2. 1 SCHEDULING FOR TODAY’S COMPUTER SYSTEMS: SCHEDULING FOR TODAY’S COMPUTER SYSTEMS: BRIDGING THEORY AND PRACTICE Adam Wierman Mor Harchol-Balter John Lafferty Bruce Maggs Alan Scheller-Wolf Ward Whitt Thesis Committee

3. Carnegie Mellon University Computer Science Department 2 “SCHEDULING SUCCESS STORIES” ARE EVERYWHERE Biersack, Rai, Urvoy-Keller, Harchol-Balter, Schroeder, Agrawal, Ganger, Petrou, Misra, Feng, Hu, Zhang, Mangharam, Sadowsky, Rawat, Dinda, McWherter, Ailamaki, & others Web Servers users Routers Internet Disks CPUs Locks Databases …also p2p, wireless, operating systems…

4. Carnegie Mellon University Computer Science Department 3 web server, edge router, etc. Goal Minimize user response times THE ESSENCE OF A “SCHEDULING SUCCESS STORY” Processor Sharing (PS) bottleneck resource Use a different scheduling policy Use a different scheduling policy

5. Carnegie Mellon University Computer Science Department 4 load 0 0.25 0.5 0.75 mean response time PS SRPT ? Sched- uling Sched- uling FCFS Assumption: M/GI/1 Queue SRPT WINS BIG Can we trust this comparison? Can we trust this comparison? ? WHAT POLICY SHOULD WE USE? SRPT

6. 5 mean response time SRPT M / GI / 1 Can’t implement pure SRPT What about multiserver systems? Real users are interactive What about fairness to large jobs? HOW DO REAL SYSTEMS DIFFER? What about QoS? What about user impatience? What about time-varying arrivals? What about power management?

7. 6 mean response time SRPT M / GI / 1 Can’t implement pure SRPT What about multiserver systems? Real users are interactive What about fairness to large jobs? What about QoS? What about user impatience? What about time-varying arrivals? What about power management? Idealized policies The idealized policies studied in theory cannot be used in practice Limited metrics Many performance metrics that are important in practice are not studied in theory Simplistic models Traditional models include many unrealistic assumptions 3 TYPES OF GAPS BETWEEN THEORY AND PRACTICE

8. Carnegie Mellon University Computer Science Department 7 7 THE GOAL OF THE THESIS: BRIDGE THE GAPS BETWEEN THEORY AND PRACTICE Moving beyond idealized policies 1 Moving beyond mean response time 2 Moving beyond the M/GI/1 3 60% of the talk 30% of the talk 10% of the talk

9. 8 SRPT Policies are hybrids of SRPT and PS How can we study all these variations at once? How can we study all these variations at once? ? Policies use only 2 levels In practice... Policies use estimates of job sizes In practice... Time-varying workloads  time-varying policies In practice... Designers adjust SRPT due to overheads In practice... IDEALIZED POLICIES

10. 9 THE IDEA Study scheduling classifications instead of idealized policies Study scheduling classifications instead of idealized policies SRPT SMART SMART formalizes the heuristic “give priority to small jobs”

11. 10 SRPT Policies are hybrids of SRPT and PS Policies use only 2 levels In practice... Policies use estimates of job sizes In practice... Time-varying workloads  time-varying policies In practice... Designers adjust SRPT due to overheads In practice... SMART SMART ε How do we define the SMART class? How do we define the SMART class? ? IDEALIZED POLICIES

12. Carnegie Mellon University Computer Science Department 11 THE SMART CLASS 1.Bias Property 2.Consistency Property 3.Transitivity Property SMAll Response Times coherency properties

13. 12 TWO NOTIONS OF “SMALL” JOBS small original sizesmall remaining size

14. 13 ab [Sigmetrics 2005a] BIAS PROPERTY If OriginalSize(a) < RemainingSize(b) then a has priority over b DON’T FORG ET

15. 14 original size 0 0 remaining size BIAS PROPERTY If OriginalSize(a) < RemainingSize(b) then a has priority over b [Sigmetrics 2005a] DON’T FORG ET

16. 15 original size 0 0 remaining size lower priority ? higher priority BIAS PROPERTY If OriginalSize(a) < RemainingSize(b) then a has priority over b [Sigmetrics 2005a] DON’T FORG ET

17. Carnegie Mellon University Computer Science Department 16 EXAMPLES PSJF original size remaining size ? RS Many others! SRPT If OrigSize(a) < RemSize(b) then a has priority over b Bias Property allows time varying policies Bias Property allows time varying policies !

