Minimizing Maximum Flow Time on Related Machines via Dynamic Posted Pricing

We consider a setting where selfish agents want to schedule jobs on related machines.  The agent submitting a job picks a server that minimizes a linear combination of the server price and the resulting response time for that job on the selected server. The manager's task is to maintain server prices to (approximately) optimize the maximum response time, which is a measure of social good.  We show that the existence of a pricing scheme with certain competitiveness is equivalent to the existence of a monotone immediate-dispatch algorithm.  Our main result is a monotone immediate-dispatch algorithm that is  O(1)-competitive with respect to the maximum response time.