Scheduling Networks of Queues: Heavy Traffic Analysis of a Bi-Criteria Problem

We consider bi-criteria scheduling problems for three queueing systems (a single queue, a two-station closed network, and a two-station network with controllable inputs) populated by various customer types. The objective is to minimize the long run expected average value of a linear combination of the customer sojourn time and the sojourn time inequity. The inequity at time t is the sum of squares of the pairwise differences of the total number of customers in the system at time t of each type divided by their respective arrival rates. Brownian approximations to these three scheduling problems are solved, and the solutions are interpreted in order to obtain scheduling policies. Simulation results show that the second objective criteria tends to equalize the mean sojourn times of the various customer types, and may lead to a reduction in sojourn time variance. The simulation results also show that in the network settings, in contrast to the single queue case, there are priority sequencing policies that significantly reduce the variance of sojourn times relative to the first-come first-served policy.