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Based on the complex network theory, we present formulae for estimating the maximum of betweenness centrality of edge in a network after the random breakdowns happen. Betweenness centrality of edge is defined as the number of shortest paths traveling through an edge given a communication protocol. The edge possessing the max betweenness tends to be congested in communication process, so providing more precise estimation of edge betweenness can give more exact estimation to the capacity of distributing traffic to a network. Finally, We confirm the formula by simulation analysis. Little attention has been paid to the effect of random breakdown on the edge betweenness, so the proposed formula provides a new solution to estimating edge betweenness and also it can be useful for traffic engineering and network planning.