A two-queue polling model with priority on one queue and heavy-tailed On/Off sources: a heavy-traffic limit

We consider a single‐server polling system consisting of two queues of fluid with arrival process generated by a big number of heavy‐tailed On/Off sources, and application in road traffic and communication systems. Class‐j fluid is assigned to queue j, $j=1,2$ . Server 2 visits both queues to process or let pass the corresponding fluid class. If there is class‐2 fluid in the system, it is processed by server 2 until the queue is empty, and only then server 2 visits queue 1, revisiting queue 2 and restarting the cycle as soon as new class‐2 fluid arrives, with zero switchover times. Server 1 is an ‘extra’ server which continuously processes class‐1 fluid (if there is any). During the visits of server 2 to queue 1, class‐1 fluid is simultaneously processed by both servers (possibly at different speeds). We prove a heavy‐traffic limit theorem for a suitable workload process associated with this model. Our limit process is a two‐dimensional reflected fractional Brownian motion living in a convex polyhedron. A key ingredient in the proof is a version of the Invariance Principle of Semimartingale reflecting Brownian motions which, in turn, is also proved.