In polling systems, $M \geq 2$ queues are visited by a single server in cyclic order. These systems model such diverse applications as token-ring communication networks and cyclic production systems. We study polling systems with exhaustive service and zero switchover (walk) times. Under standard heavy-traffic assumptions and scalings, the total unfinished work converges to a one-dimensional reflected Brownian motion, whereas the workloads of individual queues change at a rate that becomes infinite in the limit. Although it is impossible to obtain a multidimensional limit process in the usual sense, we obtain an "averaging principle" for the individual workloads. To illustrate the use of this principle, we calculate a heavy-traffic estimate of waiting times.
"Polling Systems with Zero Switchover Times: A Heavy-Traffic Averaging Principle." Ann. Appl. Probab. 5 (3) 681 - 719, August, 1995. https://doi.org/10.1214/aoap/1177004701