Stable strategies for processor sharing systems

For a processor sharing model with a Poisson arrival process and general and independent service requirement, under a standard cost structure a join/do not join stable policy for each of the following two cases is looked for: (1) when each job knows its service requirement, and (2) when the jobs belong to various classes which differ in their expected service requirement and each job knows its class. For the first case, it is shown that there exists a unique pure stable and symmetric strategy under which jobs join the system if and only if their requirement is smaller than some threshold. A similar phenomenon exists in the second case but randomization may be required. Moreover, the stable policies are computed explicitly. The case of social optimization is considered as well.