The Expected Competitive Ratio on a Kind of Stochastic-Online Flowtime Scheduling with Machine Subject to an Uncertain Breakdown

Abstract

We consider the problem of scheduling jobs on a single machine subject to an uncertain breakdown to minimize the flowtime. Assuming the machine is unavailable during the breakdown, the starting time of the breakdown is a random variable s with distribution function D(s) and the terminating time of the breakdown has no any other information; jobs are non-resumable. Under these assumptions and starting from the perspective of statistical optimization, we first establish the scheduling problem HSONRP, which contains deterministic information, stochastic information, and online information and then define the expected competitive ratio of an algorithm to find the optimized solution of the problem HSONRP. In addition, then, we propose and prove certain results on the expected competitive ratio of the SPT rule. In particular, we prove the expected competitive ratio of SPT rule is less than 1+max{pi}2P when s is the uniform distribution on interval (0,P], where pi is the processing time of job i, P=∑i=1npi, and show that it is no more than 54 under a quite loose condition. Meanwhile, we also make some discussions about our studies. What we have performed will enrich and improve the research results on the area of scheduling to minimize flowtime and advance the development of online optimization and stochastic optimization.