Precise and Efficient FIFO-Replacement Analysis Based on Static Phase Detection

Transcription

1 Precise and Efficient FIFO-Replacement Analysis Based on Static Phase Detection Daniel Grund 1 Jan Reineke 2 1 Saarland University, Saarbrücken, Germany 2 University of California, Berkeley, USA Euromicro Conference on Real-Time Systems 2010

2 Outline 1 Introduction and Problem Timing Analysis Cache Analysis Challenge FIFO Replacement 2 Predicting Hits for FIFO Idea and Theorem Must Analysis Efficient Implementation 3 Paper Contents 4 Evaluation Related Work Analysis Precision 5 Summary Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

3 Timing Analysis for Real-Time Systems Frequency Analysis-guaranteed timing bounds Possible execution times Overest. LB BCET WCET UB Execution time Need to bound execution time of programs Execution time influenced by architectural features pipelines, caches, branch prediction,... Need to analyze behavior of architectural components Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

4 Caches and Replacement Policies hit (ab) CPU Cache Main Memory Capacity: Latency: 32 KB 3 cycles 2 MB 100 cycles Caches transparently buffer memory blocks Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

5 Caches and Replacement Policies a? hit (ab) CPU Cache Main Memory Capacity: Latency: 32 KB 3 cycles 2 MB 100 cycles Caches transparently buffer memory blocks Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

6 Caches and Replacement Policies a! hit (ab) CPU Cache Main Memory Capacity: Latency: 32 KB 3 cycles 2 MB 100 cycles Caches transparently buffer memory blocks Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

7 Caches and Replacement Policies c? miss (ab) CPU Cache Main Memory Capacity: Latency: 32 KB 3 cycles 2 MB 100 cycles Caches transparently buffer memory blocks Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

8 Caches and Replacement Policies miss (ab) c? CPU Cache Main Memory Capacity: Latency: 32 KB 3 cycles 2 MB 100 cycles Caches transparently buffer memory blocks Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

9 Caches and Replacement Policies miss (ac) CPU Cache Main Memory c! Capacity: Latency: 32 KB 3 cycles 2 MB 100 cycles Caches transparently buffer memory blocks Replacement policy dynamically decides which element to replace LRU least recently used PLRU pseudo LRU FIFO first-in first-out Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

10 Caches and Replacement Policies c! miss (ac) CPU Cache Main Memory Capacity: Latency: 32 KB 3 cycles 2 MB 100 cycles Caches transparently buffer memory blocks Replacement policy dynamically decides which element to replace LRU least recently used PLRU pseudo LRU FIFO first-in first-out Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

11 Static Cache Analysis Goals & Notions Derive approximations to cache contents at each program point in order to classify memory accesses as cache hits or cache misses Must-information Underapproximation of cache contents Used to soundly classify cache hits May-information Overapproximation of cache contents Used to soundly classify cache misses Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

12 Static Cache Analysis Challenges read z read y read x write z Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

13 Static Cache Analysis Challenges read z Initial cache contents unknown read y read x write z Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

14 Static Cache Analysis Challenges read z Initial cache contents unknown read y read x Need to combine analysis information write z Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

15 Static Cache Analysis Challenges read z Initial cache contents unknown read y read x Need to combine analysis information Need to determine addresses of x, y, z write z Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

16 Static Cache Analysis Challenges read z Initial cache contents unknown read y read x Need to combine analysis information Need to determine addresses of x, y, z write z 1 Approximate accessed addresses by value analysis (not this talk) 2 Approximate cached contents by replacement analysis Cache analysis = value analysis replacement analysis Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

17 FIFO Replacement FIFO cache of size k: last-in first-in [b 1,..., b k ] Q k := B k Example updates: [d, c, b, a] c hit [d, c, b, a] [d, c, b, a] e miss [e, d, c, b] Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

18 FIFO Replacement Analysis Why Predicting Hits is Difficult Take a set of blocks B that does fit into a cache q For example, B = {a, b, e} and k = 4. B k. Access all blocks in B: q a,b,e q Must all accessed blocks be cached? q : B q? Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

