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Improved Online Scheduling in Maximizing Throughput of Equal Length Jobs Nguyen Kim Thang (university Paris-Dauphine, France) presented by Kristoffer Arnsfelt Hansen (university Aarhus, Denmark)
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Motivation Proﬁt maximization Enterprise: perishable product (electricity, ice-cream, ...). Clients: single-minded, arrive online, different demands. Goal: maximize the proﬁt.Other applicationsData broadcast: online $broadcast pages $ATM network: online packages,typically of the same length. $Objective: maximize the totalvalue.
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ModelOnline SchedulingJobs: arrive at ri, processing timepi, deadline di , value (weight) wi . Preemption is necessaryObjective: maximize the totalvalue of jobs completed on time. Preemption with restart: when a job is scheduled again,it must be executed from the beginning (e.g., databroadcast). Preemption with resume: when a job is scheduled again,the previously done work can be resumed (e.g., ATMnetwork) .
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Competitive ratioAn algorithm ALG is α-competitive if for any instance I OP T (I) ≤ α (maximization problem) ALG(I)
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Competitive ratioAn algorithm ALG is α-competitive if for any instance I OP T (I) ≤ α (maximization problem) ALG(I)What is a competitive ratio? Measure the performance of an algorithm (worst-case analysis) The price of an object (the problem): negotiation Algorithm Adversary (upper bound) (lower bound)
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Contribution equal processing bounded processing unbounded times times (by k ) processing times unit α=1 α = Θ(log k) ∞weight √ 3 3general ≤α≤5 α = Θ(k/ log k) ∞ 2 ≤ 4.24
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Contribution equal processing bounded processing unbounded times times (by k ) processing times unit α=1 α = Θ(log k) ∞weight √ 3 3general ≤α≤5 α = Θ(k/ log k) ∞ 2 ≤ 4.24 Improved algorithms for both models of preemption Weights and correlation between jobs’ deadlines
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Settling the competitivityMethods: charging scheme, potential function, etc wj j i α · wi
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Settling the competitivity Methods: charging scheme, potential function, etc wj j i α · wiALGADV
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Starting pointParadox: low weight, which jobs? higher weight, imminent deadline later deadline
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Starting pointParadox: low weight, which jobs? higher weight, imminent deadline later deadlinep : initial job length, qj (t) : length of job j at time tA job j is pending at time t if t + qj (t) ≤ dj
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Starting pointParadox: low weight, which jobs? higher weight, imminent deadline later deadlinep : initial job length, qj (t) : length of job j at time tA job j is pending at time t if t + qj (t) ≤ djA 5-competitive algorithm (preemption with restart) At any time If no currently scheduled job, schedule the pending one with highest weight If a new pending job arrive with weight at least twice that of the currently scheduled job, then schedule the new one (by interrupting the current job)
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Observations Correlation among jobs’ deadlines is ignoredTreatment: A job i is urgent at time t if di < t + qi (t) + p Some job would be delayed by new urgent jobs (even with low weight) Ensure no signiﬁcant lost if new heavy jobs arrive.
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Algorithm Initially, set Q = ∅, α = 0, 1 < β < 3/2 At time t , let i, j be a new released job and the currently scheduledjob, respectively. At any interruption, if α > 0 then α := α + 1
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Algorithm Initially, set Q = ∅, α = 0, 1 < β < 3/2 At time t , let i, j be a new released job and the currently scheduledjob, respectively. At any interruption, if α > 0 then α := α + 1 schedule i If wi ≥ 2wj , wi ≥ 2α w(Q) do set Q = ∅, α = 0
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Algorithm Initially, set Q = ∅, α = 0, 1 < β < 3/2 At time t , let i, j be a new released job and the currently scheduledjob, respectively. At any interruption, if α > 0 then α := α + 1 schedule i If wi ≥ 2wj , wi ≥ 2α w(Q) do set Q = ∅, α = 0 schedule job which is If α = 0, βwj ≤ wi ≤ 2wj do arg max{w : d < t + 2p} i urgent and dj ≥ t + 2p set Q = {j}, α = 1
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Algorithm Initially, set Q = ∅, α = 0, 1 < β < 3/2 At time t , let i, j be a new released job and the currently scheduledjob, respectively. At any interruption, if α > 0 then α := α + 1 schedule i If wi ≥ 2wj , wi ≥ 2α w(Q) do set Q = ∅, α = 0 schedule job which is If α = 0, βwj ≤ wi ≤ 2wj do arg max{w : d < t + 2p} i urgent and dj ≥ t + 2p set Q = {j}, α = 1 If i is urgent do schedule i wi ≥ 2wj + wj no job such that Sj (t) + 2p ≤ d < t + 2p, w ≥ wj
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The charging scheme phase of job i 2wf (i)ALG f (i) i wj wiADV j i √ Theorem: the algorithm is (2 + 5)-competitive √ Theorem: there is a (2 + 5) -competitive algorithm for modelof preemption with resume
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ConclusionImproved algorithms for both models of preemptionOpen questions: Settling the right competitive ratio 2.5 ≤ α ≤ 4.24 Interesting: not to reduce the gap but new methods.
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