Fundamental Characteristics of Queues with Fluctuating Load (appeared in SIGMETRICS 2006)

1. Fundamental Characteristics of Queues with Fluctuating Load(appeared in SIGMETRICS 2006) VARUN GUPTA Joint with:

2. Motivation Requests Clients Server Farm

3. Motivation Requests Clients Server Farm

4. Motivation Requests Clients Server Farm

5. Motivation Requests Clients Server Farm

6. Motivation Requests Clients Server Farm

7. Motivation Requests Clients Server Farm

8. Motivation Requests Clients Server Farm

9. Motivation Requests Real World Fluctuating arrival and service intensities  Clients Server Farm

10. A Simple Model exp(H) High Load L H Low Load exp(L)

11. H H L L A Simple Model exp() High Load H,H   L,L Low Load exp() • Poisson Arrivals • Exponential Job Size Distribution • H/H> L/L • H>Hpossible, only need stability

12. L L 0 1 2 . . . L L       H H 0 1 2 . . . H H The Markov Chain Number of jobs L Phase H Solving the Markov chain provides no behavioral insight

13. H H   L L • N = Number of jobs in the fluctuating load system • Lets try approximating N using (simpler) non-fluctuating systems

14. H H L L Method 1   Nmix

15. H H L L Method 1 ½ + Nmix ½ Q: Is Nmix≈ N? A: Only when   0  ,

16. H H L L Method 2  

17. Method 2 avg(H,L) ≡ Navg avg(H,L) Q: Is Navg ≈ N? A: When    ,

18. Example H=1, H=0.99   L=1, L=0.01 0  E[Navg] = 1 E[Nmix] ≈ 49.5

19. Observations • Fluctuating system can be worse than non-fluctuating •   0 and    asymptotes can be very far apart E[Nmix] > E[Navg] E[Nmix]  E[Navg]

20. Questions • Is fluctuation always bad? • Is E[N] monotonic in ? • Is there a simple closed form approximation for E[N] for intermediate ’s? • How do queue lengths during High Load and Low Load phase compare? How do they compare with Navg? More than 40 years of research has not addressed such fundamental questions!

21. Outline • Is E[Nmix] ≥ E[Navg], always? • Is E[N] monotonic in ? • Simple closed form approximation for E[N] • Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase • Application: Capacity Planning

22. Prior Work ? But cubic equations have a close form solution… Transforms Matrix Analytic & Spectral Analysis Fluid/Diffusion Approximations • - Clarke • - Neuts • Yechiali and Naor • - P. Harrison • Adan and Kulkarni • - Massey • Newell • Abate, Choudhary, Whitt Numerical Approaches Involves solution of cubic Limiting Behavior Involves solution of cubic

23. Good luck understanding this!

24. H=1, H=0.99 E[Nmix] > E[Navg]    L=1, L=0.01 Asymptotics for E[N] (H<H) E[Nmix] E[N] E[Navg] High fluctuation a (switching rate) Low fluctuation

25. E[Navg] E[N] E[Nmix] a Asymptotics for E[N] (H<H) E[Nmix] E[N] E[Navg] a Q: Is this behavior possible? A: Yes • Agrees with our example (H= L) • Ross’s conjecture for systems with constant service rate: • “Fluctuation increases mean delay”

26. E[N] E[N] E[N] a a a Our Results (H-H) > (L-L) (H-H) = (L-L) (H-H) < (L-L) • Define the slacks during L and H as • sL = L - L • sH = H - H

27. Our Results E[N] E[N] E[N] a a a sH > sL sH = sL sH < sL • Define the slacks during L and H as • sL = L - L • sH = H - H • Not load but slacks determine the response times! KEY IDEA

28. Outline • Is E[Nmix] ≥ E[Navg], always? • Is E[N] monotonic in ? • Simple closed form approximation for E[N] • Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase • Application: Capacity Planning

29. Outline • Is E[Nmix] ≥ E[Navg], always? No • Is E[N] monotonic in ? • Simple closed form approximation for E[N] • Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase • Application: Capacity Planning

