State space collapse for asymptotically critical multi-class fluid networks

We consider a class of fluid queueing networks with multiple fluid classes and feedback allowed, which are fed by N heavy tailed ON/OFF sources. We study the asymptotic behavior when N → ∞ of these queueing systems in a heavy traffic regime (that is, when they are asymptotically critical). As performance processes we consider the workload W N (the amount of time needed for each server to complete processing of all the fluid in queue), and the fluid queue Z (the quantity of each fluid class in the system). We show the convergence of √ N W N and √ N Z (to Ŵ and Ẑ) in heavy traffic if state space collapse (SSC) holds. (SSC) is a condition that establishes a relationship between those components of Ẑ that correspond to fluid classes processed by the same server, which implies that Ẑ = ∆ Ŵ for a deterministic lifting matrix ∆. Our main contribution is to prove that assuming that the other hypotheses are true, (SSC) is not only sufficient for this convergence, but necessary. Furthermore, we prove that processes Ŵ and Ẑ, conveniently scaled in time, converge to W (a reflected fractional Brownian motion) and Z (= ∆ W ) . We illustrate the application of our results with some examples including a tandem queue.