Stochastic networks are used as models for complex processing systems involving dynamic interactions subject to uncertainty. Applications arise in high-tech manufacturing, the service industry, telecommunications, computer systems and bioengineering. The control and analysis of such networks present challenging mathematical problems. In this talk, a concrete application will be used to illustrate a general approach to the study of stochastic processing networks based on deriving more tractable approximate models. Specifically, we will consider a model of Internet congestion control in which processing can involve the simultaneous use of several resources (or links), a phenomenon that is not well understood. Elegant fluid and diffusion approximations will be derived and used to study the performance of this model. A key insight from this analysis is a geometric representation of the consequences of using a "fair" policy for the sharing of resources. The talk will conclude with a summary of the current status and description of open problems associated with approximate models for general stochastic processing networks.