The problem of determining whether it is possible for a set of “free-running” processes to become deadlocked is considered. It is assumed that any request by a process is immediately granted as long as there are enough free resource units to satisfy the request. The question of whether or not there exists a polynomial algorithm for predicting deadlock in a “claim-limited” serially reusable resource system has been open. An algorithm employing a network flow technique is presented for this purpose. Its running time is bounded byO(mn1.5) if the system consists ofnprocesses sharingmtypes of serially reusable resources.