A tutorial on amortized local competitiveness in online scheduling

Recently the use of potential functions to analyze online scheduling algorithms has become popular [19, 7,29, 13, 31, 4, 30, 3, 21, 15, 14, 28, 12, 2, 5, 6, 9, 11, 23, 33, 24, 8, 17, 16, 25, 1, 20, 26, 22, 18]. Thesepotential functions are used to show that a particular online algorithm is locally competitive in an amortizedsense. Algorithm analyses using potential functions are sometimes criticized as seeming to be black magicas the formal proofs do not require, and commonly do not contain, any discussion of the intuition behindthe design of the potential function. Sometimes, as in the case for the first couple uses of potential functionsin the online scheduling literature, this is because the authors arrived at the potential function by trial anderror, and there was not a cohesive underlying intuitionguidingthe development. However, now that tens ofonline scheduling papers have used potential functions, one can see that a “standard” potential function hasemerged that seems to be applicable to a wide range of problems. The use of this standard potential functionto prove amortized local competitiveness can no longer be considered to be magical, and is a learnabletechnique. Our main goal here is to give a tutorialteaching this technique to readers with some modest priorknowledge of scheduling, online problems, and the concept of worst-case performance ratios.Online Scheduling: We consider online schedulingproblems where jobs/tasksarrive at a server (e.g. a webserver, a database server, an operating system, etc.) over time. Throughoutthis paper N willdenote the totalnumber of jobs and jobs are indexed J