1. Counter Stacks: Storage Workload Analysis via Streaming Algorithms Nick Harvey University of British Columbia and Coho Data Joint work with Zachary Drudi, Stephen Ingram, Jake Wires, Andy Warfield

2. Caching What data to keep in fast memory? Fast, Low-Capacity Memory Slow, High-Capacity Memory

3. Caching Historically Registers RAM Disk Belady, 1966: FIFO, RAND, MIN Denning, 1968: LRU

4. Caching Modern Registers, L1, L2, L3 RAM Disk SSD Cloud Storage Proxy CDN Associative map LRU etc. LRU Consistent Hashing... from 1968 CPUs are >1000x faster Disk latency is <10x better Cache misses are increasingly costly

5. Challenge: Provisioning How much cache should you buy to support your workload?

6. Challenge: Virtualization Modern servers are heavily virtualized How should we allocate the physical cache among virtual servers to improve overall performance? What is “marginal benefit” to giving server more cache?

7. Understanding workloads better can help – Administrators make provisioning decisions – Software make allocation decisions Storing a trace is costly: GBs per day Analyzing and distilling traces is a challenge Understanding Workloads

8. Hit Rate Curve Hit rate 0.20.10.300.40.5 MSR Cambridge “TS” Trace, LRU Policy Fix a particular workload and caching policy If cache size is x, what would hit rate be? HRCs are useful for choosing an appropriate cache size Cache Size (GB) “Elbow” “Knee” “Working Set” Not much marginal benefit of a bigger cache

9. Hit Rate Curve Real-world HRCs need not be concave or smooth “Marginal benefit” is meaningless “Working set” is a fallacy Cache Size (GB) Hit rate MSR Cambridge “Web” Trace, LRU Policy 402060080 “Elbow”? “Knee”? “Working Set”?

10. LRU Caching Policy: An LRU cache of size x always contains the x most recently requested distinct symbols. A B C A D A B … If cache size >3 then B will still be in the cache during the second request for B. – Second request for B is a hit for cache size x if x>3. Inclusive: Larger caches always include contents of smaller caches. 3 distinct symbols “Reuse Distance”

11. Mattson’s Algorithm Maintain LRU cache of size n; simulate cache of all sizes x · n. Keep list of all blocks, sorted by most recent request time. Reuse distance of a request is its position in that list. If distance is d, this request is a hit for all cache sizes ¸ d. Hit rate curve is CDF of reuse distances. A B C A D A B … List: AB AC B AA C BD A C BA D C BB A D C Requests:

12. Faster Mattson [Bennett-Kruskal 1975, Olken 1981, Almasi et al. 2001,...] Maintain table mapping block to time of last request # of blocks whose last request time is ¸ t = # of distinct blocks seen since time t Can compute this in O(log n) time with a balanced tree Can compute HRC in O(m log n) time A B C A D A B … Block A B C D Space is  (n) n = # blocks m = length of trace

13. Is linear space OK? A modern disk is 8TB, divided in 4kB blocks ) n = 2B The problem is worse in multi-disk arrays ) n = 15B If the algorithm for improving memory usage consumes 15GB of RAM, that’s counterproductive! 60TB JBOD

14. We ran an optimized C implementation of Mattson on the MSR-Cambridge traces of 13 live servers over 1 week Trace file is 20GB in size, 2.3B requests, 750M blocks (3TB) Processing time: 1 hour RAM usage: 92GB Lesson: Cannot afford linear space to process storage workloads Question:Can we estimate HRCs in sublinear space? Is linear space OK?

15. Quadratic Space A B C A D A B Requests: Set of all subsequent items: A BB CCC AAA DDDDD AA BBBBB Items seen since first request Items seen since second request Reuse distance is size of oldest set that grows. Hit rate curve is CDF of reuse distances. Reuse Distance = 2 Reuse Distance = 3 Reuse Distance = 1

16. Quadratic Space A B C A D A B Requests: For t=1,…,m Receive request b t; Find minimum j such that b t is not in j th set Let v j be cardinality of j th set Record a hit at reuse distance v j Insert b t into all previous sets Set of all subsequent items: A BB CCC AAA DDDDD AA v j = 3 j=3

17. More Abstract Version For t=1,…,m Let v j be cardinality of j th set Receive request b t Let ± j be change in j th set’s cardinality when adding b t For j=2,…,t Record ( ± j - ± j-1 ) hits at reuse distance v j A B C A D A B Requests: Set of all subsequent items: A BB CCC AAA DDDDD AA ±j:±j: 0 0 1 1 1 1 ± j - ± j-1 : 0 0 1 0 0 0 v j = 3 How should we represent these sets?Hash table? ; Insert b t into all previous sets

18. Insert Delete Member? Cardinality? Space (in bits) Random Set Data Structures Hash TableBloom FilterF 0 Estimator Yes  (n log n) Yes No Yes* No  (n) Yes No Yes* O(log n) Operations Aka “HyperLogLog” “Probabilistic Counter” “Distinct Element Estimator” * allowing some error

19. Subquadratic Space A B C A D A B Requests: Set of all subsequent items: Items seen since first request Items seen since second request Reuse distance is size of oldest set that grows (cardinality query) Hit rate curve is CDF of reuse distances. F0 Estimator Insert … For t=1,…,m Let v j be value of j th F 0 -estimator Receive request b t Let ± j be change in j th F 0 -estimator when adding b t For j=2,…,t Record ( ± j - ± j-1 ) hits at reuse distance v j

