ON THE STABILITY OF GREEDY POLLING SYSTEMS WITH GENERAL SERVICE POLICIES

We consider a polling system with a finite number of stations fed by c ompound Poisson arrival streams of customers asking for service. A ser ver 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 ran dom level C < x before the server has completed the B services. The ra ndom variable B may also take the value co so that the server has to p rovide service as long as the queue length has reached size C. The dis tribution H-i,H-x of the pair (B,C) may depend on i and x while the se rvice time distribution is allowed to depend on i. The station to be v isited next is chosen among some neighbors according to a greedy polic y. That is to say that the server always tries to walk to the fullest station in his well-defined neighborhood. Under appropriate independen ce assumptions two conditions are established that are sufficient for stability and sufficient for instability. Some examples will illustrat e the relevance of our results.