In this paper, we focus on convergence packet routing techniques in a network, obtained from an Eulerian routing in the digraph modeling the target network. Given an Eulerian circuit $\cal C$ in a digraph $G$, we consider the maximal number $diamW_{\cal C}$ of arcs that a packet has to follow on $\cal C$ from its origin to its destination (we talk about the {\em ending guarantee} of the routing). We consider the {\em Eulerian diameter of $G$} as defined by ${\cal E}(G)=\min\limits_{{\cal C}? Eul(G)} diamW_{\cal C}$, where $Eul(G)$ is the set of all the Eulerian circuits in $G$. After giving a preliminary result about the complexity of finding ${\cal E}(G)$ for any digraph $G$, we give some lower and upper bounds of this parameter. We conclude by giving some families of digraphs having good Eulerian diameter.