gpp - Math. Meth. Oper. Res (2005) 62: 99122 DOI...

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Math. Meth. Oper. Res (2005) 62: 99–122 DOI 10.1007/s00186-005-0443-4 ORIGINAL ARTICLE Eugene A. Feinberg · Michael T. Curry Generalized Pinwheel Problem Received: November 2004 / Revised: April 2005 © Springer-Verlag 2005 Abstract Thispaperstudiesanon-preemptiveinfnite-horizonschedulingproblem with a single server and a fxed set oF recurring jobs. Each job is characterized by two given positive numbers: job duration and maximum allowable time between the job completion and its next start. We show that For a Feasible problem there exists a periodic schedule. We also provide necessary conditions For the Feasibility, Formulate an algorithm based on dynamic programming, and, since this problem is NP-hard, Formulate and study heuristic algorithms. In particular, by applying the theory oF Markov Decision Process, we establish natural necessary conditions For Feasibility and develop heuristics, called Frequency based algorithms, that outper- Form standard scheduling heuristics. 1 Introduction Suppose there are n N ={ 1 , 2 ,... } recurring jobs. Each job i = 1 ,... ,n is characterized by two positive numbers: τ i , the duration oF job i , and u i , the maxi- mum amount oF the time that can transpire between epochs when job i is completed and is started again. We call them the duration and revisit time respectively. These jobs are to be completed by a single server. There is no preemption and setups are instantaneous. A schedule is a sequence in which the jobs should be perFormed. A schedule is Feasible iF each time job i is completed, it will be started again within E.A. ±einberg ( B ) Department oF Applied Mathematics and Statistics, State University oF New York, Stony Brook, NY 11794-3600, USA E-mail: eFeinberg@notes.cc.sunysb.edu M.T. Curry Department oF Applied Mathematics and Statistics, State University oF New York, Stony Brook, NY 11794-3600, USA E-mail: curry@ams.sunysb.edu
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100 E.A. Feinberg, M.T. Curry u i units of time. A problem is feasible if a feasible schedule exists. We call this problem the Generalized Pinwheel Problem, (GPP). The GPP is to ±nd a feasible schedule or to prove that it does not exist. Ourinterestinthisproblemwasinitiatedbyapplications.Inparticular,Feinberg, et al [12] studied the so-called radar sensor management problem, the mathemat- ical formulation of which is the GPP. Since the GPP is generic, there are various other engineering applications such as mobile communications, wired and wireless networks, satellite transmissions, database support. In addition, though the GPP is a deterministic problem, the methods of Markov Decision Processes (MDPs), an area to which Professor Ulrich Rieder made many important contributions, are useful to study this problem. From the methodological point of view, this paper continues the line of research initiated by Filar and Krass [14] that deals with appli- cations of MDP methods to discrete optimization problems; see also [1,5,8,11] and references therein.
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gpp - Math. Meth. Oper. Res (2005) 62: 99122 DOI...

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