Recurrence and transience of open Jackson networks with balanced nodes

Consider an open Jackson network with Poisson arrivals and exponential services, in which no node is overloaded and exactly~$k_0$ nodes have load equal to~$1$. We prove that the queue-length process is recurrent if and only if $k_0\le 2$. For that, we show that the network satisfies a sector condition, so its capacities are comparable to those of its symmetrisation; the latter is a reversible network whose recurrence/transience can be dealt with by standard means. We also discuss what the Lyapunov-function method gives in this setting; in particular, we show that every balanced queue is empty at arbitrarily large times, also in the transient case.