Markov Chains with finite convergence time

Israel Brosh*, Yigal Gerchak

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


We study the properties of finite ergodic Markov Chains whose transition probability matrix P is singular. The results establish bounds on the convergence time of Pm to a matrix where all the rows are equal to the stationary distribution of P. The results suggest a simple rule for identifying the singular matrices which do not have a finite convergence time. We next study finite convergence to the stationary distribution independent of the initial distribution. The results establish the connection between the convergence time and the order of the minimal polynomial of the transition probability matrix. A queuing problem and a maintenance Markovian decision problem which possess the property of rapid convergence are presented.

Original languageEnglish
Pages (from-to)247-253
Number of pages7
JournalStochastic Processes and their Applications
Issue number3
StatePublished - Aug 1978
Externally publishedYes


  • Markov chains
  • Markov decision problem
  • accessibility
  • convergence time
  • eigenvalues
  • leading vectors
  • minimal polynomial
  • null space


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