Multiple Class Symmetric G-networks with Phase Type Service Times

We consider a queueing network of symmetric G-queues with customers and signals. Since the seminal papers by Gelenbe in the early 1990s (Gelenbe, E. (1991) Product-form queueing networks with negative and positive customers. J. Appl. Probab., 28, 656-663; Gelenbe, E. (1993); G-networks with instantaneous customer movement. J. Appl. Probab., 30, 742-748; Gelenbe, E. (1993) G-Networks with signals and batch removal. Probab. Eng. Inform. Sci., 7, 335-342), generalized networks of queues have received considerable attention. But most papers assume to obtain a product form such that the service times follow exponential distributions. Here, we propose a new generalization of this model with phase-type service times. We also assume a new type of signal. When a signal enters a queue, it changes the phase of the customer in service if there is any. As usual, after completion of its service, a customer moves to another queue and may become a signal. We prove that the steady-state distribution for such a network of queues has a product form solution.