A Barak-Erdös graph is a directed acyclic version of an Erdös-Rényi graph. It is obtained by performing independent bond percolation with parameter p on the complete graph with vertices {1, ..., n}, in which the edge between two vertices i < j is directed from i to j. The length of the longest path in this graph grows linearly with the number of vertices, at rate C(p). In this article, we use a coupling between Barak-Erdös graphs and infinite-bin models to provide explicit estimates on C(p). More precisely, we prove that the front of an infinite-bin model grows at linear speed, and that this speed can be obtained as the sum of a series. Using these results, we prove the analyticity of C for p > 1/2, and compute its power series expansion. We also obtain the first two terms of the asymptotic expansion of C as p → 0, using a coupling with branching random walks.