The problem of stochastic deadline scheduling is considered. A constrained Markov decision process model is introduced in which jobs arrive randomly at a service center with stochastic job sizes, rewards, and completion deadlines. The service provider faces random processing costs, convex noncompletion penalties, and a capacity constraint that limits the simultaneous processing of jobs. Formulated as a restless multiarmed bandit problem, the stochastic deadline scheduling problem is shown to be in-dexable. A closed-form expression of the Whittle's index is obtained for the case when the processing costs are constant. An upper bound on the gap-to-optimality for the Whittle's index policy is obtained, and it is shown that the bound converges to zero as the job arrival rate and the number of available processors increase simultaneously to infinity.