The Variational Theory (VT) expresses the LWR model as a least cost path problem. Recent researches have shown that this problem can be simply applied on a graph with a minimal number of nodes and edges when the fundamental diagram is triangular (sufficient variational graph - SVG). Such a graph accounts for traffic signal settings on an urban arterial and leads to mean traffic states for the total arterial in free-flow or congested stationary conditions. The Macroscopic Fundamental Diagram (MFD) can then be directly estimated. In this paper, we extend this method to provide the complete distribution of deterministic travel times observed on an arterial. First, we will show how to obtain a tight estimation of the arterial capacity by properly identifying the most constraining part of the SVG. Then, we will show that a modified version of the SVG allows the exact calculation of the cumulative count curves at the entry and exit of an arterial. It is finally possible to derive the full travel time distributions for any dynamic conditions.