Fast-Service Polling Systems with Intensive Input Flows and Constant Switch Times

A queuing system is considered with a single server which serves $N$ queues in cyclic order. Fast customer arrivals and fast service are assumed as well as a finite time for the server to switch from one queue to the next. The process in $N$-dimensional space is studied with the value at any time instant being the queue lengths at that instant. This process is proved to coincide asymptotically with some nonrandom function taking values in $N$-dimensional space.