Structure-reversibility and departure functions of queueing networks with batch movements and state dependent routing

We consider characterizations of departure functions in Markovian queueing networks with batch movements and state-dependent routing in discrete-time and in continuous-time. For this purpose, the notion of structure-reversibility is introduced, which means that the time-reversed dynamics of a queueing network corresponds with the same type of queueing network. The notion is useful to derive a traffic equation. We also introduce a multi-source model, which means that there are different types of outside sources, to capture a wider range of applications. Characterizations of the departure functions are obtained for any routing mechanism of customers satisfying a recurrent condition. These results give a unified view to queueing network models with linear traffic equations. Furthermore, they enable us to consider new examples as well as show limited usages of this kind of queueing networks.