This paper offers a new physics-based approach to model and resiliently control traffic congestion. We model the traffic congestion as a mass conservation problem and provide discussions and proofs for the feasibility of the conservation-based traffic dynamics. The paper spatially discretizes the governing continuity equation by using a directed graph with the nodes classified as (i) interior nodes, (ii) boundary inlet nodes, (iii) boundary outlet nodes, and (iv) anomalous nodes. At the interior nodes, the traffic dynamics is modeled as a probabilistic process. At the inlet boundary nodes, the traffic inflow rates can be planned and controlled but they must satisfy certain equality and inequality constraints. This paper assumes that the traffic inflow and outflow rates are identical at the outlet boundary nodes, which in turn implies that outlet boundary nodes have no dynamics. Furthermore, the traffic coordination cannot be controlled at the anomalous nodes and they are modeled as disturbance sources. We apply the model predictive control approach to effectively control the traffic congestion through the inlet boundary nodes in the presence of anomalies. The objective of the control problem is to assign the boundary traffic inflow rate such that traffic density is uniformly distributed across the traffic nodes. The boundary control inputs are assigned as the solution of a constrained quadratic programming problem.