Fair queuing and other probabilistic allocation methods

A server processes one job per unit of time and randomly schedules the jobs requested by a given set of users; each user may request a different number of jobs. Fair queuing (Shenker 1989) schedules jobs in successive round-robin fashion, where each agent receives one unit in each round until his demand is met and the ordering is random in each round. Fair queuing(*), the reverse scheduling of fair queuing, serves first (with uniform probability) one of the users with the largest remaining demand. We characterize fair queuing* by the combination of lower composition-LC-(the scheduling sequence is history independent), demand monotonicity-DM-(increasing my demand cannot result in increased delay) and two equity axioms, equal treatment ex ante-ETEA (two identical demands give the same probability distribution of service) and equal treatment ex post-ETEP (two identical demands must be served in alternating fashion). The set of dual axioms (in which ETEA and ETEP are unchanged) characterizes fair queuing. We also characterize the rich family of methods satisfying LC, DM, and the familiar consistency-CSY-axiom. They work by fixing a standard of comparison (preordering) between a demand of x(t) units by agent i and one of x(l) units by agent j. The first job scheduled is drawn from the agents whose demand has the highest standard.