18. 17 SRPT Policies are hybrids of SRPT and PS Policies use only 2 levels In practice... Policies use estimates of job sizes In practice... Time-varying workloads  time-varying policies In practice... Designers adjust SRPT due to overheads In practice... SMART SMART ε How close to SRPT are SMART policies? How close to SRPT are SMART policies? ? IDEALIZED POLICIES

19. 18 Bound T(x) SMART ANALYSIS SETTING: M/GI/1 preempt-resume queueAPPROACH: E[T] SMART

20. 19 Theorem: Under the M/GI/1, for all SMART policies P, CONDITIONAL RESPONSE TIME UNDER SMART POLICIES Waiting time Residence time Waiting time Residence time Response time for a job of size x [Sigmetrics 2005a]

21. 20 PSJF SRPT remaining size original size SMART ? Picture “proof”: Waiting time Theorem: Under the M/GI/1, for all SMART policies P, CONDITIONAL RESPONSE TIME UNDER SMART POLICIES

22. 21 SMART POLICIES ARE “2-COMPETITIVE” Theorem: In the M/GI/1, mean response time [Sigmetrics 2005a]

23. Carnegie Mellon University Computer Science Department 22 SMART POLICIES ARE “2-COMPETITIVE” load, ρ 1 0 mean response time PS SMART Theorem: In the M/GI/1, These bounds are tight These bounds are tight ! SRPT

24. 23 SRPT Policies are hybrids of SRPT and PS Policies use only 2 levels In practice... Policies use estimates of job sizes In practice... Time-varying workloads  time-varying policies In practice... Designers adjust SRPT due to overheads In practice... SMART SMART ε All SMART policies are within a factor of 2 All SMART policies are within a factor of 2 ! IDEALIZED POLICIES

25. Carnegie Mellon University Computer Science Department 24 If OrigSize(a) = x and ε (x) < RemSize(b) then a has priority over b SMART ε SMART If OrigSize(a) < RemSize(b) then a has priority over b remaining size original size ? ? ε(x)ε(x)

26. Carnegie Mellon University Computer Science Department 25 ε(x) = x + error How can you characterize job size estimates? If OrigSize(a) = x and ε (x) < RemSize(b) then a has priority over b SMART ε original size ? ε(x)ε(x) ε(x) can also be defined to include 2-level policies

27. 26 SMART ε POLICIES ARE “CONSTANT COMPETITIVE” Theorem: In an M/GI/1 under SMART ε policy P SIZE φ bounds the SIZE of larger jobs that get higher priority LOAD δ bounds the LOAD of larger jobs that get higher priority orig. size ? rem. size ε(x)ε(x) x

28. Carnegie Mellon University Computer Science Department 27 load 1 0 mean response time SMART ε PS SRPT real sizes  web server trace estimates  within 50% WHAT DOES THIS TRANSLATE TO IN PRACTICE? SMART ε allows adversarial job size errors SMART ε allows adversarial job size errors !

29. 28 SRPT Policies are hybrids of SRPT and PS Policies use only 2 levels In practice... Policies use estimates of job sizes In practice... Time-varying workloads  time-varying policies In practice... Designers adjust SRPT due to overheads In practice... SMART SMART ε IDEALIZED POLICIES

30. 29 Analyzing the SMART class beyond E[T] Introducing & analyzing other classifications MUCH MORE WORK ON CLASSIFICATIONS [Sigmetrics 2005a] [Sigmetrics 2006] [Perf. Eval. Review 2006] [Operations Research 2007] [Freidman and Hurley, 2004] [Rai, Urvoy-Keller, Vernon, Biersack 2005] [Nunez-Queija, Kherani 2006] [Misra, Rubenstein, Feng 2007] [Kherani 2007] [Sigmetrics 2003] [Sigmetrics 2005b] [Perf. Eval. Review 2006] Collaborations with Zwart, Nuyens, Shakkottai, Yang, Harchol-Balter, Osogami, and others

31. Carnegie Mellon University Computer Science Department 30 THE GOAL OF THE THESIS: BRIDGE THE GAPS BETWEEN THEORY AND PRACTICE Moving beyond idealized policies 1 Moving beyond mean response time 2 Moving beyond the M/GI/1 3 60% of the talk 30% of the talk 10% of the talk

