A multiclass feedback queue in heavy traffic

We consider a single station queueing system with several customer classes. Each customer class has its own arrival process. The total service requirement of each customer is divided into a (possibly) random number of service quanta, where the distribution of each quantum may depend on the customer's class and the other quanta of that customer. The service discipline is round-robin, with random quanta. We prove a heavy traffic limit theorem for the above system which states that as the traffic intensity approaches unity, properly normalized sequences of queue length and sojourn time processes converge weakly to one-dimensional reflected Brownian motion.