A Cache Model of the Block Correlations Directed Cache Replacement Algorithm

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1 Applied atematics & Information Sciences An International Journal 211 NSP 5 (2) (211),79S-88S A Cace odel of te Block Correlations Directed Cace Replacement Algoritm Zu Xudong Scool of Computer Science and Information Engineering, Zejiang Gongsang University Hangzou, Cina Address: zuxd@zjgsu.edu.cn Received Oct 1, 21; Revised Jan 19, 211 Tis paper oposes a cace model of te Block Correlations Directed cace replacement algoritm (BCD). Since te edication success rate of accesses decides te BCD s it ratio, te cace model analysis te utilization of te cace space imoved by te correct edications and te penalty incurred by incorrect edications to estimate te it rate of BCD. Te model can also optimize te parameters of BCD to acieve teoretical optimum of suc spatial locality based likely replacement algoritms. For most workloads of real systems te model of BCD can reduce deviation by 2.1%-21.8%. Keywords: Cace model, Spatial locality. 1 Introduction Block correlations are common semantic patterns in storage systems and can be used to direct performance optimization of storage systems. Te correlated blocks usually are accessed by requests wic close to eac oter in a stream. For example, tere is a block correlation {abc}. Wen block a and b are accessed one after anoter, block c is likely to be accessed soon. References [1-2] try to explore te block correlations in a request stream. Recently some studies [4-5], like BCD [3], opose cace replacement policies wic use spatial locality in block correlations to increase te it ratio. To acieve teoretical optimum of te replacement algoritms, BCD edicts te of blocks spatial locality in te future by istory and runtime information. BCD places blocks in te stack of cace by te ediction results to imove te utilization of te cace and reduce te penalty of ediction failure. Te edication success rate decides te it rate of BCD. Tis paper oposes a cace model of BCD to acieve teoretical optimum of spatial locality based replacement algoritms. Te cace model analyses te utilization of te cace space imoved by te correct edications and te penalty incurred by incorrect edications to estimate te it rate of BCD. For most of real system workloads te model of BCD can reduce deviation

2 Zu Xudong 8 by 2.1%-21.8%. Te paper is organized as follows. In te next section we introduce te BCD. Section 3 discusses te cace model. Section 4 esents our experimental results and Section 5 concludes te paper. 2. Basic policy and key oblems Figure 1 is a block correlations directed cace system structure wic includes efetcing and cace replacement component. Te cace system does not limit location in te system. It can be used in client end, storage server or storage area network. Te efetcing component uses block correlations to direct efetcing policy [1]. Tis paper discusses te policy of cace replacement component. Cace System Prefetcing component Prefetc I/O Prefetc data Applications BCD used by Cace management component ain cace ain cace miss Prefetcing cace Prefetcing cace miss Request Access data Storage Figure 1. Block correlations directed cace system structure To indentify te access caracteristics we classify te requests as follow. 1. Prefetced Request, PR: Besides te first request requests in a block correlations are efetced requests. 2. Non-Prefetced Request, NPR: (a) request tat does not access in a pattern. (b) Te first request of a pattern. 1. How to identify request type online. Traditional spatial-locality based algoritms can identify only sequential access and random access. One metod is to get information of te current request by communication between replacement component and

