We propose an approximation algorithm for the problem of Fault-Tolerant Facility Location which is implemented in a distributed and asynchronous manner within O(n) rounds of communication. Here n is the number of vertices in the network. As far as we know, the performance guarantee of similar algorithms (centralized) remains unknown except a special case where all cities have a uniform connectivity requirement. In this paper, we assume the shortest-path routing scheme deployed, as well as a constant (given) size of R, which represents the distinct levels of fault-tolerant capability provided by the system (Le distinct connectivity requirements), and prove that the cost of our solution is no more than \R\ F* + 2 • C* in the general case, where F* and C* are respectively the facility cost and connection cost in an optimal solution. Further more, extensive numerical experiments showed that the quality of our solutions is comparable to the optimal solutions when \R\ is no more than 10.