Flow equivalence and stochastic equivalence in G-networks

G-networks are novel product form queuing networks that, in addition to ordinary customers, contain unusual entities such as ‘negative customers’ which eliminate normal customers, and ‘triggers’ that move other customers from some queue to another. Recently we introduced one more special type of customer, a ‘reset’, which may be sent out by any server at the end of a service epoch, and that will reset the queue to which it arrives into its steady state when that queue is empty. A reset which arrives to a non-empty queue has no effect at all. The sample paths of a system with resets is significantly different from that of a system without resets, because the arrival of a reset to an empty queue will provoke a finite positive jump in queue length which may be arbitrarily large, while without resets positive jumps are only of size +1 and they occur only when a positive customer arrives to a queue. In this paper we review this novel model, and then discuss its traffic equations. We introduce the concept of ‘stationary equivalence’ for queueing models, and of ‘flow equivalence’ for distinct queueing models. We show that the flow equivalence of two G-networks implies that they are also stationary equivalent. We then show that the stationary probability distribution of a G-network with resets is identical to that of a G-network without resets whose transition probabilities for positive (ordinary) customers has been increased in a specific manner. Our results show that a G-network with resets has the same form of traffic equations and the same joint stationary probability distribution of queue length as that of a G-network without resets.