Stationary Distribution Convergence for a Multiclass Single-server Queue in Heavy Traffic

In the author’s recent paper [11], it is established that in a multiclass single-server queue with regular service disciplines, the stationary distributions and their moments for workload as well as for queue length can be approximated by appropriate exponential distributions and their moments in the heavy-traffic regime. In this work, relaxing the assumption of moment generating function on the primitives in that paper to the moment condition of second-order or higher-order, we obtain the corresponding approximation result in a multiclass single-server queue. The key to our analysis is to use the framework of Budhiraja and Lee [4] in which under such weak moment assumption, the tightness of stationary scaled queue length for singleclass (i.e., generalized Jackson) queueing networks is established for their stationary heavy-traffic analysis. For a multiclass single-server queue, we obtain the tightness of stationary scaled workload to show that state-space collapse occurs in the heavy-traffic regime in stationarity, from which the desired approximation result follows.