Csr2011 june18 12_00_nguyen

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Csr2011 june18 12_00_nguyen

  1. 1. 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)
  2. 2. 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.
  3. 3. 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) .
  4. 4. Competitive ratioAn algorithm ALG is α-competitive if for any instance I OP T (I) ≤ α (maximization problem) ALG(I)
  5. 5. 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)
  6. 6. 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
  7. 7. 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
  8. 8. Settling the competitivityMethods: charging scheme, potential function, etc wj j i α · wi
  9. 9. Settling the competitivity Methods: charging scheme, potential function, etc wj j i α · wiALGADV
  10. 10. Starting pointParadox: low weight, which jobs? higher weight, imminent deadline later deadline
  11. 11. 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
  12. 12. 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)
  13. 13. 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.
  14. 14. 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
  15. 15. 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
  16. 16. 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
  17. 17. 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
  18. 18. 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
  19. 19. 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.
  20. 20. Thank you!Thank Kristoffer!
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