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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2$$\end{document}. 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.