19 FIFO Replacement Analysis Why Predicting Hits is Difficult Take a set of blocks B that does fit into a cache q For example, B = {a, b, e} and k = 4. B k. Access all blocks in B: q a,b,e q Must all accessed blocks be cached? q : B q? No. [d, c, b, a] a hit [d, c, b, a] Observation b hit [d, c, b, a] After accessing a set of fitting blocks, not all of them must be cached. e miss [e, d, c, b] a Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

20 FIFO Replacement Analysis Why Predicting Misses is Difficult Take a set of blocks B that does not fit into a cache q For example, B = {a, b, c, d, e, f} and k = 4. B k. Access all blocks in B: q a,b,c,d,e,f q Must all non-accessed blocks be evicted? q : q B? Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

21 FIFO Replacement Analysis Why Predicting Misses is Difficult Take a set of blocks B that does not fit into a cache q For example, B = {a, b, c, d, e, f} and k = 4. B k. Access all blocks in B: q a,b,c,d,e,f q Must all non-accessed blocks be evicted? q : q B? No. [x, c, b, a] Observation a,b,c d,e,f hits [x, c, b, a] [f, e, d, x] x misses After accessing a set of non-fitting blocks, other blocks may still be cached. Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

22 Outline 1 Introduction and Problem Timing Analysis Cache Analysis Challenge FIFO Replacement 2 Predicting Hits for FIFO Idea and Theorem Must Analysis Efficient Implementation 3 Paper Contents 4 Evaluation Related Work Analysis Precision 5 Summary Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

23 To the point: Anticipation & Idea Considering repeated accesses to fitting blocks B helps: B = {a, b, c} s = a, b, b, c, b, b, a, c, c, a, b,... Eventually, all blocks in B must be cached. Need to detect repetitions Partition access sequence s into phases Definition (Phase) A B-phase is an access sequence s such that the set of accessed blocks A(s) = B. a, b, b, c, b, b, a, c,... }{{}}{{} {a,b,c}-phase {a,b,c}-phase Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

24 Predicting Hits by Detecting Phases Lemma (Single Phase) Let s be a B-phase and B k. q Q k, q s q : Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

25 Predicting Hits by Detecting Phases Lemma (Single Phase) Let s be a B-phase and B k. q Q k, q s q : B q 1 Either all blocks already cached: B q only hits in s B q Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

26 Predicting Hits by Detecting Phases Lemma (Single Phase) Let s be a B-phase and B k. q Q k, q s q : B q C 1 (q ) B 1 Either all blocks already cached: B q only hits in s B q 2 Or not: B q at least one miss s C 1 (q ) B a,b,e [d, c, b, a] [ }{{} e, d, c, b] C 1(q )={e} B Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

27 Predicting Hits by Detecting Phases Theorem (Multiple Phases) Let s i be B-phases and B k and s = s 1... s j q Q k, q s q : B q C j (q ) B 1 For each individual phase the lemma applies 2 Misses, if any, accumulate in last-in positions C j (q ) b,a,e a,b,e a,b,e [d, c, b, a] [ }{{} e, d, c, b] [ a, e, d, c] [b, a, e, d] }{{}}{{} C 1 B C 2 B C 3 B Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

28 Predicting Hits by Detecting Phases Theorem (Multiple Phases) Let s i be B-phases and B k and s = s 1... s j q Q k, q s q : B q C j (q ) B 1 For each individual phase the lemma applies 2 Misses, if any, accumulate in last-in positions C j (q ) b,a,e a,b,e a,b,e [d, c, b, a] [ }{{} e, d, c, b] [ a, e, d, c] [b, a, e, d] }{{}}{{} C 1 B C 2 B C 3 B Corollary: After B B-phases, all blocks in B must be cached Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

29 The Must Analysis How to Count Phases For phase blocks B, the analysis maintains: P phase progress, blocks already accessed in current phase pc phase counter, number of detected B-phases Predicts hits for blocks in B if pc = B Example for B = {a, b} s a b b b a b P {a} {a, b} {b} {b} {a, b} {b} pc Hit Hit Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

30 The Must Analysis Dependency on Future Accesses Need B B-phases to predict hits for blocks in B How to choose B? After observing a, b, c it makes sense trying to detect 2 further {a, b, c}-phases 1 further {b, c}-phase 0 further {c}-phases Optimal B depends on future accesses a, b, c, a, b, c, a, b, c, a, b, c a, b, c, b, c, b, c, b, c, b, c Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

