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Use of a Levy Distribution for Modeling Best Case Execution Time Variation
- 1. Use of a Levy Distribution for Modeling Best Case Execution Time Variation Jonathan Beard, Roger Chamberlain SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Work also supported by: 1
- 2. Outline • Motivation! • Stream Processing! • Optimization Goals! • Methodology! • Distributions! • Results 2
- 3. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Streaming Computing 3
- 4. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Streaming Computing Kernel 3
- 5. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Streaming Computing Kernel 1 Kernel 2 Kernel 3 Kernel 2 Stream Stream Stream Stream 4
- 6. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Streaming Languages StreamIt, Auto-Pipe, Brook, Cg, S-Net, Scala-Pipe, Streams-C and many others 5
- 7. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Optimization Slow Fast Kernel Super Fast Medium 6
- 8. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Optimization Kernel 1 Kernel 2 Kernel 3 Kernel 2 multi-core A 1 2 3 4 multi-core B 1 2 3 4 More allocation choices, NUMA node A or B to allocate stream. 7
- 9. 1 2 SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Optimization Kernel 1 Kernel 2 Kernel 3 Kernel 2 multi-core A 1 2 3 4 multi-core B 1 2 3 4 More allocation choices, NUMA node A or B to allocate stream. 7
- 10. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Optimization Kernel 1 Kernel 2 Kernel 3 Kernel 2 multi-core A 1 2 3 4 multi-core B 1 2 3 4 More allocation choices, NUMA node A or B to allocate stream. 1 2 7
- 11. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Optimization A B C “Stream” is modeled as a Queue A Q1 B Q2 C 8
- 12. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Optimization A B C “Stream” is modeled as a Queue A Q1 B Q2 C 8
- 13. We want good models for streaming systems on shared multi-core systems (i.e., a cluster) Problem: Accurate measurement is very difficult. Is there a way to decide on a model without it. • Commodity multi-core timer availability and latency • Frequency scaling and core migration • Measuring modifies the application behavior SBS Streaming on Multi-core Systems Stream Based Supercomputing Lab http://sbs.wustl.edu 9
- 14. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Derived Information Expected Observed 10
- 15. SBS Is there a pattern of minimal variation within the Stream Based Supercomputing Lab http://sbs.wustl.edu Derived Information Expected Observed systems we’re running on? Avg. Service Time = E[ X ] + Error 10
- 16. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Goal Find a distribution that characterizes the minimum expected variation of a hardware and software system Use this characterization to accept or reject models 11
- 17. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Process 12 • Measurement! • Workload definition! • Find a distribution! • Utilize the distribution to aid model selection
- 18. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Timer Mechanism Timer Thread Code 13 Ask for Time Receive Time
- 19. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Timer Mechanism Timer Thread rdtsc clock_gettime 14 • x86 assembly • varying methods to serialize • relatively fast • multiple drift issues • POSIX standard • relatively accurate • portable • slower than rdtsc
- 20. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Two Timing Choices 15
- 21. SBS NUMA Node Variations Stream Based Supercomputing Lab http://sbs.wustl.edu 16
- 22. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Minimize Variation • Restricting timer to single core ! • Use the x86 rdtsc instruction with processor recommended serializers for each processor type ! • Keeping processes under test on the same NUMA node as timer ! • Run timer thread with altered priority to minimize core context swaps 17
- 23. SBS Best Case Execution Time Variation • no-op instruction implemented in most processors ! • usually takes exactly 1 cycle ! • no real functional units are involved, so least taxing ! • variation observed in execution time should be external to process Stream Based Supercomputing Lab http://sbs.wustl.edu 18
- 24. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Data Collection • no-op loops calibrated for various nominal times, tied to a single core and run thousands of times ! • Execution time measured end to end for each run, environment collected ! • Parameters include: Number of processes executing on core Number of context swaps (voluntary, involuntary) Many others 19
- 25. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Levy Distribution 20 Execution Time Error ( obs - mean )
- 26. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Levy Distribution 21 Normal Distribution
- 27. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Levy Distribution 22 Gumbel Distribution
- 28. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Levy Distribution 23 Levy Distribution
- 29. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Levy Distribution 23 Levy Distribution
- 30. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Levy Distribution • Truncation enables mean calculation, but requires fitting to each dataset to find where to truncate ! • The truncation parameters are correlated to both the number of processes per core and the expected execution time ! • Roughly linear relationship gives an approximate solution to truncation parameters without refitting 24
- 31. 1 - 5 processes 6 - 10 processes !0.000014 !0.0000145 !0.000015 !0.0000155 SBS 11 - 15 processes 16 - 20 processes !0.00002 !0.000025 !0.00003 !0.000035 !0.00004 !0.000045 !0.00005 Stream Based Supercomputing Lab http://sbs.wustl.edu Levy Fit !0.0000125 !0.000013 !0.0000135 !0.000014 !0.00001 !0.000015 !0.00002 !0.000025 !0.00003 25 !0.000025 !0.00001 !0.000025 !0.00001 0 !0.00006 !0.00003 0 !0.00005 !0.00002 0
- 32. Hypothesis: Lower Kullback-Leibler (KL) divergence SBS Question: Can we use an M/M/1 queueing model to estimate the mean queue occupancy of this system? Stream Based Supercomputing Lab http://sbs.wustl.edu Test Setup A Q1 B ! between expected and realized distribution is associated with higher model accuracy. 26
- 33. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Test Setup A Q1 B 1. Dedicated thread of execution monitors 27 queue occupancy 2. Calculate the estimated mean queue occupancy using the M/M/1 model 3. Calculate KL Divergence for the arrival process distribution using the truncated Levy distribution noise model
- 34. SBS Convolution with Exponential Stream Based Supercomputing Lab http://sbs.wustl.edu 28
- 35. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Conclusions • The truncated Levy distribution can be used to approximate BCETV ! • The distribution of BCETV can be used as a tool to accept or reject a stochastic queueing model based on distributional assumptions ! • KL divergence between the expected and convolved distribution highly correlates with queue model accuracy 29
- 36. SBS Stream Based Supercomputing Lab http://sbs.wustl.edu Parting Notes Slides available here: sbs.wust.edu ! Timer C++ template code: http://goo.gl/ItJ3jP ! Test harness used to collect data: http://goo.gl/U1VG6N 30