32. 31 Designers care about power usage In practice... Designers care about QoS – Pr(T>x) In practice... Designers care about weighted response times In practice... MEAN RESPONSE TIME Designers care about fairness In practice... Designers care about buffer overflow probabilities

33. 32 Designers care about buffer overflow probabilities Designers care about power usage In practice... Designers care about QoS – Pr(T>x) In practice... Designers care about weighted response times In practice... Designers care about fairness In practice... MEAN RESPONSE TIME [Perf Eval 2002] [Sigmetrics 2003] [Sigmetrics 2005a] [PER 2007] [Sigmetrics 2005b] [Sigmetrics 2006] [OR 2007]

34. Carnegie Mellon University Computer Science Department 33 ARE POLICIES THAT PRIORITIZE SMALL JOBS UNFAIR TO LARGE JOBS?

35. 34 WHAT DOES “FAIRNESS” MEAN?...it depends entirely on the application OUR SETTING: OUR SETTING: Are the response times of large jobs “unfairly” long? How can we formalize this? How can we formalize this? ?

36. Carnegie Mellon University Computer Science Department 35 [Sigmetrics 2003: Best student paper award] Definition: In an M/GI/1 queue, a policy P is fair if, for all x: WHY IS THIS FAIR? Aristotle’s notion of fairness Like cases should be treated alike, different cases should be treated differently, and different cases should be treated differently in proportion to their differences. DON’T FORG ET

37. Carnegie Mellon University Computer Science Department 36 WHY IS THIS FAIR? Rawls’ Theory of Social Justice All social goods should be distributed equally, unless unequal distribution is to the advantage of the least favored Definition: In an M/GI/1 queue, a policy P is fair if, for all x: [Sigmetrics 2003: Best student paper award] DON’T FORG ET

38. Carnegie Mellon University Computer Science Department 37 WHY IS THIS FAIR? Min-Max fairness (Pareto Efficiency) All jobs deserve an equal share of the resources... but if some jobs can use more without hurting others, that’s okay Definition: In an M/GI/1 queue, a policy P is fair if, for all x: [Sigmetrics 2003: Best student paper award] DON’T FORG ET

39. 38 HOW UNFAIR ARE SMART POLICIES? x E[T(x)] / x 1/(1-ρ) SRPT Theorem: For all service distributions, SRPT is fair if ρ≤0.5. Theorem: For all power law (α) service distributions with α < 1.5, all SMART policies are fair. 1/(1-ρ) SMART

40. 39 x E[T(x)] / x 1/(1-ρ) Theorem: For all service distributions with finite variance, all SMART policies are unfair for high enough load. the largest jobs are treated the same as under PS small degree of unfairness BUT SMART POLICIES CAN BE UNFAIR SMART <4% of the job sizes What about other classifications? What about other classifications? ?

41. 40 FAIRNESS AND CLASSIFICATIONS Always Fair Sometimes Fair Always Unfair PS [Sigmetrics 2003 Best student paper award] SMART PSJF LRPT PLJF FOOLISH SRPT

42. 41 Always Fair Sometimes Fair Always Unfair Remaining size based LRPT Preemptive size based PSJF PLJF SJF LJF Non-preemptive size based Non-preemptive non-size based LCFS FCFS Age based LAS PLCFS PS SMART FOOLISH SRPT FAIRNESS AND CLASSIFICATIONS FSP [Sigmetrics 2003 Best student paper award] SYMMETRIC PROTECTIVE

43. 42 MUCH MORE WORK ON FAIRNESS [Williamson, Gong 2003, 2004] [Brown 2006] and others. analyzing policies & classifications extending definition to higher moments defining other types of fairness Many other papers by: Henderson, Friedman, Biersack, Rai, Ayesta, Aalto, Nunez-Queija, Misra, Feng, Vernon, Williamson, Brown, Bansal, and others [Raz, Avi-Itzhak 2004] [Levy, Raz, Avi-Itzhak 04] [Sandmann 2005] [Perf. Eval 2002] [Sigmetrics 2003] [Sigmetrics 2005b] [Under submission]

44. Carnegie Mellon University Computer Science Department 43 THE GOAL OF THE THESIS: BRIDGE THE GAPS BETWEEN THEORY AND PRACTICE Moving beyond idealized policies 1 Moving beyond mean response time 2 Moving beyond the M/GI/1 3 60% of the talk 30% of the talk 10% of the talk

45. 44 Multiserver designs are prevalent In practice... Job sizes may be correlated Arrivals are bursty In practice... Users are interactive In practice... The arrival process is time-varying In practice...M/GI/1

46. 45 Job sizes may be correlated Arrivals are bursty In practice... Real users are interactive In practice... The arrival process is time-varying In practice... Multiserver designs are prevalent In practice... M/GI/1 [PER 2004] [QUESTA 2005] [Perf Eval 2006] [NSDI 2006]

47. Carnegie Mellon University Computer Science Department 46 REAL USERS ARE INTERACTIVE How does this difference affect scheduling? How does this difference affect scheduling? ?

