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  • Probability in the Engineering and Informational Sciences, Volume 26, Issue 4
  • October 2012, pp. 483-508


  • Michael N. Katehakis (a1) and Laurens C. Smit (a2)
  • DOI:
  • Published online: 01 July 2012

A class of Markov chains we call successively lumbaple is specified for which it is shown that the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has a(typically much) smaller state space and this yields significant computational improvements. We discuss how the results for discrete time Markov chains extend to semi-Markov processes and continuous time Markov processes. Finally, we will study applications of successively lumbaple Markov chains to classical reliability and queueing models.

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1.C. Derman , G.J. Lieberman , & S.M. Ross (1980). On the optimal assignment of servers and a repairman. Journal of Applied Probability 17: 577581.

2.B.N. Feinberg , & S.S. Chui (1987). A method to calculate steady state distributions of large Markov chains. Operations Research 35: 282290.

3.E. Frostig (1999). Jointly optimal allocation of a repairman and optimal control of service rate for machine repairman problem. European Journal of Operational Research 116: 274280.

4.G. Hooghiemstra , & G. Koole (2000). On the convergence of the power series algorithm. Performance Evaluation 42: 2139.

5.M.N. Katehakis , & C. Derman (1984). Optimal repair allocation in a series system. Mathematics of Operations Research 9: 615623.

6.M.N. Katehakis , & C. Derman (1989). On the maintenance of systems composed of highly reliable components. Management Science 35(5): 551560.

7.M.N. Katehakis , & C. Melolidakis (1995). On the optimal maintenance of systems and control of arrivals in queues. Stochastic Analysis and Applications 13: 137164.

11.G.M. Koole , & F.M. Spieksma (2001). On deviation matrices for birth–death processes. Probability in the Engineering and Informational Sciences 15: 239258.

12.W.L. Miranker , & V.Ya. Pan (1980). Methods of aggregation. Linear Algebra and its Application 29: 231257.

16.P.J. Schweitzer , M.L. Puterman , & W.L. Kindle (1984). Iterative aggregation–disaggregation procedures for discounted semi-markov reward processes. Operations Research 33: 589605.

17.V. Ungureanu , B. Melamed , M.N. Katehakis , & P.G. Bradford (2006). Deferred assignment scheduling in cluster-based servers. Cluster Computing 9: 5765.

18.V.B. Yap (2009). Similar states in continuous time Markov chains. Journal of Applied Probability 46: 497506.

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Probability in the Engineering and Informational Sciences
  • ISSN: 0269-9648
  • EISSN: 1469-8951
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