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Alan Scheller-Wolf

Dimensionality Reduction for the analysis of Cycle Stealing, Task Assignment, Priority Queueing, and Threshold Policies (PART 2). Alan Scheller-Wolf. Joint with: Mor Harchol-Balter, Taka Osogami, Adam Wierman, and Li Zhang. Affinity Scheduling. m 12. m 11. m 22. Fluid or Diffusion.

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Alan Scheller-Wolf

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  1. Dimensionality Reduction for the analysis of Cycle Stealing, Task Assignment, Priority Queueing, and Threshold Policies (PART 2) Alan Scheller-Wolf Joint with: Mor Harchol-Balter, Taka Osogami, Adam Wierman, and Li Zhang.

  2. Affinity Scheduling m12 m11 m22

  3. Fluid or Diffusion Bell and Williams Harrison Harrison and Lopez Squillante, Xia, Zhang Williams Prior Work: Affinity Scheduling Applications (cycle stealing) Threshold policies Squillante, Xia, Yao and Zhang Williams Green Schumsky Stanford and Grassman No accurate analysis for non-limiting behavior.

  4. Situation 1: Self-Affinities m12 m11 m22 Optimal control policy: Cycle Stealing.

  5. m12 m11 m22 Situation 2:Eager to Help If server 2 overzealous, a brake is needed.

  6. m12 m22 m11 Why? Potential Instability • Maybe server two is too eager to help: • Take too much work from server 1,leaving her idle, • Neglect own work, letting it build up.

  7. N1 T1 1 N2 The Brake: T1Policy “Come help, but only when I call you.” Asymptotically optimal, robustness concerns. We provide first easy, accurate analysis.

  8. T1 Policy: Performance vs. T1

  9. T1 Policy: Performance vs. r2 T1 Performance II

  10. What is the Dream? Switching Curve N1 N2 Optimal?

  11. T1(2) T2 New Control Policy: The ADTPolicy “Come help when I call you.” “If you are very busy and I am not, do not come.” “But if I really need you, you have to come.” N1 T1(1) N2 Performs like best of T1(1) and T1(2). We propose and analyze.

  12. T1 Policy: Performance vs. T1

  13. ADT Policy: Performance vs T1(1)

  14. T1 Policy: Performance vs. r2 T1 Performance II

  15. ADT Policy: Performance vs r2

  16. 1D-infinite chain nD-infinite chain HARD EASY Priority Scheduling in M/PH/k H L L M H H RDR and Priority Scheduling Goal: Mean response time per job type.

  17. Two job classes, exponential Two job classes, hyper- exponential Multi-class simple approx. Two job classes, exponential Aggregation or Truncation Matrix Analytic or Gen. Functions Iterative sol to balance equations Scaling as Single-server: Buzen and Bondi Kao and Narayanan Kao and Wilson Kapadia et al Nishida Ngo and Lee Cidon and Sidi Feng el at Gail et al Miller Sleptchenko et al Aggregation into Two classes: Mitrani and King Nishida Prior Work: Multi-Server Priority Queues Little work for > 2 classes or non-exponential.

  18. What’s so Hard? Low Med Hi Now chain grows infinitely in 3 dimensions!

  19. Recursive Dimensionality Reduction(RDR) • Apply standard dimensionality reduction (DR) to two highest classes (Mor’s talk). • Aggregate these classes -- carefully -- into single higher class. Many types of busy periods. • Apply DR to two-class system made up of aggregated classes and third class. • Recurse. Chain for class m used to calculate busy periods for next lower class (m+1).

  20. L L L L H H M or L L L L L L L L H H H L L L Representative Types of Busy Periods Becomes… Becomes… M or L L

  21. lM lM lM lM lM lM lM 0,1 1,1 2,1 2,0 3,1 3,0 mM mM mM mM 2mM 2mM 2mM mH mH mH lH lH lH lH lH lH lH lH What are these busy periods? lM 0,0 1,0 mM mH BH BH BH BH lM lM lM lM 0,2+ 1,2+ 2,2+ 3,2+ Neuts[1978]

  22. 0,0 1,0 mM mH lH 0,1 The Low Job Chain lM

  23. lM 5,0,0 5,1,0 mM mH lH 5,0,1 The Low Job Chain

  24. lM 5,0,0 5,1,0 mM mH lH 5,0,1 The Low Job Chain

  25. lM 5,0,0 5,1,0 mM mH lH 5,0,1 5,1,1M 5,1,1H 5,2,0M 5,2,0H The Low Job Chain 5,0,2H 5,0,2M

  26. lM 5,0,0 5,1,0 mM mH lH 5,0,1 5,1,1M 5,1,1H 5,2,0M 5,2,0H The Low Job Chain 5,0,2H 5,0,2M

  27. M/M/2 Four Priority Classes: Accuracy

  28. M/M/2 Four Priority Classes: Resp.

  29. M/M/2 Four Priority Classes: Perf

  30. Generalizations and Extensions • Phase-type service times. • More classes, more servers. • Number of different busy periods grows with complexity of system (service times, servers, classes). • RDR-A approximation for these more complex systems, within 5% error for four class problem.

  31. m12 m11 m22 m21 DR and RDR, future directions We solve problems where one class depends on the other, but the dependencies can be solved sequentially (H,M,L). What about systems that do not decouple?

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