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 q_ij) if we can find a set of numbers q'_ij and π_i summing to 1 where[1]

${\displaystyle {\begin{aligned}\sum _{j\neq i}\pi _{i}q'_{ij}&=\sum _{j\neq i}q_{ij}\quad \forall i\in S\\\pi _{i}q_{ij}&=\pi _{j}q_{ji}'\quad \forall i,j\in S\end{aligned}}}$

then q'_ij are the rates for the reversed process and π_i are the stationary distribution for both processes.

Proof

Given the assumptions made on the q_ij and π_i we can see

${\displaystyle \sum _{i\neq j}\pi _{i}q_{ij}=\sum _{i\neq j}\pi _{j}q'_{ji}=\pi _{j}\sum _{i\neq j}q_{ji}=-\pi _{j}q_{jj}}$

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

References

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.