The Internet composes of thousands of Autonomous System (ASes). The Border Gateway Protocol (BGP) is the standard protocol for sharing inter-domain routing information. Unlike OSPF and IS-IS, BGP allows an AS to use a lot of attributes to express semantic rich routing policies that are consistent with its desired economic, business, performance, and security goals. However, the expressiveness could cause to delay convergence or even divergence in BGP. Recent work do not rigorously analyze the impact of the general routing policies on the convergence condition and convergence time of BGP, especially considering the widely used Multi-Exit Discriminator (MED) attribute. In this paper, we will fill this gap and give the rigorous analysis on the impact of the general routing policies on the convergence condition and convergence time of BGP, including MED attribute. We first introduce a timeless model to represent BGP with the general routing policies including the MED attribute. By incorporating the timeless model we derive a sufficient condition on these general routing policies for robust convergence of BGP. We then extend the timeless model to the real-time model by adding the edge delay. Finally, we find an upper bound on convergence time of BGP by incorporating the real-time model.