On the Smoothness of Paging Algorithms

Transcription

1 On the Smoothness of Paging Algorithms Jan Reineke and Alejandro Salinger Technische Universität Dortmund November 17, 2016

2 MoHvaHon: Real-Hme Systems Side airbag in car Side airbag in car ReacHon in < 10 ms CrankshaT-synchronous tasks ReacHon in < 45 microsec Controllers must finish their tasks within given Hme bounds à Need to determine Worst-Case Response Times (WCRT)

3 Influence of Caches on ExecuHon Times x=a+b; LOAD LOAD ADD r2, _a r1, _b r3,r2,r1 PowerPC 755 Courte 3

4 Basics: Caching/Paging Cache Slow memory Is p i in the cache? σ = p 6 p 3 p 2 p 4 p 4 p 2 p 10 p 11 p 5 p 4 -Yes Hit -No Fault (miss) Fetch p i from slow memory, evict one page from cache à Replacement policy determines page to evict à Access sequence + policy determine cache state 4

5 Why are Caches a Challenge? Core Task 1 Task Task... n Core Task 1 Task Task... n Cache Main Memory 1. Input-dependent memory accesses 2. Interference due to preempted tasks 3. Interference due to co-running tasks 5

6 Two Approaches to Timing Analysis 1. StaHc Analysis: Uncertainty about input-dependent memory accesses How does uncertainty about memory accesses Uncertainty about memory accesses of preemphng tasks Uncertainty affect uncertainty about memory about accesses number of co-running of faults? tasks 2. Measurement-based Analysis: Covers only subset of inputs How representahve are measurements Covers only subset of preemphng task scenarios Covers on only a subset subset of of co-running the possible task scenarios cases? 6

7 How does uncertainty about memory accesses affect uncertainty about execuhon Hme? number of faults A(σ') A(σ) QuanHfy uncertainty by edit distance σ σ' access sequences 7

8 DefiniHon of Smoothness dist(σ,σ ') = edit distance between σ and σ ' For generic δ: We call A smooth if it is (1,β,1)-smooth for some β. 8

9 Key QuesHons How smooth are known paging algorithms? Are there fundamental bounds on the smoothness of paging algorithms? Are smoothness and high performance contradictory goals? Can randomizahon help? 9

10 DeterminisHc Replacement Policies / Paging Algorithms Online Algorithms: LRU: Least-Recently-Used FIFO: First-In-First-Out FWF: Flush-When-Full Offline Algorithm: FITF: Furthest-In-The-Future (OPT, LFD, Belady s) 10

11 Smoothness of LRU... [d c b a] [d c b a] [a d c b] [b a d c] [d c b a] [p d c b] [a p d c] [b a p d] miss miss miss [c b a d] [c b a p] [d c b a] [d c b a] Same state miss miss 11

12 Smoothness of LRU 12

13 For any replacement policy A: If A is (1,c,1)-smooth, then A is (1,δc)-smooth 13

14 FIFO is not smooth miss [a b c d] [e a b c ] [e a b c] [e a b c] [d c b a] [e d c b] [a e d c] [b a e d] miss miss miss [e a b c] [c b a e] miss [d e a b] [d c b a] [d e a b] [e d c b] [d e a b] [a e d c] miss miss miss miss 14

15 Lower Bounds (1 of 2) An algorithm is demand paging if it only evicts pages when needed e.g., LRU, FIFO But not all algorithms are demand paging: How about compehhve algorithms? 15

16 CompeHHve Analysis 16

17 Lower Bounds (2 of 2) An algorithm is demand paging if it only evicts pages when needed e.g., LRU, FIFO But not all algorithms are demand paging e.g., FWF 17

18 DeterminisHc Algorithms Smoothness c-compehhve FIFO DeterminisHc strongly-compehhve SMOOTH FWF LRU OPT CompeHHveness 18

19 Can randomizahon help? 19

20 Randomized Replacement Policies RAND: Evict page chosen uniformly at random MARK: Evict only unmarked pages PARTITION, EQUITABLE: define state probability distribuhon based on OPT s cache contents Evict-On-Access: like RAND, but evict on hits too! 20

21 Randomized Replacement Policies Algorithm Compe//ve ra/o Smoothness RANDOM MARK PARTITION EQUITABLE EOA 21

22 Bound extra misses by distance between distribuhons [a b c] 1/3 [a b d] 1/3 [a c d] 1/3 1/3 1/3 1/3 [a b c] 2/3 [a b e] 1/3 2 Claims: 22

23 Smoothness Randomized Algorithms c-compehhve FIFO PARTITION EQUITABLE MARK DeterminisHc strongly-compehhve SMOOTH Rand. strongly-comp. FWF RANDOM, LRU OPT EOA CompeHHveness 23

24 Can we design smoother algorithms? 24

25 Smoothed-LRU 25

26 Smoothed-LRU Smoothed-LRU is -smooth 2k 1 k + i 1 (1,δ min( +1, 2i +1 2i )) i = 0 à as smooth as LRU i = k-1 à as smooth as OPT Is it compehhve? As compehhve as LRU for size k-i. 26

27 LRU-Random Smoothed-LRU and EOA are very smooth, but not compehhve LRU-Random: evict the i th oldest page with probability 1 ih k 27

28 LRU-Random: Conjectures LRU-Random is (1, Θ(H k2 ) δ)-smooth LRU-Random is Θ(H k2 )-compehhve against an oblivious adversary 28

29 Smoothness The Whole Picture c-compehhve PARTITION EQUITABLE MARK FIFO * With resource augmentahon DeterminisHc strongly-compehhve SMOOTH Rand. strongly-comp. FWF RANDOM, LRU LRU-Random (k=2) OPT Smoothed-LRU* EOA CompeHHveness 29

30 Open Problems (Generalize smoothness proof for LRU-Random) Is there a randomized LRU-sibling? Are there randomized algorithms that are smooth with high probability? Are there less pessimishc nohons of smoothness? 30

31 Thank you for your a~enhon! 31