The duration of time for which each application locks each shared resource is critically important in composing multiple independently-developed applications upon a shared "open" platform. The concept of resource hold time (RHT) -- the largest length of time that may elapse between the instant that an application system locks a resource and the instant that it subsequently releases the resource--is formally defined and studied in this paper. An algorithm is presented for computing resource hold times for every resource in an application that is scheduled using Earliest Deadline First scheduling, with resource access arbitrated using the Stack Resource Policy. An algorithm is presented for decreasing these RHT?s without changing the semantics of the application or compromising application feasibility.