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Unformatted text preview: Optimal scheduling of jobs with a DHR tail in the M/G/1 queue Samuli Aalto TKK Helsinki University of Technology Department of Communications and Networking P.O.Box 3000 02015 TKK Finland firstname.lastname@example.org Urtzi Ayesta LAAS-CNRS Universit de Toulouse 7 Avenue Colonel Roche 31077 Toulouse Cedex France email@example.com ABSTRACT We consider the mean delay optimization in the M/G/1 queue for jobs with a service time distribution that has a tail with decreasing hazard rate (DHR). If the DHR property is valid for the whole distribution, then it is known that the Foreground-Background (FB) discipline, which gives prior- ity to the job with least amount of attained service, is opti- mal among nonanticipating scheduling disciplines. However, FB may fail to be optimal if the DHR property is valid only for the tail of the distribution. An important example is the Pareto distribution bounded away from zero. In this paper we show that for a class of service time distributions with a DHR tail (including the Pareto distribution), the optimal nonanticipating discipline is a combination of FCFS and FB disciplines, which gives priority to the jobs with attained service less than some fixed threshold * . These priority jobs are served in the FCFS manner. If there are no jobs with attained service less than * , priority is given to the job with least amount of attained service. Keywords M/G/1, scheduling, mean delay, Pareto distribution, Gittins index 1. INTRODUCTION We consider the optimal scheduling problem in the M/G/1 queue with the objective to minimize the mean delay ( i.e. , sojourn time). We assume that jobs are served according to a preemptive, work conserving and nonanticipating schedul- ing discipline. A discipline is work conserving if it does not idle when there are jobs waiting, and nonanticipating if the remaining service times of jobs are unknown for the scheduler. Nonanticipating disciplines may utilize the at- tained service (age) information, but the remaining service times are unknown to such a scheduler. Thus, for example, the Shortest-Remaining-Processing-Time (SRPT) discipline Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ValueTools 2008, October 21-23, 2008, Athens, GREECE. Copyright 2008 ICST 978-963-9799-31-8. does not belong to these disciplines, while the Foreground- Background (FB) discipline, which gives priority to the job with least amount of attained service , or any other age- based discipline is nonanticipating. Our motivation comes from the recent literature that deals with the performance of age-based scheduling on the Internet, see [3, 4, 6, 8, 10, 11, 13, 14, 15].11, 13, 14, 15]....
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