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Effect of higher moments of job size distribution on the performance of an M/G/k system

Effect of higher moments of job size distribution on the performance of an M/G/k system. VARUN GUPTA Joint work with:. Mor Harchol-Balter Carnegie Mellon University. Jim Dai, Bert Zwart Georgia Institute of Technology. Multi-server/resource sharing systems are the norm today. Server Farms.

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Effect of higher moments of job size distribution on the performance of an M/G/k system

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  1. Effect of higher moments of job size distribution on the performance of an M/G/k system VARUN GUPTA Joint work with: Mor Harchol-Balter Carnegie Mellon University Jim Dai, Bert Zwart Georgia Institute of Technology

  2. Multi-server/resource sharing systems are the norm today Server Farms Multicore chips Call centers

  3. Poisson arrivals (rate ) M/G/k: the classical multi-server model Ji+2 Ji+1 Ji J2 J1 GOAL : Analysis of mean delay (time spent in buffer)

  4. M/G/k model assumptions and notation • Poisson arrivals • Service requirements (job sizes) are i.i.d. • S≡ random variable for job sizes • Define • Define Per server utilization or load: 0 < < 1 Squared coefficient of variability (SCV) of job sizes: C2 0

  5. M/G/k mean delay analysis • Lets take a step back: M/G/1

  6. M/G/k mean delay analysis • Lets take a step back: M/G/1

  7. M/G/k mean delay analysis • Lee and Longton (1959) • Simple and closed-form • Involves only first two moments of S • Exact for k=1 • Asymptotically exact in heavy traffic [Köllerström[74]] • No exact analysis exists • All closed-form approximations involve only the first two moments of S • Takahashi[77], Hokstad[78], Nozaki Ross[78], Boxma Cohen Huffels[79], Whitt [93], Kimura[94]

  8. But… Q1: Are 2 moments of S enough to reasonably approximate E[Delay]? • Does the third moment have no/negligible effect? Q2: How inaccurate can a 2-moment approximation be? Q3: Are 3 moments enough? 4 moments?

  9. Outline Q1: Are 2 moments of S enough to reasonably approximate E[Delay]? • Does the third moment have no/negligible effect? Q2: How inaccurate can a 2-moment approximation be? Q3: Are 3 moments enough? 4 moments?

  10. H2 • The H2 class has three degrees of freedom • Can vary E[S3] while keeping first two moments constant • Can numerically evaluate M/H2/k using the matrix analytic method

  11. E[Delay] vs. E[S3] k=10, E[S]=1, C2=19, =0.9 2-moment approx E[Delay] E[Delay]M/H2/k X104 E[S3]

  12. E[S3] can have a huge impact on mean delay! The mean delay decreases as E[S3] increases! 2-moment approx E[Delay] E[Delay]M/H2/k X104 E[S3]

  13. Intuition for the effect of E[S3] (x) x (x) = load due to jobs smaller than x E[S]=1 C2=19

  14. Intuition for the effect of E[S3] (x) E[S3]=600 x (x) = load due to jobs smaller than x E[S]=1 C2=19

  15. Intuition for the effect of E[S3] (x) E[S3]=700 E[S3]=600 x (x) = load due to jobs smaller than x E[S]=1 C2=19

  16. Intuition for the effect of E[S3] E[S3]=1200 (x) E[S3]=700 E[S3]=600 x (x) = load due to jobs smaller than x E[S]=1 C2=19

  17. Intuition for the effect of E[S3] E[S3]=15000 E[S3]=1200 (x) E[S3]=700 E[S3]=600 x (x) = load due to jobs smaller than x E[S]=1 C2=19

  18. Intuition for the effect of E[S3] (x) Increasing E[S3] x As E[S3] increases (with fixed E[S] and E[S2]): • Load gets ‘concentrated’ on small jobs • Load due to ‘big’ jobs vanishes • Bigs become rarer, usually see small jobs only • Causes drop in E[Delay]M/H2/k

  19. E[Delay] vs. E[S3] k=10, E[S]=1, C2=19, =0.9 2-moment approx E[Delay] E[Delay]M/H2/k X104 E[S3]

  20. Outline Q1: Are 2 moments of S enough to reasonably approximate E[Delay]? • Does the third moment have no/negligible effect? Q2: How inaccurate can a 2-moment approximation be? Q3: Are 3 moments enough? 4 moments?