3 A Cace odel of te Block Correlations 81 efetcing component. It is not general for a cace system. A simple and efficient request identification mecanism is an issue to be solved in te study. Table 1. Influence of request-type ediction on cace-miss ratio {ABCD} is frequent access pattern and uses LRU and RU policies respectively for te blocks wit NPR and PR edicted LRU BCD (inaccurate ediction) BCD (accurate ediction) Req Pref Stack state iss times Stack state iss times Type Stack state iss times Type E 1 1 NPR 1 NPR F E 2 E 2 NPR E 2 NPR A FE 3 FE 3 NPR FE 3 NPR B B AFE 3 AFE 3 PR AFE 3 PR C C BAFE 3 AFEB 3 PR AFEB 3 PR D D CBAF 3 AFEC 3 PR AFEC 3 PR E DCBA 4 AFED 3 NPR AFED 3 NPR F EDCB 5 EAFD 3 NPR EAFD 3 NPR B FEDC 6 FEAD 4 PR FEAD 4 NPR C BFED 7 FEAB 5 PR BFEA 5 NPR B CBFE 8 FEAC 6 PR CBFE 5 NPR C BDFE 9 FEAB 7 PR BCFE 5 NPR 2. How to edict accurately te future request type of te block. A block can be accessed by different types of requests. Accurate ediction of te future-request type for eac block can effectively reduce te cace-miss ratio. On te contrary wrong ediction increases te cace-miss ratio. As sown in Table 1, if blocks to be accessed by NPR in future are mistakenly edicted to be accessed by PR, te block is soon replaced from te cace. It directly results in cace miss. Similarly, if a PR is edicted as NPR, te block accessed by it stays in te cace for long time and reduces te utilization of cace space. Table 1 sows te influence of accurate ediction of request type on cace-miss ratio. In addition ediction of te algoritm must ave very low overead. How to edict te future access type of blocks accurately and rapidly is a key oblem to be solved. 3. How to reduce penalty incurred by failure of ediction. If BCD cooses LRU or RU policy simply based on ediction results, failure in ediction results in ig cace-miss ratio. BCD sould consider ow to reduce te penalty in case of ediction failure. Te overead of te algoritm sould be also taken into account. For example, DULO s sorting operation oduces a larger overead. How to place blocks based on te ediction results and tolerate te failure of ediction is anoter important goal of tis paper 3. Te BCD Sceme BCD adjusts te residence time of te block in te cace to maximize te NPR it ratio and to reduce te penalty of failure of ediction. Te blocks tat are edicted being accessed by different request types sould ave different residence times in te cace.

4 Zu Xudong 82 Te block is replaced out of te cace from te stack bottom (LRU end). If te block is not accessed again, te minimum residence time of a block in te cace is equal to te distance from te current position to te stack bottom. Tere are two sections in BCD s stack and te depts of tem are respectively T and T l. Te blocks depart troug te bottom of ig section and enter te top of te low one. Te blocks are replaced at te bottom of te low section. Terefore blocks at te ig section ave longer residence times tan blocks at te low section. No move e PR NPR ove to top of ig section e NPR Hig section start NPR replace end PR ove to top of low section e PR Low section replace Figure 3. State conversion of a block in BCD If a block is edicted to be accessed by NPR, BCD places it at te top of te ig section. BCD uses two policies to reduce te penalty of te ediction failure. Firstly BCD keeps temporal locality of NPR requests. If te block is it at te ig section and is edicted to be accessed by PR next, BCD does not cange te position of te block in te stack. If te type of next request for tis block is NPR, te ediction fails, but te block still can stay in te cace for a longer time. Secondly, If te block is not in te ig section and it is edicted to be accessed by PR next time, BCD places it at te top of te low section to keep te block for a period to reduce te penalty of ediction failure. In general Figure 3 sows te state conversion of a block in te BCD cace. 4. Cace model of BCD In tis section we build te BCD model to analyze te influence of various factors on cace effect and direct configuration of BCD parameter. Let cace capacity T be a constant greater tan. Let d be a reference distance of te block wic reesents te number of oter blocks accessed between two adjacent accessors to te same block (te block repeatedly accessed during te interval is counted once only). Teorem 1: Let T be te size of te cace, reference tem at a distance d. Ten miss ratio of LRU algoritm is lru T 1 f ( d)d( d). f (d) be te oportion of requests tat

5 A Cace odel of te Block Correlations 83 Proof: According to LRU, te block can be it in te cace wen te reference distance is smaller tan T. Terefore te it ratio of LRU is H = T lru f ( d) Wen te number of requests is large enoug, f (d) tends to be, so H lru T = T f ( d) f ( d)d( d) and accordingly lru H lru T 1 f ( d)d( d). Let S be te total count of requests in te storage system. Let main be te it times in te main cace, be te it times of PRs in main cace and let be te it main main times of NPRs in te main cace. Ten = main + main. Let main S be te number of PRs in S. Te cace system miss ratio is = ( S ( S )) / S ( main main + S ) / S main S Since is irrelevant to te replacement policy, is decided by NPRs resides in te main cace as long as possible to increase (1) main. Te block of main. It is more complicated to use BCD in te storage system. Te following oblems and callenges are encountered. f ( ) All te PRs are it in te main cace. Let d be te obability density function P S / S of te reference distance. Let PRs of te workloads be = P S / S, as. According to Formula (1), if T l = and te edictions are all correct, BCD as te teoretical optimum cace miss ratio (OPT): opt ( S + main ) / S T (2) P P f ( d)d( d) Te reference distance distribution of workloads and NPR sequence can be apoximately obtained by linear conversion of te exponential distribution. Figure 4 and Figure 5 are respectively reference distance distribution of Cello92 and OLTP. Analysis of Cello96, Cello99 and TCP-C can also obtain te similar results. Let te obability 1d density function of reference distance of I/O stream be f ( d) = b f '( d) = b λ e λ 1 1 1, te distribution function of reference distance of I/O stream can be obtained as:

