On the equivalence between multiclass PS-type scheduling policies

Consider a single server queue serving a multiclass population. Some popular scheduling policies for such a system (and of interest in this paper) are the discriminatory processor sharing (DPS), discriminatory random order service (DROS), generalized processor sharing (GPS) and weighted fair queueing (WFQ). The aim of this paper is to show a certain equivalence between these scheduling policies for the special case when the multiclass population have identical and exponential service requirements. In fact, we show the equivalence between two broader classes of policies that generalize the above mentioned four policies. We specifically show that the sojourn time distribution for a customer of a particular class in a system with the DPS (GPS) scheduling policy is a constant multiple of the waiting time distribution of a customer of the same class in a system with the DROS (respectively WFQ) policy.