Sojourn times in (discrete) time shared systems and their continuous time limits

We study the mean sojourn times in two M/G/1 weighted round–robin systems: the weight of a customer at any given point in time in the first system is a function of its age (imparted service), while in the second system the weight is a function of the customer's remaining processing time (RPT). We provide a sufficient condition under which the sojourn time of a customer with large service requirement (say, x) and that arrives in the steady state is close to that of a customer which starts a busy period and has the same service requirement. A sufficient condition is then provided for continuity of the performance metric (the mean sojourn time) as the quanta size in the discrete time system converges to 0. We then consider a multi–class system and provide relative ordering of the mean sojourn times among the various classes.