A single machine is available to process a collection of stochastic jobs. Processing is preemptive and so (for example) the machine is allowed to switch away from a job before completion, should that prove advantageous. The jobs are deteriorating in the sense that their processing requirements grow (at job-specific rates) as they await processing. This phenomenon might be expected to enhance the status of non-preemptive policies. The primary objective of the paper is to find conditions which are sufficient to ensure the existence of a permutation policy to minimise the expected makespan. We also derive results for a weighted flowtime criterion. Applications of such models to the control of queues and to communication systems have been cited by other authors.