Asymptotic convergence of scheduling policies with respect to slowdown

Abstract

We explore the performance of an M/GI/1 queue under various scheduling policies from the perspective of a new metric: the slowdown experienced by largest jobs. We consider scheduling policies that bias against large jobs, towards large jobs, and those that are fair, e.g., Processor-Sharing. We prove that as job size increases to infinity, all work conserving policies converge almost surely with respect to this metric to no more than 1=(1 ? ), where denotes load. We also find that the expected slowdown under any work conserving policy can be made arbitrarily close to that under Processor-Sharing, for all job sizes that are sufficiently large.

Decoupled speed scaling: Analysis and evaluation

“…For example, the slowdown of SRPT for large jobs remains above that of PS, and does not exhibit the "asymptotic convergence of slowdown" property from the non-speed-scaling world [32]. However, one additional empirical observation from our simulation results is that the unfairness of SRPT for large jobs appears to be upper bounded by the results for PS-SRPT with decoupled speed scaling (i.e., PS scheduling based on SRPT's speed scaling function for the same workload).…”

“…Similarly, T p (x, σ) and S p (x, σ) = T p (x, σ)/x denote the response time and slowdown [31], [32], respectively, under policy p for a job chosen at random from all jobs of size x on sample path σ. For brevity, we omit σ and write T p , T p (x), S p (x), and ε p , when the sample path is apparent from the context.…”

Lookahead actions in dispatching to parallel queues

“…However, B ( j) may still depend on the corresponding job size X ( j) . For example, the slowdown of Job j is defined as the ratio of the sojourn time T ( j) to the job size X ( j) [24] …”

Fairness and scheduling in single server queues

“…For E[T (x)] P , 1/x is an appropriate scaling factor because E[T (x)] P = Θ(x) under all work conserving scheduling policies [21], and thus we need to normalize by 1/x to provide non-trivial comparisons between small and large job sizes. Note that slowdown is also used as a measure of fairness in a variety of other, earlier, works, e.g.…”

“…In fact, this bound converges to 1 as ρ → 1, which signifies that the size of the smallest job that might be treated unfairly is increasing unboundedly as ρ increases. Note though that although we have focused the above discussion on SRPT, a wide variety of other common policies have been analyzed with respect to Definition 1 in [5,10,16,21,36,37,53]. The reader may find illustrations of the fairness behavior of FB, PSJF, and many other common policies in Figures 1 and 2.…”