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Some Systems, Applications and Models I Have Known

Some Systems, Applications and Models I Have Known. Ken Sevcik University of Toronto. Overview. In the past 35 years, … Systems Have Changed Applications Have Grown Models Have Matured and Adapted … and some interesting problems have been encountered.

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Some Systems, Applications and Models I Have Known

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  1. Some Systems, Applicationsand Models I Have Known Ken Sevcik University of Toronto Sigmetrics and Performance 2004

  2. Overview • In the past 35 years, … • Systems Have Changed • Applications Have Grown • Models Have Matured and Adapted • … and some interesting problems have been encountered Sigmetrics and Performance 2004

  3. First Research Application: Probability of a Voters’ Paradox • C candidates for election • V voters with strict preference orderings • Can one candidate beat each other pairwise? Example: V = 3 & C = 3 V1 : X > Y > Z V2 : Y > Z > X V3 : Z > X > Y Then, in pair-wise elections, X beats Y ; and Y beats Z ; yet Z beats X ! Paradox occurs in 12 of the (3!)3 = 216 possible configurations. In general, there are (C!)V voting configurations. Sigmetrics and Performance 2004

  4. My first “personal” computer: IBM System 360 Model 30 with BOS Sigmetrics and Performance 2004

  5. Exact Probabilities of Voters’ Paradox • V = 3 & C = 3  12 cycles in 216 configs. • V = 7 & C = 7  26,295,386,028,643,902,475,468,800 cycles in 82,606,411,253,903,523,840,000,000 configs. (Computed in approximately 40 hours of CPU time.) C = 3 5 7 ~ 40 V = 3 .0555… .1600… .238798185941 ~ .61 V = 5 .06944… .19999525 .295755170299 ~ .71 V = 7 .075017 .215334 .318321370333~ .74 V ~ 40 ~ .09 ~ .24 ~ .36 ~ .80 Sigmetrics and Performance 2004

  6. Job Sequencing on a Single Processor (using service time distribution knowledge) “Smallest Rank” (SR) Scheduling: Minimize Investment (quantum length) Payoff (Pr [Completion]) = Service Time Knowledge exact average distribution No SPT SEPT SEPT Preemption Allowed? Yes SRPT SERPT SR Sigmetrics and Performance 2004

  7. Job Sequencing with Two Processors & Two Customers Extending “Shortest First” to Multiple Resources SBT-RSBT -- Based on average service time per visit of each customer at each resource SBT:  A gets priority at k RSBT:  A gets priority at 1 Sigmetrics and Performance 2004

  8. In the Beginning … • Single Server Queue • Many variations • arrival process, service process • multiple servers, finite buffer size • scheduling discipline • FCFS, RR, FBn, PS, SRPT, … N , Z S RR, FBn, and PS increased relevance of models Sigmetrics and Performance 2004

  9. Z avg. think time K centers Dj demand at j Queuing Network Models “Central Server” Model “Separable” (or “product form”) models N customers and efficient computational algorithms Variants: Open, Closed, Mixed scheduling disciplines Sigmetrics and Performance 2004

  10. The “Great Debate”:Operational Analysis vs. Stochastic Modeling • SM • Ergodic stationary Markov process in equilibrium • Coxian distributions of service times • independence in service times and routing • OA • finite time interval • measurable quantities • testable assumptions OAmade analytic modelling accessible to capacity planners in large computing environments Sigmetrics and Performance 2004

  11. Uses and Analysis of Queuing Network Models • Applications • System Sizing; Capacity Planning; Tuning • Analysis Techniques • Global Balance Solution • Massive sets of Simultaneous Linear Equations • Bounds Analysis • Asymptotic Bounds (ABA), Balanced System Bounds (BSB) • Solutions of “Separable” Models • Exact (Convolution, eMVA) • Approximate (aMVA) • Generalizations beyond “Separable” Models • aMVA with extended equations Sigmetrics and Performance 2004

  12. Bounding Analysis Case Study: Insurance Company with 20 sites • Upgrade alternatives: Upgrade Dcpu Dio Dtot Improvement Current 4.6 4.0 10.6 ----- # 1 5.1 1.9 7.0 1.5 to 2.0 # 2 3.1 1.9 5.0 2.0 to 3.5 ABA Inputs:N, Z, Dtot,Dmax Throughput Bound: Response Time Bound: Sigmetrics and Performance 2004

  13. Bounding Analysis Case Study: Insurance Company with 20 sites • Upgrade alternatives: Upgrade Dcpu Dio Dtot Improvement Current 4.6 4.0 10.6 # 1 5.1 1.9 7.0 1.5 to 2.0 # 2 3.1 1.9 5.0 2.0 to 3.5 .4 #2 .3 X Cur .2 #1 .1 2 4 6 8 10 N Sigmetrics and Performance 2004

