Geometric equilibrium distributions for queues with interactive batch departures

Gelenbe et al. consider single server Jackson networks for queues which contain both positive and negative customers. A negative customer arriving to a non-empty queue causes the number of customers in that queue to decrease by one, and has no effect on any empty queue, whereas a positive customer arriving at a queue will always increase the queue length by one. Gelenbe et al. show that a geometric product form equilibrium distribution prevails for this network. Applications for these types of networks can be found in systems incorporating resource allocations and in the modelling of decision making algorithms, neural networks and communications protocols. In this paper the authors extend the results by allowing customer arrivals to the network, or the transfer between queues of a single positive customer in the network to trigger the creation of a batch of negative customers at the destination queue. This causes the length of the queue to decrease by the size of the created batch or the size of the queue, whichever is the smallest. The probability of creating a batch of negative customers of a particular size due to the transfer of a positive customer can depend on both the source and destination queue. The authors give a criterion for the validity of a geometric product form equilibrium distribution for these extended networks. When such a distribution holds it satisfies partial balance equations which are enforced by the boundaries of the state space. Furthermore it will be shown that these partial balance equations relate to traffic equations for the throughputs of the individual queues.