Networks of Queues with Customers, Signals and Arbitrary Service Time Distributions

We consider a network of queues with multiple classes of customers, signals, and arbitrary service time distributions. The signals bring commands to the service nodes and may trigger customers to move instantly within the network. We consider symmetric service disciplines (e.g., processor sharing, LIFO preemptive, infinite server) similar to those in F. Kelly (Kelly, F. 1979. Reversibility and Stochastic Networks. John Wiley, New York.), and show that when each node has a single server operating under a symmetric service discipline, the stationary distribution has a product-form solution. The product form, however, depends on the service time distributions of customers at each node beyond their means. This is in contrast to the insensitivity result of the conventional Jackson networks and can be attributed to the effect of signals. When the nodes have multiple servers in parallel, the usual definition of signals does not lead to a product-form solution. We propose a new definition of signals that depends on the service effort and provides a product-form solution.