In this article, we prove that for any probability distribution $μ$ on $\mathbb{N}$ one can construct a two-sided stationary version of the infinite-bin model (an interacting particle system introduced by Foss and Konstantopoulos) with move distribution $μ$. Using this result, we obtain a new formula for the speed of the front of infinite-bin models, as a series of positive terms. This implies that the growth rate $C(p)$ of the longest path in a Barak-Erdős graph of parameter $p$ is analytic on $(0,1]$.