Parameters characterizing safety critical systems are generally assigned very conservative values for reasons of safety assurance. Provisioning computing resources on the basis of such conservatively assigned parameter values can lead to system implementations that make inefficient use of platform resources during run time. We address the problem of achieving more efficient implementations of sporadic task systems where, in addition to a conservatively assigned value for the period parameter of each task, we also have a more optimistic (i.e., larger), but perhaps incorrect, prediction of this value. We devise an algorithm that executes the system more efficiently during runtime if the prediction is correct, without compromising safety if it turns out to be incorrect.