  14. Bounding Analysis Case Study: Insurance Company with 20 sites • Upgrade alternatives: Upgrade Dcpu Dio Dtot Improvement Current 4.6 4.0 10.6 # 1 5.1 1.9 7.0 1.5 to 2.0 # 2 3.1 1.9 5.0 2.0 to 3.5 # 1 Cur 20 # 2 15 R 10 5 N 2 4 6 8 10 Sigmetrics and Performance 2004

  15. Exact Mean Value Analysis Algorithm Initialize (for zero customers): Iterate up to N customers: for n = 1, … , N Set Arrival Instant Queue Lengths: Set Residence Time: Understandable and Easy to Implement Sigmetrics and Performance 2004

  16. Approximate Mean Value Analysis Initialize to Equal Queue Lengths: Iterate until convergence: loop until Qk ( N ) are stable Revise Arrival Instant Queue Lengths: Revise Residence Times: Substantial time savings; Little loss of accuracy Sigmetrics and Performance 2004

  17. “Details” of Real Systems • Going beyond “Separable” models • Priority Scheduling • Alter Residence Time equation • FCFS with high variance service times • Reflect coefficient of variation in service times • Memory Constraints • Alter MPL limit N , or Dpaging • I/O Subsystems (simultaneous resource possession) • Reflect contention by inflating Ddisk • Enhanced Utility of QNM’s for Real Systems Sigmetrics and Performance 2004

  18. System Sizing Case Study:NASA Numerical Aerodynamic Simulator GOAL: to attain a sustainable Gigaflop Work Stations Data Mgmt Cray 1 Cray 2 Cray 3 Graphics QNM’s proved more useful than a simulation model Sigmetrics and Performance 2004

  19. QNM’s for Capacity Planning & Tuning • Existing system with measurable workload • “What if …” • … the workload volume increases? • … the workload mix changes? • … the processor is upgraded? • … memory is added? • … the I/O configuration is enhanced? • … class priorities are adjusted? • … file placements are changed? • … changing usage of memory? • Answer by changing model parameters CAPACITY PLANNING TUNING Sigmetrics and Performance 2004

  20. Capacity Planning Case Study: FAA Air Traffic Control System • ~ 40 distributed air traffic control centers • Each with the SAME: • software • hardware family • 35 transaction types • But DIFFERENT: • transaction volumes and mixes • Single QNM (one class per transaction type) supports capacity planning for all sites Sigmetrics and Performance 2004

  21. QNM’s for System and Architecture Analysis • Architectures • caching structures • Communication networks • Local Area Networks • Rings, buses • Store and Forward • flow control • end to end response time • Interconnection networks • omega, shuffle-exchange, … Sigmetrics and Performance 2004

  22. SE&EU Interconnection Network Source 000 001 010 011 100 101 110 111 Destination 000 001 010 011 100 101 110 111 Exchange Unshuffle Shuffle Exchange Sigmetrics and Performance 2004

  23. Sn Sn-1 Sn-2 S4 S3 S2 S1 Bn B1 Bn-1 B2 Bn-2 B3 B4 Bn-3 (Longest Matching Bit String) Dn Dn-1 Dn-2 D4 D3 D2 D1 SE&EU operation Combination Lock Algorithm: SE: Left 3 EU: Right 5 SE: Left 2 Up to 40% increase in throughput Sigmetrics and Performance 2004

  24. Space Station Orbital Platform Tethered Platform Shuttle Extra-Vehicular Activity Network for NASA’s Space Station (circa 1984) • Distributed LAN for many components Results: Some properties of the FDDI Protocol Ground Station Sigmetrics and Performance 2004

  25. Continuing vs. Upward Exiting vs. Entering Architectural Analysis Case Study: NUMAchine • 4 x 4 x 4 Hierarchical Ring Architecture Setting Routing Priorities: Message Handling: Contiguous vs. Interleaved Shortest First ? Sigmetrics and Performance 2004

  26. Job Scheduling for Parallel Processing Variants: Static Moldable Malleable Dynamic Job j = ( tj , pj ) 1 2 3 processors P time Sigmetrics and Performance 2004

  27. j1 j2 j1 j2 Parallelism: Early or Late ? • Problem • Schedule N jobs of two tasks each on two processors to minimize average residence time • Each pair of jobs can be executed as … PARALLEL: SEQUENTIAL: overhead of parallel execution Sigmetrics and Performance 2004

  28. Results of two similar studies: [RN et al.] Start parallel; Finish sequential Parallelism: Early or Late ? S S S P P P P P P S S S Sigmetrics and Performance 2004

  29. Results of two similar studies: [RN et al.] Start parallel; Finish sequential [KCS] Start sequential; Finish parallel Parallelism: Early or Late ? S S S P P P P P P S S S S S S P P P P P P S S S Sigmetrics and Performance 2004

  30. Results of two similar studies: [RN et al.] Start parallel; Finish sequential [KCS] Start sequential; Finish parallel Differences in assumptions: Some variability in task service times ( or ) [RN] Some overhead of parallelism ( ) [KCS] Parallelism: Early or Late ? S S S P P P P P P S S S S S S P P P P P P S S S Sigmetrics and Performance 2004

