Kelly's lemma

Statement

For a continuous time Markov chain with an at most countable 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 \pi that satisfy the following conditions: ::\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} then q_{ij}' are the rates for the reversed process and \pi is proportional to the stationary distribution for both processes.

Proof

Given the assumptions made on the q_{ij} and \pi we have :: \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, so the global balance equations are satisfied and the measure \pi is proportional to the stationary distribution of the original process. By symmetry, the same argument shows that \pi is also proportional to the stationary distribution of the reversed process.

References

References

1. (2011). "Queueing Networks: A Fundamental Approach". Springer.
2. Kelly, Frank P.. (1979). "Reversibility and Stochastic Networks". J. Wiley.
3. Walrand, Jean. (1988). "An introduction to queueing networks". Prentice Hall.
4. (1976). "Networks of Queues". Advances in Applied Probability.
5. Asmussen, S. R.. (2003). "Applied Probability and Queues".