48. 47 Open System Send Receive [NSDI 2006] Closed System 0.25.5.75 1 load mean response time 300 200 100 mean response time 300 200 100 load 0.25.5.75 1 SRPT PS FCFS # users=75 PS SRPT FCFS

49. 48 REAL USERS ARE NOT OPEN OR CLOSED [NSDI 2006] mean response time mean number of requests per session 300 200 100 0 0 5 10 15 20 OPEN CLOSED PS SRPT load = 0.7 Where do real workloads fall? Where do real workloads fall? ?

50. 49 [NSDI 2006] mean response time mean number of requests per session 300 200 100 0 0 5 10 15 20 OPEN CLOSED slashdotted site CMU web server Kasparov vs. Deep Blue online shopping world cup site online gaming site Where do real workloads fall? Where do real workloads fall? ?

51. Carnegie Mellon University Computer Science Department 50 USER BEHAVIOR IMPACTS SYSTEM DESIGN When evaluating new designs, choose a workload generator carefully When evaluating new designs, choose a workload generator carefully !

52. Carnegie Mellon University Computer Science Department 51 THE GOAL OF THE THESIS: BRIDGE THE GAPS BETWEEN THEORY AND PRACTICE Moving beyond idealized policies 1 Moving beyond mean response time 2 Moving beyond the M/GI/1 3 60% of the talk 30% of the talk 10% of the talk

53. 52 THE GOAL OF THE THESIS: BRIDGE THE GAPS BETWEEN THEORY AND PRACTICE Moving beyond idealized policies 1 Moving beyond mean response time 2 Moving beyond the M/GI/1 3 Scheduling classifications Fairness & QoS – Pr(T>x) Interactive users & Multiserver systems

54. 53 Adam Wierman Carnegie Mellon University acw@cs.cmu.edu The thesis is available at: http://www.cs.cmu.edu/~acw/thesis

55. 54

56. 55 Non-preemptive SMART LCFS SMART* Remaining size based SRPT RS SMART Є Preemptive size based PSJF FOOLISH FOOLISH* LRPT PLJF ROS Blind Age based PLCFS SYMMETRIC PROTECTIVE PS FSP SJF LJF Non-preemptive size based FCFS FB

57. 56 THE BIAS PROPERTY ISN’T ENOUGH orig. size remaining size ? CONSISTENCY TRANSITIVITY [Sigmetrics 2005a] at most 1 has higher priority + If a is served ahead of b then a will always have priority over b If an arriving job b preempts c, then until b leaves, every arriving job a with original size smaller than b has priority over c.

58. 57 Theorem: Proof sketch: If there are two, one was the first to have priority over the tagged job. x 1 2a 2b By Consistency 2a can’t receive service 2b has lower priority than x (Bias). If 2b is run, then 1 has lower priority than 2b (Consistency). So, 1 has lower priority than x (Transitivity). at most 1 has higher priority ?

59. 58 WEB WORKLOAD GENERATORS Surge SPECWeb TPC-W Sclient RUBiS WebBench Webjamma DO YOU USE AN OPEN OR CLOSED MODEL? Open System Closed System httperf

60. 59 Theorem: In an M/GI/1 with an unbounded, continuous service distribution having finite E[X 2 ], under any non-idling policy we have and further [Wierman and Harchol-Balter 2003] Is dividing by “x” the right scaling? Is dividing by “x” the right scaling? ? Is 1/(1-ρ) really a min-max criteria Is 1/(1-ρ) really a min-max criteria ?

61. 60 First Come First Served x Under a Pareto with ρ=0.8, this is >80% of the jobs The unfairness can be unbounded PS FCFS E[T(x)] / x

62. 61 SJF LJF SRPT LRPT ROS LCFS FCFS FB PS PLCFS PSJF PLJF Always Fair Always Unfair Sometimes Fair FAIRNESS VS. EFFICIENCY more circles  better mean response time Is there a fair policy with near optimal performance? Is there a fair policy with near optimal performance? ?

