In the seminal paper of Gamarnik and Zeevi , the authors justify the steady-state diffusion approximation of a generalized Jackson network (GJN) in heavy traffic. Their approach involves the so-called limit interchange argument, which has since become a popular tool employed by many others who study diffusion approximations. In this paper we illustrate a novel approach by using it to justify the steady-state approximation of a GJN in heavy traffic. Our approach involves working directly with the basic adjoint relationship (BAR), an integral equation that characterizes the stationary distribution of a Markov process. As we will show, the BAR approach is a more natural choice than the limit interchange approach for justifying steady-state approximations, and can potentially be applied to the study of other stochastic processing networks such as multiclass queueing networks.
"Heavy traffic approximation for the stationary distribution of a generalized Jackson network: The BAR approach." Stoch. Syst. 7 (1) 143 - 196, 2017. https://doi.org/10.1214/15-SSY199