31 The Must Analysis Resolving the Dependency Perform multiple analyses for different B sets For which? B already determines sensible contents of B For B = 2, after a, b, c already detected 1 {b, c}-phase no advantage in trying to detect 2 {x, y}-phases Perform k analyses for different B sets for each phase size n = 1... k B n consists of the n most-recently-used blocks Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

32 The Must Analysis Subanalyses for n = a b c c b c a a c a b a n = 1 : n = 2 : n = 3 : Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

33 The Must Analysis Subanalyses for n = n = 1 : a b c c b c a a c a b a H {c} n = 2 : n = 3 : Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

34 The Must Analysis Subanalyses for n = n = 1 : a b c c b c a a c a b a H H {c} n = 2 : {b, c} {b, c} n = 3 : Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

35 The Must Analysis Subanalyses for n = n = 1 : a b c c b c a a c a b a H H H {c} {a} n = 2 : {b, c} {b, c} n = 3 : Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

36 The Must Analysis Subanalyses for n = n = 1 : a b c c b c a a c a b a H H H H {c} {a} n = 2 : {b, c} {b, c} {a, c} {a, c} n = 3 : Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

37 The Must Analysis Subanalyses for n = n = 1 : a b c c b c a a c a b a H H H H H {c} {a} n = 2 : {b, c} {b, c} {a, c} {a, c} n = 3 : {a, b, c} {a, b, c} {a, b, c} Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

38 Efficient Implementation Observation For n = 1... k, analysis needs to maintain: phase blocks B n 2 B phase progress P n 2 B phase counter pc n N Phase blocks B n are the n most-recently-used blocks For i < j : B i B j Encode all B n in a single LRU-stack For all i : P i B i Encode all P n as pointers into the stack Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

39 Efficient Implementation Encoding For phase blocks B n : pc n complete B n -phases were detected current phase progress is B ppn B 1 pc 1, pp 1 B 2 \ B 1 pc 2, pp 2 B 3 \ B 2 pc 3, pp 3 B 4 \ B 3 pc 4, pp 4 B 1 {b} P 1 pc 1 1 B 2 {b, c} P 2 {b} pc 2 2 B 3 {a, b, c} P 3 {b, c} pc 3 1 = {b} 1, 0 {c} 2, 1 {a} 1, 2 0, 3 Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

40 Outline 1 Introduction and Problem Timing Analysis Cache Analysis Challenge FIFO Replacement 2 Predicting Hits for FIFO Idea and Theorem Must Analysis Efficient Implementation 3 Paper Contents 4 Evaluation Related Work Analysis Precision 5 Summary Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

41 Contents of the Paper So far, we have seen parts of the must-analysis The paper contains, for must- and may-analysis, basic theorem generalization to arbitrary control-flow formalization as abstract interpretation Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

42 Outline 1 Introduction and Problem Timing Analysis Cache Analysis Challenge FIFO Replacement 2 Predicting Hits for FIFO Idea and Theorem Must Analysis Efficient Implementation 3 Paper Contents 4 Evaluation Related Work Analysis Precision 5 Summary Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

43 Brief History of Replacement Analysis Before 97 LRU analyses LCTRTS 97 Precise and efficient must- and may-analysis for LRU [1] LCTES 08 Generic analyses for FIFO and PLRU [2] SAS 09 Cache analysis framework and FIFO analysis [3] WCET 10 Toward precise analysis for PLRU [4] ECRTS 10 Precise and efficient must- and may-analysis for FIFO Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

44 Evaluation Setup Analyses: HAM Must-analysis of SAS 09 RC Generic analyses of LCTES 08 PD Phase detecting analyses Collecting semantics: CS Limit for any static analysis Spectrum of synthetic benchmarks: Random access sequences and program fragments With varying locality Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

45 Evaluation Results k=8, random sequences % 100 CS hits n.c. misses n n is number of distinct elements that get accessed Average guaranteed hit- and miss-rates Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

46 Evaluation Results k=8, random sequences % CS RC n n is number of distinct elements that get accessed Average guaranteed hit- and miss-rates Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

47 Evaluation Results k=8, random sequences % CS RC+HAM n n is number of distinct elements that get accessed Average guaranteed hit- and miss-rates Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