63. 62 Fair Sojourn Protocol (FSP) “Do SRPT on the PS remaining times” FSP did the same thing as SRPT FSP did the same thing as SRPT !

64. 63 BEYOND EXPECTATION: BEYOND EXPECTATION: Higher Moments Raw moments E[T(x) i ] Central moments Var[T(x)], etc Cumulant moments X X [Wierman and Harchol-Balter 2005] E[T(x)]  ?

65. 64 CUMULANTS Cumulants are a descriptive statistic, similar to the moments. They can be found as a function of the moments: or from the log of the moment generating function: Do these look familiar? Do these look familiar? ?

66. 65 WHY CUMULANTS? Cumulants have many nice properties: additivity: homogeneity: 1 st cumulant is shift-equivariant & the rest are shift-invariant

67. 66 Why is this the right generalization? Why is this the right generalization? ? MIN-MAX FAIRNESS Definition: Consider an M/GI/1 queue. A policy P is min-max fair if, for all i: Wierman and Harchol-Balter 2005 Lots of open questions here Lots of open questions here !

68. 67 TEMPORAL FAIRNESS Definition: The politeness experienced by a job of size x under policy P, Pol(x) P, is the fraction of the response time during which the seniority of the job is respected. It is unfair to violate the seniority of a job [Wierman 2004]

69. 68 Min-max Fairness Politeness less fair more fair less polite more polite PS PLCFS FCFS SRPT FSP FCFS PS PLCFS SRPT FSP LRPT Theorem In an M/GI/1 any Always Fair policy has

70. 69 Min-max Fairness Politeness less fair more fair less polite more polite FCFS PS PLCFS LRPT more circles  better mean response time FSP SRPT

71. 70 MANY OTHER INTERESTING FAIRNESS METRICS percent of service given to job i fair service percentage DiscFreq = n i + c∙m i n i = number of jobs that arrived later and completed earlier than job i m i = number of larger jobs (at the arrival of job i) that complete earlier than job i [Levy, Raz, Avi-Itzhak 04] [Sandmann 2005]

72. 71 SMART POLICIES ARE 2-COMPETITIVE Theorem: In the M/GI/1, These bounds are tight These bounds are tight ! Consider the M/D/1 SRPT does FCFS (only in M/D/1). So as ρ  1 As ρ  1, E[T] PLCFS  2 E[T] SRPT PLCFS is in SMART (only in M/D/1)

73. 72 ONLINE MULTI-OBJECTIVE SCHEDULING USING SMART 1.Use a parameterized policy set that is (nearly) dense in SMART, e.g. iR j + S 2.Search (i,j) space for policy that optimizes secondary objectives, e.g. fairness and predictability Partial ordering allows time varying policies Partial ordering allows time varying policies !

74. 73 A B C D Partial ordering allows time varying policies Partial ordering allows time varying policies ! time remaining size AB C D *PSJF

75. 74 A B C D *PSJF *SRPT time remaining size *PSJF A B D ? Partial ordering allows time varying policies Partial ordering allows time varying policies !

76. 75 TAIL BEHAVIOR OF SMART Pr(T>y) is difficult to study directly so it is typically it is studied asymptotically Large buffer Many sources NμNμ NB λ1λ1 λ2λ2 λNλN SMART policies are asymptotically equivalent in both SMART policies are asymptotically equivalent in both !

77. 76 LARGE BUFFER SCALING X is of intermediate regular variation if X is light-tailed if for some s>0 For this talk, assume no mass in the upper bound. LIGHT-TAILED JOB SIZES HEAVY-TAILED JOB SIZES

78. 77 SMART POLICIES ARE ASYMPTOTICALLY EQUIVALENT [Nuyens, Wierman, Zwart 2005] Theorem: Under the GI/GI/1, for all SMART policies: when the service distribution is light-tailed with no mass in the endpoint when the service distribution is of intermediate regular variation service distribution busy period length

79. 78 [Nunez-Queija, Boxma, Zwart, Borst, Nuyens, and many others] Pr(T>y) ~ busy period SMART worse SMART POLICIES ARE ASYMPTOTICALLY EQUIVALENT LIGHT-TAILED JOB SIZES HEAVY-TAILED JOB SIZES Log Pr(T>y) ~ busy period SMART LCFS SJF Log Pr(T>y) ~ workload Pr(T>y) ~ workload FCFS... FCFS LCFS SJF...

