A mathematical framework for delay analysis in single source networks

This article presents a mathematical framework for modeling heterogeneous flow networks with a single source and multiple sinks. The traffic is differentiated by its destination (i.e. Lagrangian flow) and different flow groups are assumed to satisfy the first-in-first-out (FIFO) condition at each junction. We show that our model leads to a well-posed problem for computing the dynamics of the system and prove that the solution is unique through a mathematical derivation of the model properties. The framework is then used to analytically prescribe the delays at each junction of the network and across any sub-path, which is one of the contributions of the article. This is a critical requirement when solving control and optimization problems over the network, such as system optimal network routing and solving for equilibrium behavior. In fact, the framework provides analytical expressions for the delay at any node or sub-path as a function of the inflow at any upstream node. Furthermore, the model can be solved numerically using a very simple and efficient feed forward algorithm. We demonstrate the versatility of the framework by applying it to a diverge junction with complex junction dynamics.