Top-k query processing techniques provide two main advantages for unstructured peer-to-peer (P2P) systems. First they avoid overwhelming users with too many results. Second they reduce significantly network resources consumption. However, existing approaches suffer from long waiting times. This is because top-k results are returned only when all queried peers have finished processing the query. As a result, query response time is dominated by the slowest queried peer. In this paper, we address this users' waiting time problem. For this, we revisit top-k query processing in P2P systems by introducing two novel notions in addition to response time: the stabilization time and the cumulative quality gap. Using these notions, we formally define the as-soonas-possible (ASAP) top-k processing problem. Then, we propose a family of algorithms called ASAP to deal with this problem. We validate our solution through implementation and extensive experimentation. The results show that ASAP significantly outperforms baseline algorithms by returning final top-k result to users in much better times.