Diffusion models or mean-field models have been used to approximate many stochastic dynamical systems. A functional strong law of large numbers or a functional central limit theorem justifies such an approximation. Such a result, however, does not justify the convergence of the equilibria of pre-limit systems to the equilibrium of a limit system. In this talk, I will touch on three recently developed methods for justifying equilibrium convergence in the setting of bandwidth sharing networks and multiclass queueing networks, with a focus on moment generating function method and the state-space-collapse.