6 Zu Xudong a1 b1e F( d) =, NPR NPR-EXP TOTAL TOTAL-EXP λ1d, d > d =.4 NPR-EXP: R 2 =.9896 y = e x.2 TOTAL-EXP: R 2 =.9868 y = 1-e -.125x 2K 4K 6K 8K 1K 12K 14K 16K rereference distance Figure 4. Reference distance distribution of Cello NPR TOTAL NPR-EXP TOTAL-EXP NPR-EXP: R 2 =.9955 y = e x TOTAL-EXP: R 2 =.9972 y = e x 1K 2K 3K 4K rereference distance Figure 5. Reference distance distribution of OLTP Similarly let te distribution function of Reference distance of NPR sequence be F Te LRU s miss ratio is OPT s miss ratio is opt a2 b2e ( d) =, lru λ2d 1 ( a b e 2 P P ( a b e 2 1 2, 1 λ T d > d = λ T ) ) (4) (3) according to Teorem 1 and te according to Formula 2. Considering failure of ediction in te edicted NRP request sequence te obability density function of NPR reference distance is f ( γ, d ), of γ = P *(1 α) /(P *(1 α) + P * β ) wic, reesenting te oportion of correctly edicted NPR in all te edicted NRP requests. In te sequence formed wit edicted NPR requests, simply suppose reference distance distribution of te real NPR as an apoximately linear relationsip wit te oportion of real NPRs in te sequence:

7 A Cace odel of te Block Correlations 85 T F( γ, T ) = ( γ P = f ( γ,d) = F ) * F 1 γ ( T ) 1 P 1 P *( F ( T ) + (1 γ ) * F( T ) ( T ) F( T )) In case of ediction failure let Type 1 error be a mistake of edicting NPR by PR and let Type 2 error be a mistake of edicting PR by NPR. Let te obability of Type 1 error be α and te obability of Type 2 error be β. Te requests it in BCD cace are from four parts: PRs, correctly edicted NPR requests, NPR request tat is edicted RP, but te block was not moved and NPR request tat is edicted RP, but te block was placed in low section. Te BCD s miss ratio is BCD ( T ) 1 P l P (1 α)( F( γ, T ) + F( T ) F( T )) P αp F( T ) P αp F( T )(1 P F( T )) Wit Formula 5 te low section cace capacity can be calculated to minimize te miss ratio of BCD. 5. Test and evaluation Our simulator uses block correlations explored by C-iner [1] and uses LRU to manage te main cace. Te simulated disk specification is similar to tat of te 1 RP IB Ultrastar 36Z15. In our experiments we use only te first alf part of te trace to mine block correlations. We uses block relations to direct efetcing and evaluate te performance of BCD. Tis section discusses influence of edicting on BCD performance and validate BCD model wit te example of Cello92 and OLTP. According to discussion in Section 4, tresold p is used to distinguis block type. Let defined as LPB. If pi > (1 p), block i is defined as PB. ediction error ratio Cello92 α Cello92 β OLTP α OLTP β Tresold p (a) Influence of tresold p on ediction error l iss Ratio p [,.5]. If p i (5) p, block i is Cello92 real miss ratio Cello92 teoretical miss ratio OLTP real miss ratio OLTP teoretical miss ratio Tresold p (b) Influence of tresold p on miss ratio Figure 6. Influence of ediction on BCD miss ratio.

