Optimal and equilibrium allocations in a discriminatory processor sharing system

In this paper, we consider the problem of assigning heterogeneous customers to a discriminatory processor sharing (DPS) system with M service classes. A type of a customer is determined by its cost per unit waiting time. In the first part of this paper, we consider the problem of optimally assigning the heterogeneous customers to the service classes to minimize a social cost function. We show that when there is a continuum of customer types, the optimal allocation is of a threshold type. For the special case of M = 2 we also explicitly characterize the threshold. When the customer types are finite, we show that at the optimal allocation, customers with the highest (resp. lowest) delay cost will always have to be routed to the service class with highest (resp. lowest) weight. Further, when there are three customer types, the middle class is indifferent to the choice of service class. Next we consider a DPS system that has different admission charges for different service classes and the customers can choose their service class for individual optimization. Here we show that if the prices are fixed, the individually optimally routing policy is a threshold policy. Finally, we investigate the use of admission price so that the resulting equilibrium allocation coincides with the optimum allocation that minimizes the social cost.