30. Mean Queue Length Mean Queue Length H L H L  : ’ : Monotonicity of E[N]

31. Mean Queue Length H L  : Monotonicity of E[N] Mean Queue Length H L ’ : Not obvious that true for all ,’ with ’< ! • We show : E[N] is monotonic in 

32. Outline • Is E[Nmix] ≥ E[Navg], always? No • Is E[N] monotonic in ? • Simple closed form approximation for E[N] • Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase • Application: Capacity Planning

33. Outline • Is E[Nmix] ≥ E[Navg], always? No • Is E[N] monotonic in ? Yes • Simple closed form approximation for E[N] • Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase • Application: Capacity Planning

34. Approximating E[N] KEY IDEA • Express the first moment as * E[N] = E[Nmix]r+E[Navg](1-r) • Approximate r by the root of a quadratic KEY IDEA * True for H< H; a similar expression exists for case of transient overload

35. 9 Exact Approx. 7 5 3 1 10-5 10-4 10-3 10-2 10-1 100 10 Approximating E[N] E[N] H=L=1, H=0.95, L=0.2 

36. 9 Exact Approx. 7 5 3 1 10-5 10-4 10-3 10-2 10-1 100 10 Approximating E[N] E[N] H=L=1, H=0.95, L=0.2 

37. 18 Exact Approx. 14 10 6 2 10-2 10-1 100 10 Approximating E[N] E[N] H=L=1, H=1.2, L=0.2 

38. 18 Exact Approx. 14 10 6 2 10-2 10-1 100 10 Approximating E[N] E[N] H=L=1, H=1.2, L=0.2 

39. Outline • Is E[Nmix] ≥ E[Navg], always? No • Is E[N] monotonic in ? Yes • Simple closed form approximation for E[N] • Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase • Application: Capacity Planning

40. Outline • Is E[Nmix] ≥ E[Navg], always? No • Is E[N] monotonic in ? Yes • Simple closed form approximation for E[N] • Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase • Application: Capacity Planning

41. Stochastic Ordering refresher • For random variables X and Y X st Y  Pr{Xi}  Pr{Yi} for all i. • X stY  E[f(X)]  E[f(Y)] for all increasing f • E[Xk]  E[Yk] for all k  0.

42. NL NH exp() H,H L,L exp() Notation • NH: Number of jobs in system during H phase • NL: Number of jobs in system during L phase • N = (NH+NL)/2

43. NL NH exp() H,H Conjecture: NH increases stochastically as ↓ L,L exp() Stochastic Orderings for NL, NH ? • NL≥st NM/M/1/L • NH ≤st NM/M/1/H • NH ≥st NL • NH ≥st Navg • NLst Navg ? ? ? ?

44. Outline • Is E[Nmix] ≥ E[Navg], always? No • Is E[N] monotonic in ? Yes • Simple closed form approximation for E[N] • Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase • Application: Capacity Planning

45. Outline • Is E[Nmix] ≥ E[Navg], always? No • Is E[N] monotonic in ? Yes • Simple closed form approximation for E[N] • Stochastic orderings for the number of jobs seen by an arrival into an H phase, L phase • Application: Capacity Planning

46. 2H H H H L L 2L L Application: Capacity Provisioning Scenario     Aim: To keep the mean response times same

47. 2H H 2H H 2L L 2L L Application: Capacity Provisioning Scenario     Question: What is the effect of doubling the arrival and service rates on the mean response time?

48. What happens to the mean response time when , are doubled in the fluctuating load queue? A: Halves B: Reduces by more than half C: Reduces by less than half D: Remains almost the same

49. What happens to the mean response time when , are doubled in the fluctuating load queue? A: Halves B: Reduces by more than half C: Reduces by less than half D: Remains almost the same

50. Look at slacks! A: sH = sL B: sH > sL C: sH < sL D:sH < 0, 0  reduces by half  more than half  less than half  remains same What happens to the mean response time when , are doubled in the fluctuating load queue? A: Halves B: Reduces by more than half C: Reduces by less than half D: Remains almost the same