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Kelly's lemma

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In probability theory, Kelly's lemma states that for a stationary continuous-time Markov chain, a process defined as the time-reversed process has the same stationary distribution as the forward-time process.[1] The theorem is named after Frank Kelly.[2][3][4][5]

Statement

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For a continuous time Markov chain with state space S and transition-rate matrix Q (with elements qij) if we can find a set of non-negative numbers q'ij and a positive measure π that satisfy the following conditions:[1]

then q'ij are the rates for the reversed process and π is proportional to the stationary distribution for both processes.

Proof

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Given the assumptions made on the qij and π we have

so the global balance equations are satisfied and the measure π is proportional to the stationary distribution of the original process. By symmetry, the same argument shows that π is also proportional to the stationary distribution of the reversed process.

References

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  1. ^ a b Boucherie, Richard J.; van Dijk, N. M. (2011). Queueing Networks: A Fundamental Approach. Springer. p. 222. ISBN 144196472X.
  2. ^ Kelly, Frank P. (1979). Reversibility and Stochastic Networks. J. Wiley. p. 22. ISBN 0471276014.
  3. ^ Walrand, Jean (1988). An introduction to queueing networks. Prentice Hall. p. 63 (Lemma 2.8.5). ISBN 013474487X.
  4. ^ Kelly, F. P. (1976). "Networks of Queues". Advances in Applied Probability. 8 (2): 416–432. doi:10.2307/1425912. JSTOR 1425912.
  5. ^ Asmussen, S. R. (2003). "Markov Jump Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. Vol. 51. pp. 39–59. doi:10.1007/0-387-21525-5_2. ISBN 978-0-387-00211-8.