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 non-negative numbers q'ij and a positive measure π that satisfy the following conditions:[1]
- [math]\displaystyle{ \begin{align} \sum_{j \in S} q_{ij} &= \sum_{j \in S} q'_{ij} \quad \forall i\in S\\ \pi_i q_{ij} &= \pi_jq_{ji}' \quad \forall i,j \in S, \end{align} }[/math]
then q'ij 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 qij and π we have
- [math]\displaystyle{ \sum_{i \in S} \pi_i q_{ij} = \sum_{i \in S} \pi_j q'_{ji} = \pi_j \sum_{i \in S} q'_{ji} = \pi_j \sum_{i \in S} q_{ji} =\pi_j, }[/math]
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
- ↑ 1.0 1.1 Boucherie, Richard J.; van Dijk, N. M. (2011). Queueing Networks: A Fundamental Approach. Springer. p. 222. ISBN 144196472X.
- ↑ Kelly, Frank P. (1979). Reversibility and Stochastic Networks. J. Wiley. p. 22. ISBN 0471276014. http://www.statslab.cam.ac.uk/~frank/BOOKS/kelly_book.html.
- ↑ Walrand, Jean (1988). An introduction to queueing networks. Prentice Hall. p. 63 (Lemma 2.8.5). ISBN 013474487X.
- ↑ Kelly, F. P. (1976). "Networks of Queues". Advances in Applied Probability 8 (2): 416–432. doi:10.2307/1425912.
- ↑ 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.
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