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A Maiden Analysis of Longest Wait First

A Maiden Analysis of Longest Wait First. Jeff Edmonds York University Kirk Pruhs University of Pittsburgh. Client-Server System. Requests for page transmission “pull”. Server. Clients. Client-Server System. Transmit page. Server scheduling problem: How does the server decide

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A Maiden Analysis of Longest Wait First

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  1. A Maiden Analysis of Longest Wait First Jeff Edmonds York University Kirk Pruhs University of Pittsburgh

  2. Client-Server System Requests for page transmission “pull” Server Clients

  3. Client-Server System Transmit page Server scheduling problem: How does the server decide which requests to respond to first? Server Clients

  4. Content Richness Today’s Internet Audience Size The Big Problem Movie Distribution Olympics Database Replication via Internet Software Download Harry Potter Book Download Pay-Per-View Movies

  5. The 1-1 communication is not scalable

  6. Broadcast Common Pages From www.direcpc.com From Newsweek magazine

  7. Time Requests Pages:

  8. Given requests deciding when to broadcast to minimize total “wait” (flow) time Time NP-complete [EH, 2002] Scheduling Problem:

  9. All Knowing All Powerful Optimal: ? Future Online: Time Time Requests Requests Optimal Online O(1)-Approximate Algorithms

  10. All Knowing All Powerful Optimal: ? Future Online: no online O(1)-comp. Alg. [KPV ‘00, EP ‘02] O(1)-Approximate Algorithms

  11. All Knowing All Powerful Optimal: ? Future Online: Time Time Requests Requests Optimal Online O(1)-Approximate Algorithms

  12. Resource Augmentation Analysis Algorithm is s-speed c-competitive if maxI Onlines(I)/Opt1(I) < c Time Time Requests Requests Optimal Online

  13. Not O(1)-competitive O(1)-speed O(1)-competitive Classic Server QoS Curves Online Optimal Average response time High load Low load Slow Processor Fast processor

  14. Scheduling Algorithms 2-speed 2-competitive [KPV ‘01, EH ‘02, GKKW ‘02, GKPS ‘02] Difficult Off-line Linear Programming Algorithms

  15. Scheduling Algorithms First In First Out (FIFO) 2-competitive for Max-Waitbut bad for Total-Wait. Time Requests

  16. Scheduling Algorithms Most Requests First (MRF) not O(1)-speed O(1)-competitive. [KPV ‘00] Time Requests

  17. Scheduling Algorithms B-Equipoise Proportional to number of requests Not 2-speed O(1)-competitive. (4+e)-speed O(1)-competitivefor any page lengths [EP 2002] Time Requests

  18. Scheduling Algorithms B-Equipoise-EDF Non-preemptive ~ number of requests (8+e)-speed O(1)-competitive for unit sized files [EP 2002] Time Requests

  19. New Not 1.6-speed O(1)-competitive. New 6-speed O(1)-competitive. Scheduling Algorithms Longest Wait First(LWF) Best Experimentally [AM] Efficient implementation, [KTT ‘01] Was hoped to be (1+e)-speed O(1)-competitive. Time Requests

  20. LWF is not 1.6-speed O(1)-competitive.

  21. LWF is not 1.6-speed O(1)-competitive.

  22. LWF is not 1.6-speed O(1)-competitive. With s=1.6 LWF catches up. LWF is competitive.

  23. > LWF is not 1.6-speed O(1)-competitive.

  24. < < LWF is not 1.6-speed O(1)-competitive.

  25. xc LWF is 6-speed O(1)-competitive. LWF6(I) < Opt1(I) x c Time Time Requests Requests Optimal LWF

  26. LWF is 6-speed O(1)-competitive. LWF6(I) < Opt1(I) x c xc Time Time Requests Requests Optimal LWF

  27. LWF is 6-speed O(1)-competitive. Optimal LWF

  28. = LWF is 6-speed O(1)-competitive. Optimal LWF

  29. LWF is 6-speed O(1)-competitive. xc Optimal LWF

  30. LWF is 6-speed O(1)-competitive. Optimal LWF

  31. LWF is 6-speed O(1)-competitive. LWF LWF

  32. ? £ LWF is 6-speed O(1)-competitive. LWF LWF

  33. LWF is 6-speed O(1)-competitive. LWF LWF

  34. Hall’s Theorem LWF is 6-speed O(1)-competitive. Needs to be paid Able to pay

  35. LWF One of s. Optimal LWF LWF is 6-speed O(1)-competitive. LWF

  36. LWF is 6-speed O(1)-competitive. LWF LWF LWF

  37. LWF is 6-speed O(1)-competitive. £ LWF LWF LWF

  38. LWF is 6-speed O(1)-competitive. £ LWF LWF LWF

  39. LWF is 6-speed O(1)-competitive. £ + LWF LWF LWF

  40. LWF is 6-speed O(1)-competitive. LWF6(I) < Opt1(I) x c xc Time Time Requests Requests Optimal LWF Everyone paid enough. No one pays to much.

  41. New LWF is not 1.6-speed O(1)-competitive. New LWF is 6-speed O(1)-competitive. Conclusion A Maiden Analysis of Longest Wait First LWF is best experimentally [AM] Efficient implementation, [KTT ‘01] Was hoped to be (1+e)-speed O(1)-competitive.

  42. (2+e) ? No Online is Conclusion Future A Maiden Analysis of Longest Wait First LWF is best experimentally [AM] Efficient implementation, [KTT ‘01] Was hoped to be (1+e)-speed O(1)-competitive. LWF is not 1.6-speed O(1)-competitive. LWF is 6-speed O(1)-competitive. for any file lengths The End

  43. Multicast Pull Scheduling: When Fairness is Fine Jeff Edmonds York University Kirk Pruhs University of Pittsburgh

  44. Scheduling Algorithms B-Equipoise Proportional to number of requests Not 2-speed O(1)-competitive. (4+e)-speed O(1)-competitivefor any page lengths [EP 2002] Time Requests

  45. The Power of the Adversary in Multicast Pull • Basic idea of the proof that there is no O(1)-competitive online algorithm • Immediately after the online algorithm broadcasts a document, the adversary requests that document • The adversary broadcasts the document after the second request to the document utilizing the power of broadcast • After a while the online algorithm still has a lot of work left while the adversary has little work left • Then a high load stream of work that requires the full processing power of the server arrives

  46. More on the Power of the Adversary in Multicast Pull • Hence, the adversary forces the online algorithm to labor on sequential work • Sequential work = increasing the processing power devoted to the work does not change the rate at which the remaining work decreases • Parallel work = doubling the processing power devoted to work doubles that rate at which that work is completed • IMHO, the main contribution of this paper is the insight that • Multicast pull scheduling = scheduling of jobs with arbitrary speed-ups

  47. Scheduling Jobs with Variable Speed-up Curves In the Context of Parallel Processing • Equipoise (Round Robin) = Give each job equal processing time • Equipoise is a 3-speed 6-competitive algorithm for jobs with arbitrary speed-up curves [E, 1999] • Formally means that Equipose with a speed 3 processor has average flow time at most 6 times the optimal average flow time for a speed 1 processor • Intuitively means that Equipoise will perform reasonably well at low loads

  48. Proof by picture that Bequi is O(1)-speed O(1)-approximation algorithm

  49. More proof by picture

  50. More proof by picture

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