  21. Outline Q1: Are 2 moments of S enough to reasonably approximate E[Delay]? • Does the third moment have no/negligible effect? Q2: How inaccurate can a 2-moment approximation be? Q3: Are 3 moments enough? 4 moments?

  22. {G|C2} ≡ positive distributions with mean 1 and SCV C2 G1 GAP  Error of 2-moment approx G2 E[Delay]

  23. Our Theorems • Upper bound • Lower bound • <1-1/k •  1-1/k

  24. GAP E[Delay]

  25. D* has the smallest third moment in {G|C2} E[Delay] third moment  as  0  0

  26. Conjecture E[Delay]

  27. Outline Q1: Are 2 moments of S enough to reasonably approximate E[Delay]? • Does the third moment have no/negligible effect? Q2: How inaccurate can a 2-moment approximation be? Q3: Are 3 moments enough? 4 moments?

  28. Outline Q1: Are 2 moments of S enough to reasonably approximate E[Delay]? • Does the third moment have no/negligible effect? Q2: How inaccurate can a 2-moment approximation be? Q3: Are 3 moments enough? 4 moments?

  29. What about higher moments? {G|C2} • H*3 class has four degrees of freedom • Can vary E[S4] while keeping first three moments constant H*3 H2

  30. Increasing fourth moment E[Delay] vs. E[S4] k=10, E[S]=1, C2=19, =0.9 2-moment approx E[Delay] E[Delay]M/H2/k X104 E[S3]

  31. Increasing fourth moment • Even E[S4] can have a significant impact on mean delay! • High E[S4] can nullify the effect of E[S3]! 2-moment approx E[Delay] E[Delay]M/H2/k X104 E[S3]

  32. The BIG picture Odd/Even moments refine the Lower/Upper bounds on mean delay UB1,2=(C2+1)E[Delay]M/D/k UB1,2,3,4 E[Delay] LB1,2,3,4,5 LB1,2,3 LB1=E[Delay]M/D/k

  33. Outline Q1: Are 2 moments of S enough to reasonably approximate E[Delay]? • Does the third moment have no/negligible effect? Q2: How inaccurate can a 2-moment approximation be? Q3: Are 3 moments enough? 4 moments?

  34. Conclusions • Shown that 2-moment approximations for M/G/k are insufficient • Shown bounds on inaccuracy of 2-moment only approximations • (C2+1) inaccuracy factor • Observed alternating effects of odd and even moments

  35. Thank you!

  36. Open Questions • Proof (or counter-example) of conjectures on bounds • Are there other attributes of service distribution that characterize it better than moments? • For example, mean and variability of small and big jobs • Where do real world service distributions sit with respect to these attributes?

  37. {G|C2} ≡ positive distributions with mean 1 and SCV C2 • The H2 class has three degrees of freedom (s, p, ps) • Can vary E[S3] while holding first two moments constant H2

  38. Look at the moments of H2 … • Load due to big jobs vanishes as E[S3] increases • When k>1, a big job does not block small jobs • This reduces the effect of variability (C2) as third moment increases

  39. Observations •  < 1-1/k : UB/LB  (C2+1) • No 2-moment approximation can be accurate in this case • [Kiefer Wolfowitz] [Scheller-Wolf]: When  > 1-1/k, E[Delay] is finite iff C2 is finite. • Matches with the conjectured lower bound • Also popular as the “0 spare server” case

  40. Proof outline: Upper bound • THEOREM: • PROOF: Consider the following service distribution • Intuition for conjecture: k>1 should mitigate the effect of variability; D* exposes it completely • Note: D* has the smallest third moment in {G|C2}

  41. Proof outline: Lower bound • THEOREM: •  < 1-1/k •  1-1/k • PROOF: Consider the following sequence of service distributions in {G|C2} as  0

  42. What about higher moments? {G|C2} H*3 • H*3 allows control over fourth moment while holding first three moments fixed • The fourth moment is minimized when p0=0 (H2 distribution) H*3 H2

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