Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris


  • Saul I. Gass
  • Carl M. Harris
Reference work entry
DOI: https://doi.org/10.1007/1-4020-0611-X_50

In a continuous-time Markov chain with state X(t) at time t, define pij(t) as the probability that X(t + s) = j, given that X(s) = i, s, t ≥ 0, and rij as the transition rate out of state i to state j. Then Kolmogorov's backward equations say that, for all states i, j and times t ≥ 0, the derivatives dpij(t)/dt = Σkirikpkj(t) − vipij(t), where vi is the transition rate out of state i, vi = Σjrij. Markov chains; Markov processes.

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Saul I. Gass
    • 1
  • Carl M. Harris
    • 2
  1. 1.Robert H. Smith School of BusinessUniversity of MarylandCollege PartUSA
  2. 2.School of Information Technology & EngineeringGeorge Mason UniversityFairfaxUSA