This thesis explores the BAR approach applied on various of stochastic processing networks. The weighted-workload-task-allocation (WWTA) load-balancing policy is known to be throughput optimal for parallel-server systems. The first part of the thesis concerns the steady-state performance approximation of WWTA policy in heavy traffic. Instead of proving a stochastic process limit followed by a limit interchange -- a method that dominates the literature, our method works directly with pre-limit BAR that characterizes the stationary distribution of each pre-limit system. Under a complete-resource-pooling condition, we prove that WWTA achieves a "strong form" of state-space collapse in heavy traffic and that each scaled workload converges in distribution to an exponential random variable, whose parameter is explicitly given by system primitives. Various steady-state performance measures are shown to be approximated from this exponential random variable. In the second part, we prove that under a multi-scale heavy traffic condition, the stationary distribution of the multi-scaled queue length process in any generalized Jackson network has a product-form limit. Each component in the product-form has an exponential distribution, corresponding to the Brownian approximation of a single station queue. The "single station" can be constructed precisely and has a good intuitive interpretation.