Generalized network tomography (GNT) deals with estimation of link performance parameters for networks with arbitrary topologies using only end-to-end path measurements of pure unicast probe packets. In this paper, by taking advantage of the properties of generalized hyperexponential distributions and polynomial systems, a novel algorithm to infer the complete link metric distributions under the framework of GNT is developed. The significant advantages of this algorithm are that it does not require: i) the path measurements to be synchronous and ii) any prior knowledge of the link metric distributions. Moreover, if the path-link matrix of the network has the property that every pair of its columns are linearly independent, then it is shown that the algorithm can uniquely identify the link metric distributions up to any desired accuracy. Matlab based simulations have been included to illustrate the potential of the proposed scheme.