Average stretch without migration

We study the problem of scheduling parallel machines online, allowing preemptions while disallowing migration of jobs that have been scheduled on one machine to another. For a given job, we measure the quality of service provided by an algorithm by the stretch of the job, defined as the ratio between the amount of time spent by the job in the system (the response time) and its processing time. For a sequence of jobs, we measure the performance of an algorithm by the average stretch achieved over all jobs. The scheduling goal is to minimize the average stretch. This problem is of relevance in many applications. e.g., wireless data servers and distributed server systems in wired networks. We prove an O(1) competitive ratio for this problem. The algorithm for which we prove this result is the one proposed in Awerbuch et al. (Proceedings of the ACM Symposium on the Theory of Computing (STOC '99), 1999, pp. 198-205) that has (tight) logarithmic competitive ratio for minimizing the average response time. Thus, the algorithm in Awerbuch et al. (Proceedings of the ACM Symposium on the Theory of Computing (STOC '99), 1999. pp. 198-205) simultaneously performs well for average response time as well as average stretch. We prove the O(1) competitive ratio against an adversary who not only knows the entire input ahead of time, but is also allowed to migrate jobs. Thus, our result shows that migration is not necessary to be competitive for minimizing average stretch; in contrast, we prove that preemption is essential, even if randomization is allowed.