Markov Analysis for Optimum Caching as an Alternative to Belady s Algorithm

Transcription

1 arov Analysis for Otiu Caching as an Alternative to Belady s Algorith, Deutsche Teleo, Darstadt, Gerany gerhard.hasslinger@teleo.de Analytic Results on LRU, LFU, Otiu Caching Belady s Princile for Otiu Caching [L.A. Belady 66] arov Analysis: Cache of size for a single obect Coarison to nown results [A. Sith 76], [D.E. Knuth 85] General arov Analysis coared to Belady s Algorith arov Analysis of Otiu Caching

2 arov Analysis of Otiu Caching Hit Rate for Least Recently Used (LRU) Hit rate analysis for obects, cache size, assuing indeendent requests (IR) with rob.,, er obect Exact hit rate result, see e.g. [Dan & Towsley 999]: i n i C o o h,...,, ;...,, LRU, },..., Pr{ Colexity: O (State cover all obects in the cache) Che aroxiation [Che0] or siulation is used for LRU hit rate evaluation for oderate and large,.

3 Hit rate for Least Frequently Used (LFU) Hit rate analysis for obects, cache size, assuing indeendent requests (IR) with robabilities The general result is: h LFU, assuing obects with axiu request robabilities being constantly et in the cache axiu IR hit rate without rediction or loo-ahead LFU statistics over few hours or a day is ost relevant arov Analysis of Otiu Caching

4 Belady s Algorith for Otiu Caching Based on nowledge of future requests: Relace the obect with farthest next request Kee/insert obects with the next request (i.e. hit) in a sorted list Effort er request for sorting: O(ln( )) Precoute sequence of next request er request r T r T+ r T+ r T+3 r T+4 r T+5 r T+6 r T+7 r T+8 r T+9 Future requests o o o 5 o 3 o o 3 o o o 5 o 4 o ext request T + (o ) T+3 T+7 T+4 T+6 T+9 List of cached obects sorted by T + (o ) after request r T : o 5 o 3 o o o 4 ext request T + (o ) T+ T+ T+3 T+5 T+8 arov Analysis of Otiu Caching

5 Analysis of Otiu Caching: Cache Size arov analysis for : - State indicates the obect with the lastest hit - State changes, if another obect is requested twice before o - Inter hit tie: Request sequence to airwise different obects - State transition atrix via cobinatorial forulas Request Sequence o 3 o 3 o o 7 o 5 o 6 o 3 o o o 4 o 9 o 5 o o 3 o 8 o... Hit on o 3 Hit on o 3 Hit on o ; Hit on o State: 3 3 arov Analysis of Otiu Caching

6 Analysis of Otiu Caching: Cache Size A arov odel with states is sufficient for obects and for indeendent requests with rob.,, er obect The state ars the obect o addressed by the last cache hit π ( O q n ) Pr{ requests to airwise different obects fro the set O! 0,..., l n l π ( O /{ o }); q... ; π 0 ( n) π ( On ) + π ( On q O ) q π ( O /{ o}) Pr{ext hit on o after + requests} q ( + ) π ( O /{ o, o }) Pr{ext hit on o after + req.} Transition atrix of the arov chain: for Steady state solution r Pr{A hit is addressing o } Hit rate via ean inter hit tie: h π ( O /{ ). r ( + }) Ot o n n } r q arov Analysis of Otiu Caching

7 Analysis of Otiu Caching: Cache Size The revious solution of the state arov chain has colexity O( 4 ) for couting all transition robabilities Fortunately, we can rove a sile steady state solution: r ; h OPT ( ) ( + E R π ( O /{ o }) ) i.e. the robability of a hit on o is roortional to This reduces the coutational colexity to O( ) arov Analysis of Otiu Caching

8 Result for Paging Systes with Unifor Requests & Conversion: Paging systes Web cache results: h PagingCache (+, +) [( + h WebCache (, )] / (+) h Ot, WebCache / 0 ( )!/ ( )! arov Analysis of Otiu Caching

9 Analysis of Otiu Caching: Cache Size 0,8 0,6. Zif Distribution: ß -.0 Zif Distribution: ß -0.8 Zif Distribution: ß -0.5 Unifor Distribution: ß 0 0,4 0, 0,0 uber of Obects Hit Rate for a Cache of Size arov Analysis of Otiu Caching

10 Exale of the General arov odel for Otiu Caching Analysis arov odel states include all obects which can lead to a hit in the next request (Cached obects + loading otions) Request Sequence Indices of the requested obects ( 3) arov Analysis of Otiu Caching

11 arov Analysis Belady s Algorith arov Analysis of Otiu Caching

12 Conclusions on arov Analysis for Otiu Caching Size of the arov odel is large: S T ( ) ( + ) ( + ) Coarable analysis odel by A. Sith [Si76]: S Sith! ( ) ( + ) ( + ) Direct analysis aroaches are liited to 0 arov Analysis is an alternative to Belady s algorith Effort er request is siilar for both aroaches arov analysis decides about hits er request without loo-ahead (!) L.A. Belady: The otial solution can be found by storing the rogra s entire sequence of references and then woring bacward to reconstruct a iniu relaceent sequence. Liited loo-ahead is relevant for caches in large CDs S T arov Analysis of Otiu Caching

13 Results for indeendent unifor requests Hit Rates for Otiu Caching, LFU and LRU 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0, 0, 0,0 Otiu Caching: Otiu Caching: Otiu Caching: Otiu Caching: 000 LFU & LRU: All Catalogue Sizes Unifor requests: Request Probabilities /Ν for all Ν obects Cache Size as a Percentage / of the Catalogue Size For otiu caching we obtain.37 / > hot >.5 / For LRU, SG-LRU, LFU we obtain h / as worst case scenario arov Analysis of Otiu Caching

14 Results for Zif law requests Z() α β with β -0.5 Hit Rates for Otiu Caching, LFU and LRU 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0, 0, 0,0 Otiu Caching: Otiu Caching: Otiu Caching: Otiu Caching: 000 LFU: LFU: LFU: LFU: 000 LRU: LRU: LRU: LRU: 000 Cache Size as a Percentage / of the Catalogue Size arov Analysis of Otiu Caching

15 References [Bel66] L.A. Belady, A study of relaceent algoriths for a virtual-storage couter, IB Systes Journal (966) 78-0 [BS7]. Becann and D. Sanchez, axiizing cache erforance under uncertainty, Proc. 3 rd Syosiu on High Perfor. Co. Arch. (07) [Che0] H. Che, Y. Tung, and Z. Wang, Hierarchic web caching systes: odeling, design and exeriental results, IEEE JSAC 0(7) (00) [Ha7a] G. et al., Perforance evaluation for new web caching strategies cobining LRU with score based selection, Couter etwors (07) [Ha7b] G. et al., Web Caching Evaluation fro Wiiedia Request Statistics, Syosiu on Otiization in Wireless etwors, Paris (07) [Ha7c] G., arov Analysis for Otiu Caching as an Alternative to Belady s Algorith, subitted to B 8 < [Knu85] D.E. Knuth, An analysis of otiu caching, Journal of algoriths (985) [JL6] A. Jain and C. Lin, Bac to the future: Leveraging Belady s algorith for iroved cache relaceent, in Proc. ISCA-4,(06) [Si76] A.J. Sith, Analysis of the otial, loo-ahead deand aging algoriths, SIA Journal Coutation 5/4 (976) arov Analysis of Otiu Caching