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 urban arterial and provides mean traffic states for the whole arterial in free-flow and congestion for stationary situations. The Macroscopic Fundamental Diagram (MFD) can then be directly estimated. In this paper, we extend this method to provide the complete distribution of travel times observed on an arterial for stationary conditions. First, we will show how to obtain a tight estimation of an arterial capacity by properly identifying the most constraining part of the SVG. Then, we will show that a slightly modified version of the SVG makes it possible to calculate the cumulative vehicle curves at the entry and exit of an arterial and finally to determine the full travel time distributions for either free-flow or congested conditions.