Instability of FIFO Queueing Networks

Consider a queueing network with customers arriving according to a rate-1 Poisson process. Each customer proceeds along the same prescribed route, waiting at the different queues until exiting from the system. The service times are assumed to be independent and exponentially distributed. Individual queues may be visited more than once by a customer, with the mean service time perhaps depending on the stage along the route. The network is assumed to be first-in, first-out. An obvious necessary condition for such a queueing network to have an equilibrium distribution is that the sum of the mean service times at each queue be less than 1. We show by means of a class of examples that this condition does not suffice, these networks being unstable. Each such network possesses two queues, the first with one slow and one quick stage, and the other with one slow and numerous quick stages.