef Saad & William J. Stewart, 1992. "Numerical Methods in Markov Chain Modeling," Operations Research, INFORMS, vol. 40(6), pages 1156-1179, December.
    • Handle: RePEc:inm:oropre:v:40:y:1992:i:6:p:1156-1179
    DOI: 10.1287/opre.40.6.1156
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    Citations

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    Cited by:

    1. Kirkavak, Nureddin & Dincer, Cemal, 1999. "The general behavior of pull production systems: The allocation problems," European Journal of Operational Research, Elsevier, vol. 119(2), pages 479-494, December.
    2. Moghaddass, Ramin & Zuo, Ming J. & Wang, Wenbin, 2011. "Availability of a general k-out-of-n:G system with non-identical components considering shut-off rules using quasi-birth–death process," Reliability Engineering and System Safety, Elsevier, vol. 96(4), pages 489-496.
    3. Abderezak Touzene, 2008. "A Tensor Sum Preconditioner for Stochastic Automata Networks," INFORMS Journal on Computing, INFORMS, vol. 20(2), pages 234-242, May.
    4. Lin, Yan-Hui & Yam, Richard C.M., 2017. "Uncertainty importance measures of dependent transition rates for transient and steady state probabilities," Reliability Engineering and System Safety, Elsevier, vol. 165(C), pages 402-409.
    5. Amy N. Langville & William J. Stewart, 2004. "Testing the Nearest Kronecker Product Preconditioner on Markov Chains and Stochastic Automata Networks," INFORMS Journal on Computing, INFORMS, vol. 16(3), pages 300-315, August.
    6. Ahmad Hanbali & Roland Haan & Richard Boucherie & Jan-Kees Ommeren, 2012. "Time-limited polling systems with batch arrivals and phase-type service times," Annals of Operations Research, Springer, vol. 198(1), pages 57-82, September.

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