On quasi-reversibility and local balance: An alternative derivation of the product-form results

We are concerned with a queueing network, described by a continuous-time Markov chain, in which each node is quasi-reversible. A new class of local balance equations is derived for the Markov chain with respect to a product-form distribution, which simultaneously provides an alternative and short proof for product form results of queueing networks with customers and signals. Furthermore, if each node is internally balanced, i.e., the total arrival rate equals the total departure rate for each node, the Markov chain is locally balanced in the ordinary sense. Our arguments reveal a close relationship between quasi-reversibility and local balance, and provide further insights into how transition rates are balanced in product form queueing networks.