Mor Harchol-Balter

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

2. H L L H H Many multiserver scheduling problems FIFO FIFO Goal: Mean response time per job type

3. H L L H H Many multiserver scheduling problems FIFO FIFO Common Problem: 2D-infinite Markov chain (or nD-infinite)

4. Dimensionality Reduction (2D) Recursive Dimensionality Reduction(nD) • Numerical technique • (not closed form) • Problem-dependent. • Doesn’t always apply. • Provides accurate performance • numbers for many common problems. • Very fast (less than 1 sec). • < 1% error typically, and can improve. • Can analyze any load (non-limiting). • Allows PH service time distributions. • Adding thresholds and switching costs • is often easy.

5. Betty the beneficiary Dan the donor Cycle Stealing Problem lBjobs/sec lDjobs/sec Load rB Load rD

6. switch When donor is idle, donor helps beneficiary. Cycle Stealing Problem lBjobs/sec lDjobs/sec Load rB Load rD

7. Cycle Stealing Problem lBjobs/sec lDjobs/sec Load rB Load rD switch back When new donor job arrives, donor switches back to donor queue.

8. lBjobs/sec PHBjob size PHD job size time KSW NBthresh NDthresh time KBA Only switch if Betty has NBthreshjobs queued. Only switch back if Dan has NDthreshjobs queued. Generalized Cycle Stealing [Sigmetrics 03] lDjobs/sec Load rB Load rD

9. Cycle Stealing Problem lBjobs/sec lDjobs/sec Load rB Load rD What is Betty’s/Dan’s mean response time?

10. What’s so hard? Even simplest-case chain grows infinitely in 2D.

11. Prior work: cycle stealing & coupled-processor Truncate the chain Exponential job sizes (80’s) General Job sizes (80,90,00’s) Tail Asymp or Heavy traff. Bob Foley McDonald Mike Harrison Boxma Borst van Uitert Williams Convert to Riemann-Hilbert problem Convert to Wiener-Hopf boundary problem Very complex integrals. No numbers Very complex integrals for WORKLOAD. No numbers. Fayolle, Iasnogoradski, Konheim, Meilijson, Melkman ... Cohen, Boxma, Borst, Uitert, Jelenkovic ...

12. Dimensionality Reduction Key idea: 2D-infinite chain 1D-infinite chain VERY HARD EASY

13. What’s so hard? Even simplest-case chain grows infinitely in 2D.

14. Solution • New type of transition -- Busy period transition: BD • 1D-infinite chain

15. Solution Approximation. But can be made as close to exact as desired. [Tools03a,Tools03b] g = BD b a

16. lBjobs/sec lDjobs/sec Load rB LoadrD PHB job size PHDjob size time KSW NBthresh NDthresh time KBA Markov chain is easy to generalize • Generalize to PH service distribution • Generalize to switching times • Generalize to include thresholds

17. MODIFIED 1D-chain: cycle stealing with switching costs and NBth = 3

18. A: For rB>1, always pays. For rB<1, sometimes pays. orig cs E[TB] E[TD] cs orig rB rB A: Hardly, for rB < 1 ! Exp E[TB] C2=8 C2=50 rB Some interesting questions Q: When does cycle stealing pay? Q: How does donor job size variability affect benef. resp. time?

19. A: Want NBth high. E[TB] E[TD] Increasing NBth doesn’t hurt benef! Increasing NBth helps donor A: Want NDth low if rB<1. Want NDth high if rB>1. E[TB] E[TD] Increasing NDth helps benef only in overload! Increasing NDth hurts donor Some interesting questions Q: How should we set NBth? Q: How should we set NDth?

20. FIFO High variability job sizes FIFO Supercomputing Applications: Non-preemptive service Task Assignment Problem What is a good task assignment policy for minimizing mean response time?

21. FIFO FIFO Task Assignment Problem SHORTS HERE! High variability job sizes LONGS HERE! Supercomputing Applications: Non-preemptive service Want to isolate short jobs [JPDC 99, JACM 02]

22. FIFO FIFO Task Assignment Problem SHORTS High variability job sizes LONGS. But short if idle. Supercomputing Applications: Non-preemptive service Even better!

23. Immediate-Dispatch model Central-Queue model Shorts only. Shorts only Longs. Short if idle. Longs. Short if no long. Smart Task Assignment 1D-infinite chain 2D-infinite chain DR VERY HARD EASY

24. Immediate-Dispatch model Central-Queue model Shorts Shorts Longs. Short if idle. Longs. Short if no long. Results E[TS] Immed. Disp. with sharing No sharing Central Queue with sharing [ICDCS 03] [SPAA 03] rS

25. So far ... • Examples of some problems: • - cycle stealing with switching cost • - task assignment • where Dimensionality Reduction(DR) is very useful. • Next: More problems: • a) Affinity model and more complex threshold policies. • b) Priority queueing in multiserver system and • Recursive Dimensionality Reduction(RDR). • Finally: Current limitations of DR and RDR.