On the stability of greedy polling systems with general service policies

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. Upon arrival at a nonempty station i, say, with x > 0 waiting customers, the server tries to serve there a random number B of customers if the queue length has not reached a random level C < x before the server has completed the B services. The random variable B may also take the value ∞ so that the server has to provide service as long as the queue length has reached size C. The distribution H_{i, x} of the air (B, C) may depend on i and x while the service time distribution is allowed to depend on i. The station to be visited next is chosen among some neighbors according to a greedy policy. That is to say that the server always tries to walk to the fullest station in his well-defined neighborhood. Under appropriate independence assumptions two conditions are established that are sufficient for stability and sufficient for instability. Some examples will illustrate the relevance of our results.