A new average case analysis for completion time scheduling

Abstract

We present a new average case analysis for the problem of scheduling n jobs on m machines so that the sum of job completion times is minimized. Our goal is to use the concept of competitive ratio-which is a typical worst case notion-also within an average case analysis. We show that the classic SEPT scheduling strategy with (n) worst-case competitive ratio achieves an average of O(1) under several natural distributions, among them the exponential distribution. Our analysis technique allows to also roughly estimate the probability distribution of the competitive ratio. Thus, our result bridges the gap between worst case and average case performance guarantee.

Average-Case and Smoothed Competitive Analysis of the Multilevel Feedback Algorithm

“…An alternative is to define it as the ratio between the expected costs of the algorithm and of the optimum, see also [18]. We point out that we obtain the same results under this alternative, weaker, definition.…”

“…To the best of our knowledge, this is the first average case analysis of MLF. Recently, Scharbrodt et al [18] performed the analysis of the average competitive ratio of the Shortest Expected Processing Time First heuristic to minimize the average completion time where the processing times of the jobs follow a gamma distribution. Our result is stronger in the following aspects: (a) the analysis of [18] applies when the algorithm knows the distribution of the processing times, while in our analysis we require no knowledge about the distribution of the processing times, and (b) our result applies to average flow time, a measure of optimality much stronger than average completion time.…”

“…Recently, Scharbrodt et al [18] performed the analysis of the average competitive ratio of the Shortest Expected Processing Time First heuristic to minimize the average completion time where the processing times of the jobs follow a gamma distribution. Our result is stronger in the following aspects: (a) the analysis of [18] applies when the algorithm knows the distribution of the processing times, while in our analysis we require no knowledge about the distribution of the processing times, and (b) our result applies to average flow time, a measure of optimality much stronger than average completion time. Early work by Michel and Coffman [14] only considered the problem of synthesizing a feedback queue system under Poisson arrivals and a known discrete probability distribution on processing times so that pre-specified mean flow time criteria are met.…”

“…As mentioned in the introduction, one could alternatively consider a weaker definition of smoothed competitiveness as the ratio between the expected costs of the algorithm and of the optimum, see also [18], rather than the expected competitive ratio. We remark that from Lemmas 1, 2, 5 and 6 we obtain the same bound under this alternative definition, without the need for any high probability argument.…”

Probabilistic Analysis of Online Bin Coloring Algorithms Via Stochastic Comparison

“…One drawback of competitive analysis is that it often provides rather pessimistic results due to its worst-case character. This is partly overcome by more elaborate variants like averagecase competitive analysis [22] and smoothed competitive analysis [3,21].…”