Kelly's lemma

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

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 numbers q'ij and πi summing to 1 where[1]

\begin{align} \sum_{j \neq i} \pi_i q'_{ij} &= \sum_{j \neq i} q_{ij} \quad \forall i\in S\\ \pi_i q_{ij} &= \pi_jq_{ji}' \quad \forall i,j \in S \end{align}

then q'ij are the rates for the reversed process and πi are the stationary distribution for both processes.

Proof

Given the assumptions made on the qij and πi we can see

\sum_{i \neq j} \pi_i q_{ij} = \sum_{i \neq j} \pi_j q'_{ji} = \pi_j \sum_{i \neq j} q_{ji} = -\pi_j q_{jj}

so the global balance equations are satisfied and the πi are a stationary distribution for both processes.

References

  1. ↑ 1.0 1.1 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 51. pp. 39–59. doi:10.1007/0-387-21525-5_2. ISBN 978-0-387-00211-8.