Stability and weakly convergent approximations of queueing systems on a circle

We rst consider a `non-greedy' queueing system on a circle. We present a new and very simple proof of the stability of this system (under the appropriate condition) based on the average travel times between customers. Next we show that the same non-greedy system, with a restricted number of customers, converges weakly to this system when the restricted number goes to innnity. Finally we consider a polling network with nitely many service stations, in which the server has a `greedy' service strategy. Under the appropriate condition, we give a new simple proof of the stability of this system.