Stability of polling systems with exhaustive service policies and state-dependent routing

We consider a polling system with a finite number of stations fed by compound Poisson arrival streams of customers asking for service. A server travels through the system and upon arrival at a station the server serves all waiting customers until the queue is empty, where the service time distribution depends on the station. The choice of the station to be visited next as well as the corresponding walking time may depend on the whole current state. Examples are systems with a greedy-type routing mechanism. Under appropriate independence assumptions it is proved that the system is stable if and only if the workload is less than 1.