48 Evaluation Results k=8, random sequences % CS RC+HAM PD+HAM n n is number of distinct elements that get accessed Average guaranteed hit- and miss-rates Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

49 Outline 1 Introduction and Problem Timing Analysis Cache Analysis Challenge FIFO Replacement 2 Predicting Hits for FIFO Idea and Theorem Must Analysis Efficient Implementation 3 Paper Contents 4 Evaluation Related Work Analysis Precision 5 Summary Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

50 Summary Precise and Efficient FIFO-Replacement Analysis based on Static Phase Detection Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

51 Summary % B1 {b} 100 P1 80 pc1 1 B2 {b, c} 60 P2 {b} 40 pc2 2 B3 {a, b, c} 20 P3 {b, c} 0 pc n = {b} 1, 0 {c} 2, 1 {a} 1, 2 0, 3 Precise and Efficient FIFO-Replacement Analysis based on Static Phase Detection { {a,b,c}-phase }} {{ {a,b,c}-phase }} { a, b, c }{{}, c, b }{{}, b, a,... {b,c}-phase {b,c}-phase Two theorems on FIFO-contents bound on number of phases must be cached / evicted Must- and may-analysis static phase detection multiple sub-analyses Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

52 Further Reading C. Ferdinand Cache Behaviour Prediction for Real-Time Systems PhD Thesis, Saarland University, 1997 J. Reineke and D. Grund Relative competitive analysis of cache replacement policies LCTES 2008 D. Grund and J. Reineke Abstract Interpretation of FIFO Replacement SAS 2009 D. Grund and J. Reineke Toward Precise PLRU Cache Analysis WCET 2010 Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

53 Related Work: LRU Analyses Analyses directed at worst-case execution-time analysis Mueller By static cache simulation Li By integer linear programming Ferdinand By abstract interpretation Other than that Ghosh Cache Miss Equations, loop nests Chatterjee Exact model of cache behavior for loop nests All for LRU caches only Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

54 Static Timing-Analysis Framework Binary Executable Legend: Data CFG Reconstruction Action Micro-architectural analysis Control-flow Graph models pipeline, caches, buses, etc. derives bounds on BB exec. times Value Analysis Loop Bound Analysis Control-flow Analysis is an abstract interpretation with a huge domain Annotated CFG is the computationally most expensive module Microarchitectural Analysis Basic Block Timing Info Global Bound Analysis Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

55 Applicability Any buffer with transparent FIFO replacement: Individual cache sets of instruction of data caches (I$, D$) Branch target buffers (BTB, BTIC) Translation lookaside buffers (TLB) Instances: I$ D$ ARM 1136, 1156, 1176, 920T, 922T, 926EJ-S (k {2, 4, 64}) I$ D$ Marvell (Intel) XScale(s) (k = 32) BTB Freescale (Motorola) MPC 56x, 7450-Family (k {4, 8})... Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

56 Must Analysis Full Example for k = 3 For 1 n k maintain B n, P n, pc n Example s a b c c b c a B 1 {a} {b} {c} {c} {b} {c} {a} P 1 pc B 2 {a} {a, b} {b, c} {b, c} {b, c} {b, c} {a, c} P 2 {a} {c} {c} pc B 3 {a} {a, b} {a, b, c} {a, b, c} {a, b, c} {a, b, c} {a, b, c} P 3 {a} {a, b} {c} {b, c} {b, c} pc Hit Hit Hit Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

57 Must Analysis Abstraction and Join Analysis domain is Lru k PC k PP k (2 B ) k N k N k Lru k PC k PP k LRU must-analysis, under-approximates accessed blocks lower bounds on number of phases lower bounds on phase progress Reuse abstract transformer and join of Lru k Define appropriately for PC k and PP k Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34

58 May-Analysis Similar to must-analysis Difference: Phases may be of different lengths and contents Theorem (Multiple Phases) s = s 1... s j, i : A(s i ) = n i k: q Q k, q s q : C j i=1 (n i k+1) (q ) A(s) = i A(s i ) More simultaneous sub-analyses Similar implementation employing LRU may-analysis Daniel Grund and Jan Reineke FIFO-Replacement Analysis ECRTS / 34