80. 79 TAIL BEHAVIOR OF SMART Pr(T>y) is difficult to study directly so it is typically it is studied asymptotically Large buffer Many sources NμNμ NB λ1λ1 λ2λ2 λNλN SMART policies are asymptotically equivalent in both SMART policies are asymptotically equivalent in both !

81. 80 MANY SOURCES SCALING decay rate The same under all SMART policies μ B λ NμNμ NB λ1λ1 λ2λ2 λNλN [Yang, Wierman, Shakkottai, Harchol-Balter, 2006]

82. 81 T(x) RESULT, plot for E[T(x)] Theorem: For all ε, x, y > 0 where PRIO is a 2 class priority queueing policy. SMART POLICIES ARE ASYMPTOTICALLY EQUIVALENT [Yang, Wierman, Shakkottai, Harchol-Balter, 2006] original size ? remaining size Empty! Picture “proof”:

83. 82 T(x) RESULT, plot for E[T(x)] Theorem: Under the M/GI/1, for all SMART ε : when the service distribution is unbounded and light-tailed with no mass in the endpoint when the service distribution is of intermediate regular variation and Nuyens, Wierman, under preparation busy period length TAIL BEHAVIOR OF SMART ε POLICIES

84. 83 low variabilityhigh variability mean response time 1500 1000 500 Open Closed (MPL=10) Closed (MPL=100) Closed (MPL=1000) Web Workloads HOW QUICKLY DOES CLOSED  OPEN?

85. 84 CHOOSING A SYSTEM MODEL 1.A site being “Slashdotted” 2.Online gaming site 3.Science Institute - USGS 4.Online dept. store 5.Financial service provider 6.Kasparov vs Deep Blue 7.CMU web server 8.World cup site Web workloads Open or closed? Use a partly-open model...

86. 85 FITTING A PARTLY-OPEN MODEL 12 ip1 GET a.gif HTTP/1.0 20 ip2 GET b.htm HTTP/1.0 25 ip1 GET c.jpg HTTP/1.0 27 ip1 GET d.txt HTTP/1.0 28 ip3 GET a.htm HTTP/1.0 35 ip4 GET d.gif HTTP/1.0 45 ip2 GET e.htm HTTP/1.0 : Trace service demands file sizes from trace PARTLY-OPEN PARTLY-OPEN

87. 86 FITTING A PARTLY-OPEN MODEL 12 ip1 GET a.gif HTTP/1.0 20 ip2 GET b.htm HTTP/1.0 25 ip1 GET c.jpg HTTP/1.0 27 ip1 GET d.txt HTTP/1.0 28 ip3 GET a.htm HTTP/1.0 35 ip4 GET d.gif HTTP/1.0 45 ip2 GET e.htm HTTP/1.0 : Trace PARTLY-OPEN PARTLY-OPEN Fitting the interarrival times Distinguish users e.g. use ip address in a web trace Identify user session boundaries  Use periods of inactivity of length > timeout Can’t use trace directly because dependencies between completions and follow-up requests would be lost!

88. 87 CHOOSING A TIMEOUT VALUE Number of sessions 2e5 1e5 0 0 30min Timeout length financial world cup dept store

89. 88 HOW TO CHOOSE A SYSTEM MODEL Gather a trace How many simult. users are there? Fit a partly open model to the trace OPEN ≈ CLOSED >>1000 else What is the expected num. of visits? OPENCLOSED??? <55-10 >10 Mean num. of visits 15 10 5 0 0 30min Timeout length world cup dept store financial

90. 89 MULTISERVER QUEUES Preemptive-Resume Priority Homogeneous hosts jobs L H H L H H

91. 90 HOW MANY SERVERS ARE BEST? 1 best 2 best 3 best 4 best 1 best 2 3 4 best 1 fast (rate 1) vs. k slow (rate 1/k)

92. 91 mean response time SRPT M / GI / 1 What about QoS? Can’t implement pure SRPT What about multiserver systems? Real users are interactive What about fairness to large jobs? What about time-varying workloads? What about user impatience? What about power usage? GAPS BETWEEN THEORY AND PRACTICE

93. 92 Remember Important ! Idea warning DON’T FORG ET question ?