Minimizing the worst slowdown: off-line and on-line

Minimizing the slowdown (expected sojourn time divided by job size) is a key concern of fairness in scheduling and queuing problems where job sizes are very heterogeneous. We look for protocols (service disciplines) capping the worst slowdown a job may face no matter how large (or small) the other jobs are. We call this worst slowdown the liability of the job in question. In the scheduling problem (all jobs released at the same time),we show that allowing the server to randomize the order of service cuts almost in half the liability profiles feasible under deterministic protocols. The same statement holds if cash transfers are feasible and users have linear waiting costs. In a queuing problem (release times of jobs are arbitrary), a deterministic on-line (non anticipative) protocol may guarantee the liability θr to job i, where r is the number of jobs in the queue when i was released, if and only if P∞ 1 1 θr ≤ 1. When the arrival of new jobs is Poisson with rate λ, the liability of a job of size x may not be smaller than its slowdown when all other jobs are equally long, namely 1 1−λ·x . We conjecture that this liability is feasible on-line, and identify a probabilistic protocol achieving the liability 1.45 1−λ·x .