20. Towards Sublinear Space A B C A Requests: Set of all subsequent items: Note that an earlier F 0 -estimator is a superset of later one Can this be leveraged to achieve sublinear space? F0 Estimator … ¶¶¶

21. F 0 Estimation [Flajolet-Martin ‘83, Alon-Mattias-Szegedy ‘99, …, Kane-Nelson-Woodruff ‘10] Operations: Insert(x) Cardinality(), with (1+ ² ) multiplicative error Space: log(n)/ ² 2 bits £ ( ² -2 +log n) is optimal log n rows ² -2 columns

22. F 0 Estimation A B C A D A B … Hash function h (uniform) Hash function g (geometric) Operations: Insert(x), Cardinality() ² -2 columns 1 1 log n rows

23. F 0 Estimation 11 1 A B C A D A B … Hash function h (uniform) Hash function g (geometric) Operations: Insert(x), Cardinality() ² -2 columns log n rows

24. F 0 Estimation 111 1 A B C A D A B … Hash function h (uniform) Hash function g (geometric) Operations: Insert(x), Cardinality() ² -2 columns log n rows

25. F 0 Estimation 111 1 A B C A D A B … Hash function h (uniform) Hash function g (geometric) Operations: Insert(x), Cardinality() ² -2 columns log n rows

26. F 0 Estimation 1111 11 1 A B C A D A B … Hash function h (uniform) Hash function g (geometric) Operations: Insert(x), Cardinality() ² -2 columns log n rows

27. F 0 Estimation Suppose we insert n distinct elements # of 1 s in a column is max of ¼ n ² 2 geometric RVs, so ¼ log(n ² 2 ) Averaging over all columns gives a concentrated estimate for log(n ² 2 ) Exponentiating and scaling gives concentrated estimate for n 1111 11 1 Operations: Insert(x), Cardinality() ² -2 columns log n rows

28. F 0 Estimation for a chain word ² -2 columns Operations: Insert(x) Cardinality(t), estimate # distinct elements since t th insert Space: log(n)/ ² 2 words log n rows

29. F 0 Estimation for a chain 1 1 A B C A D A B … Hash function h (uniform) Hash function g (geometric) ² -2 columns Operations: Insert(x), Cardinality(t) Space: log(n)/ ² 2 words log n rows

30. 21 1 A B C A D A B … Hash function h (uniform) Hash function g (geometric) ² -2 columns F 0 Estimation for a chain Operations: Insert(x), Cardinality(t) log n rows

31. 213 1 A B C A D A B … Hash function h (uniform) Hash function g (geometric) ² -2 columns F 0 Estimation for a chain Operations: Insert(x), Cardinality(t) log n rows

32. 243 4 A B C A D A B … Hash function h (uniform) Hash function g (geometric) ² -2 columns F 0 Estimation for a chain Operations: Insert(x), Cardinality(t) log n rows

33. 2453 45 5 A B C A D A B … Hash function h (uniform) Hash function g (geometric) ² -2 columns F 0 Estimation for a chain Operations: Insert(x), Cardinality(t) log n rows

34. 2453 45 5 ² -2 columns F 0 Estimation for a chain The {0,1}-matrix consisting of all entries ¸ t is the same as the matrix for an F 0 estimator that started at time t. So, for any t, we can estimate # distinct elements since time t. Operations: Insert(x), Cardinality(t) log n rows

35. Theorem: Let n=B ¢ W. Let C : [n] ! [0,1] be true HRC. Let Ĉ : [n] ! [0,1] be estimated HRC. Using O(B 2 ¢ log(n) ¢ log(m)/ ² 2 ) words of space, can get C((j-1) ¢ W) - ² · Ĉ (j ¢ W) · C(j ¢ W)+ ² 8 j =1,…,B Vertical error Horizontal error n = # distinct blocks m = # requests B = # “bins” W = width of each “bin” 1 0 Hit Rate 0 n W B bins C Ĉ C(j ¢ W) Ĉ (j ¢ W) C((j-1) ¢ W) C((j-1) ¢ W)- ²

36. Experiments: MSR-Cambridge traces of 13 live servers over 1 week Trace file is 20GB in size, 2.3B requests, 750M blocks Optimized C implementation of Mattson’s algorithm – Processing time: ~1 hour – RAM usage: ~92GB Java implementation of our algorithm – Processing time: 17 minutes (2M requests per second) – RAM usage: 80MB (mostly the garbage collector)

37. Experiments: MSR-Cambridge traces of 13 live servers over 1 week Trace file has m=2.3B requests, n=750M blocks heuristic counter stacks

38. Experiments: MSR-Cambridge traces of 13 live servers over 1 week Trace file has m=585M requests, n=62M blocks heuristic counter stacks

39. Experiments: MSR-Cambridge traces of 13 live servers over 1 week Trace file has m=75M requests, n=20M blocks heuristic counter stacks

40. Conclusions Workload analysis by measuring uniqueness over time. Notion of “working set” can be replaced by “hit rate curve”. Can estimate HRCs in sublinear space, quickly and accurately. On some real-world data sets, its accuracy is noticeably better than heuristics that have been proposed in the literature.

41. Open Questions Does algorithm use optimal amount of space? Can it be improved to O(B ¢ log(n) ¢ log(m)/ ² 2 ) words of space? We did not discuss runtime. Can we get runtime independent of B and ² ? We are taking difference of F 0 -estimators by subtraction. This seems crude. Is there a better approach? Streaming has been used in networks, databases, etc. To date, not used much in storage. Potentially more uses here.