Kelly's lemma explained

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 non-negative numbers q'_ij and a positive measure π that satisfy the following conditions:

\begin{align} \sum _{j} q _{ij} & = \sum _{j} q' _{ij} \forall i\in S\\ \pi _{i} q _{ij} & = \pi _{jq} _{ji} ' \forall i,j \in S, \end{align} then qij are the rates for the reversed process and π is proportional to the stationary distribution for both processes.

Proof

Given the assumptions made on the q_ij and π we have

\sum _{i} \pi _{i} q _{ij} = \sum _{i} \pi _{j} q' _{ji} = \pi _{j} \sum _{i} q' _{ji} = \pi _{j} \sum _{i} q _{ji} = \pi _{j,} 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.

Notes and References

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