8 Zu Xudong 86 BCD can accurately edict request type. Te ratio of Type 1 ediction error α and te ratio of Type 2 ediction error β can bot be smaller tan.2. Figure 6(a) sows ediction error ratio for Cello92 and OLTP under different tresolds p. It can be seen in te figure tat α and β of Cello92 decrease wit increase of p wen p <.1. Wen p is greater tan.1, α continues decreasing to.1 and below. ost of NPR requests are accurately edicted. Wen p is greater tan.1, β rises to 44% wit increase of p. Lots of PRs are edicted as NPR so tat blocks accessed by tese requests are placed to te ig section of te cace and te residence time of tese blocks is reduced. α of OLTP slowly decreases wit te increase of p and β rapidly increases wit te increase of p. iss ratio of BCD rises wit increase of ediction error ratio. Figure 6(b) sows miss ratio of BCD under different values of p. iss ratio of Cello92 is te lowest wen p =.1 because α is decreased wit te increase of p wen p.1. Wen p.1, α decreases slowly wile β increases rapidly wit increase of p. Since p becomes te dominant factor affecting miss ratio, te miss ratio of BCD rises. Similarly miss ratio of OLTP is te lowest wen p =.1. Figure 7 sows te influence of low section size on BCD miss ratio. Wit te increase of low section capacity te miss ratio of Cello92 rises because BCD can accurately edict request type on Cello92 and te failure possibility is very small. Terefore increase of low section capacity cannot reduces many miss times in case of ediction failure wile decrease of ig section capacity reduces residence time of blocks accessed by NPR, resulting in an increase of miss ratio. Wen low section capacity is equal to main cace capacity, BCD retrogresses into an apoximate LRU algoritm. Differently from Cello92, wen low section capacity for OLTP is, miss ratio of BCD is greater tan tat of LRU and rapidly decreases wit increase of low section capacity. Tis is because many reference distances of OLTP are all small. Type 1 ediction error directly results in cace miss wile a smaller capacity of low section can significantly decrease l misses caused by Type 1 ediction error. Wen low section capacity continues to increase, te miss ratio of BCD decreases more slowly because te number of requests te reference distances of wic are smaller tan low section capacity increases slowly. Wen capacity of low section is greater tan 1/8 of te main cace capacity, miss resulting from decrease of ig section cace capacity becomes te dominant factor and miss ratio begins increasing again.

9 A Cace odel of te Block Correlations 87 iss Ratio Cello92-LRU OLTP-LRU Cello92-BCD OLTP-BCD Cello92-BCD teoretical value OLTP-BCD teoretical value /64 1/32 1/16 1/8 1/4 1/2 1 low section capacity of main cace capacity Figure 7. Influence of low section capacity on BCD To validate BCD model Figure 6 and Figure 7 sow te teoretical miss ratio calculated by te model. It can be seen tat maximum relative errors between teoretical and actual test values are no more tan 21.8% and mean relative errors are 2.1%. BCD model can effectively describe relations between factors of BCD cace replacement algoritm and miss ratio. BCD model can be used to direct te optimization of te low section capacity. In Figure 1, wen capacity of low section is, tere is a large error between te OLTP teoretical miss ratio and actual miss ratio because lots of reference distance is very sort in OLTP. Wen size of low section apoaces zero, reference distance distribution is no longer fitted wit te exponential distribution function. 6. Conclusions and future work Tis paper oposes a cace model of a Block Correlations Directed cace replacement algoritm. To acieve teoretical optimum of spatial locality based replacement algoritms BCD tries to increase NPR s it ratio by edicting te request type and reducing te penalty caused by te failure of ediction. Experiments sow tat BCD can effectively reduce miss ratio of cace and average I/O response time. Tere is still muc work in te future: Firstly we will carry out te BCD on a actical system and verify its performance by more applications. Secondly self-adaption of te low section capacity to accommodate te caracteristics of te workload sould be learned. Finally researc to reduce te average response time of BCD wit optimized data layout. Acknowledgements Tis work was supported by Zejiang Sci & Tec Project (21C3345), Zejiang Provincial NSF Cina (Y111316).

10 Zu Xudong 88 References [1] Zenmin L., Zifeng C., Sudarsan. S., Yuanyuan Z. and C-iner: ining block correlations in storage systems. FAST 4, San Francisco, 24, [2] Hsu Windsor W., Smit Alan Jay and Young Honesty C., Te automatic imovement of locality in storage systems. AC Trans. on Computer Systems (4): [3] Zu Xudong, Ke Jian, Xu Lu, BCD: To acieve te teoretical optimum of spatial locality based cace replacement algoritm. NAS9, Zang Jia Jie, 29, [4] Kim J.., Coi J., Kim J. et al. A low-overead ig-performance unified buffer management sceme tat exploits sequential and looping references. OSDI San Diego, CA, Oct. 2, [5] S. Jiang, X. N. Ding, F. Cen et al. DULO: An effective buffer cace management sceme to exploit bot temporal and spatial locality. FAST 5, 25. Zu Xudong received te PD degree in Computer Arcitecture from te Institute of Computing Tecnology, Cinese Academy of Sciences (ICT/CAS) in 29. He is currently an Assistant Professor in te Zejiang Gongsang University. His researc interests are in te areas of Computer Arcitecture, Distributed Systems and Storage Systems.