  31. P P P S P P S S S P S S S S S P P P P P P P P P P P P P P P P P P P P P S S S S S S S S S P P P P P S P P S S P P S S S P P P S P P S S P P S S S P P P P P P P S P S S S S S S S S S S S S S S S S S S S S S S S Parallelism: Early or Late ? • Resolution increasing increasing Sigmetrics and Performance 2004

  32. Distributed Processing Models • Processor selection strategies • local vs. global execution • Load Sharing • sender-initiated vs. receiver-initiated Sigmetrics and Performance 2004

  33. Small example: Individual Versus Social Optimum • Arriving customers must pick one of two processors, one fast and one slow: pF F pS S Individual Optimum: Pick server with lower response time (  response times are equalized) Social Optimum: Control pF to minimize avg. response time Sigmetrics and Performance 2004

  34. Resolution of Social and Individual Goals Individual Optimum: Social Optimum minimizes: Toll on F: Rebate on S: RESULT: Everybody Wins !!! Sigmetrics and Performance 2004

  35. +2 0 -2 -2 0 +2 Anomaly of High Dimensional Spaces 2k Spheres (radius = 1) in Cube (vol. 4k & 2 k sides) and an Inner sphere 1. Pointy-ness Property 2. Radius of Inner Sphere R2 = .414 R10 = 2.16 !!! 3. Volume Ratio Sigmetrics and Performance 2004

  36. Diagonal of a k-dimensional Cube (Example: k = 25 ) Corners = Red = Blues = Sigmetrics and Performance 2004

  37. Blue width = Red width = Corner width = Diagonals of Cube K = 1 K = 2 K = 3 K = 4 Sigmetrics and Performance 2004

  38. Diagonals of Cube K = 9 K = 121 (There are 2121 blue spheres) Sigmetrics and Performance 2004

  39. A1 A2 A3 A4 … Ak-1 Ak A1 A3 A2 Multidimensional Databases Relational View: (Records of k Attributes) Multidimensional View: (Points in k-dimensional space) Indexing Support for: -- point search -- range search -- similarity search -- clustering Sigmetrics and Performance 2004

  40. Bounding Spheres and Rectangles circumscribed inscribed ratio of Dim k sphere cube sphere volumes -------- ---------------- ---------- --------------- ------------- 2 1.57 1.00 .785 2 4 4.93 1.00 .308 16 8 64.94 1.00 .0159 4096 16 15422.64 1.00 .000004 4294967296 Sigmetrics and Performance 2004

  41. Edge Density in High-Dimensions • Proportion of points near some side: Fraction near some edge: 1 k eps = .002 .020 .200 ---- ------ ------ ----- 1 .004 .040 .400 2 .007 .078 .640 4 .015 .150 .870 8 .031 .278 .983 16 .062 .479 .999 Sigmetrics and Performance 2004

  42. Lessons and Conclusions • Exact answers are overrated • accurate approximate answers often suffice • (e.g., Voters’ Paradox and aMVA ) • Analytic models have an important role • quick, inexpensive answers in many situations • (e.g., Insurance Co., NAS System, and FAA System ) • Assumptions matter • subtle differences can have big effects • (e.g., in Early or Late Parallelism, NUMAchine analysis and PRI vs. FCFS or PS) Sigmetrics and Performance 2004

  43. What is the “best” way to attain largeimprovements in computer performance? • -- Analysis? • -- Simulation? • -- Experimentation? Sigmetrics and Performance 2004

  44. What is the “best” way to attain largeimprovements in computer performance? • -- Analysis? • -- Simulation? • -- Experimentation? • None of the above … Just wait 30 years!!! Sigmetrics and Performance 2004

  45. ACM Sigmetrics & IFIP W.G. 7.3 , Thanks for the memories … Sigmetrics and Performance 2004

  46. Problems with Voting Systems • Problems have occurred recently in .. • France (lowest eliminated) • R > M > L 40% • L > M > L 40% • M > (R, L) 20% • Middle eliminated in first round though rank score (2.2) • Beats rank score of others (1.9) • USA (primaries, and electoral college) • E.g., McCain loses to Bush in primaries although he • Might be both candidates in a final election Sigmetrics and Performance 2004

  47. Exact Mean Value Analysis Algorithm for n = 1, … , N -- Understandable -- Easy to implement -- Arrival Instant Theorem end for Sigmetrics and Performance 2004

  48. Approximate Mean Value Analysis loop -- Substantial time savings -- Little loss of accuracy exit when X(N) and R(N) converge end loop Sigmetrics and Performance 2004

  49. The Case for Popt = 1 : • (Assume p > 1 Ej (p) < 1 ) • Argument: • Demand is insatiable (unbounded backlog) • Economies of scale (100’s of users) • “Good” systems will be heavily used • Parallelism overhead decreases throughput and increases queuing times Sigmetrics and Performance 2004

  50. System Sizing Case Study:NASA Numerical Aerodynamic Simulator Sigmetrics and Performance 2004

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