G-Networks with multiple class negative and positive customers

A new class of queueing networks with “negative and positive” customers was introduced and shown to have a non-standard product form (Gelenbe E., Product form queueing networks with negative and positive customers, Journal of Applied Probability, Vol. 28, p. 656-63, 1991); since then this model has undergone several generalizations. Positive customers are identical to the usual customers of a queueing network, while a negative customer which arrives at a queue simply destroys a positive customer. We call these generalized queueing networks G-Networks. In this paper we extend the basic model to the case of multiple class of positive customers, and multiple class of negative customers. As in known multiple class queueing networks a positive customer class is characterized by the routing probabilities and the service rate parameter at each service center while negative customers of different classes may have different “customer destruction” capabilities. All service time distributions are exponential and the service centers can be of the following types: FIFO (first-in-first-out), LIFO/PR (last-in-first-out with preemption), and PS (processor sharing), with class dependent service rates