# 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. ^ 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. edit
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. edit