Queueing Systems: Theory and Applications
State space collapse with application to heavy traffic limits for multiclass queueing networks
Queueing Systems: Theory and Applications
Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic
Mathematics of Operations Research
Diffusion limit of a two-class network: stationary distributions and interchange of limits
ACM SIGMETRICS Performance Evaluation Review
Queueing Systems: Theory and Applications
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Recently Gamarnik and Zeevi (Ann. Appl. Probab. 16:56---90, 2006) and Budhiraja and Lee (Math. Oper. Res. 34:45---56, 2009) established that, under suitable conditions, a sequence of the stationary scaled queue lengths in a generalized Jackson queueing network converges to the stationary distribution of multidimensional reflected Brownian motion in the heavy-traffic regime. In this work we study the corresponding problem in multiclass queueing networks (MQNs).In the first part of this work we consider the MQNs for which the fluid stability is valid, state-space collapse is exhibited under suitable initial conditions and a heavy traffic limit theorem holds. For such MQNs we establish that, under the assumption of the tightness of a sequence of stationary scaled workloads, the sequence converges to the stationary distribution of semimartingale reflecting Brownian motion in the heavy-traffic regime. The key to the proof is to show that state-space collapse occurs in the heavy-traffic regime in stationarity under the assumption of tightness.In the second part, using the result obtained, it is shown that such a convergence of stationary workload holds for a multiclass single-server queue with feedback routing, where the tightness is proved by the Lyapunov function method developed in (Gamarnik and Zeevi in Ann. Appl. Probab. 16:56---90, 2006).