Stability of multi-class queueing networks with infinite virtual queues

We generalize the standard multi-class queueing network model by allowing both standard queues and infinite virtual queues which have infinite supply of work. We pose the general problem of finding policies which allow some of the nodes of the network to work with full utilization, and yet keep all the standard queues in the system stable. Towards this end we show that re-entrant lines, systems of two re-entrant lines through two service stations, and rings of service stations can be stabilized with priority policies under certain parameter restrictions. The analysis throughout the paper depends on model and policy and illustrates the difficulty in solving the general problem.