A Framework for Efficient Evaluation of the Fault Tolerance of Deduplicated Storage Systems

Transcription

1 A Framwork for Efficint Evaluation of th Fault Tolranc of Dduplicatd Storag Sytm Eric William Davi Rozir, William H. Sandr Coordinatd Scinc Laboratory Univrity of Illinoi at Urbana-Champaign Urbana, Illinoi, USA Abtract In thi papr w prnt a framwork for analyzing th fault tolranc of dduplicatd torag ytm. W dicu mthod for building modl of dduplicatd torag ytm by analyzing mpirical data on a fil catgory bai. W provid an algorithm for gnrating componnt-bad modl from thi information and a pcification of th torag ytm architctur. Givn th complx natur of dtaild modl of dduplicatd torag ytm, finding a olution uing traditional dicrt vnt imulation or numrical olvr can b difficult. W introduc an algorithm which allow for a mor fficint olution by xploiting th undrlying tructur of dpndnci to dcompo th modl of th torag ytm. W prnt a ca tudy of our framwork for a ral ytm. W analyz a production dduplicatd torag ytm and propo xtnion which improv fault tolranc whil maintaining high torag fficincy. Kyword-torag, dduplication, rliability, imulation, dcompoition I. INTRODUCTION Modrn torag ytm hav bcom incraingly larg and complx, both bcau of th growth of ur data, and in rpon to rcnt lgilation mandating th lngth of tim data mut b tord for rtrival. In 2002, ovr fiv xabyt of data wr producd [1], rprnting an incra of 30% from By 2007 th figur had incrad to 281 xabyt, 10% mor than xpctd bcau of fatr growth in camra, digital TV hipmnt, and othr mdia ctor [2]. In 2010, th total data producd pad th zttabyt barrir. Forcat put th total iz of tord data for 2011 at 10 tim th iz for 2006, roughly 1.8 zttabyt [3]. Storing thi data ha bcom incraingly problmatic. In 2007, a forcat, th amount of data cratd xcdd availabl torag for th firt tim [2]. In ordr to rduc th footprint of backup and archival torag, ytm architct hav bgun uing a nw mthod to improv torag fficincy, calld data dduplication. At a high lvl, data dduplication i a mthod for liminating rdundant data in a torag ytm to improv torag fficincy. Sub-fil gmnt ar fingrprintd and compard to a data ba of idntifid gmnt to find duplicat data. Duplicat data i thn rplacd with rfrnc to th tord intanc. Thi papr prnt a framwork for valuating th fault tolranc of larg-cal ytm which utiliz dduplication. Th framwork upport th analyi of xiting dduplication torag profil in ordr to build accurat modl of th rlationhip btwn dduplicatd fil a th rlationhip can diffr nough, vn btwn catgori of fil on a ingl torag ytm, to hift th impact of dduplication on fault tolranc from poitiv to ngativ. Additionally thi framwork u knowldg of th dpndnc rlationhip caud by dduplication and RAID to dynamically dcompo th ytm modl, bad on th rward variabl dfind ovr th ytm, to improv olution fficincy. Our framwork conit of vral part: A t of componnt-bad modl of th undrlying torag ytm. A modl of dduplication rlationhip in th torag ytm, gnratd tochatically uing mpirical data from a ral torag ytm. A mthod for idntifying dpndnc rlationhip in th rulting modl, important vnt which tmporarily rmov or rturn dpndnc rlationhip in a modl (fault, fault propagation, and fault mitigation), and a mthod to automatically dcompo a modl bad on thi information to improv th fficincy of modl olution. Our framwork formaliz th tchniqu prntd in [4] for a 7TB ytm. In thi papr w apply our framwork to a modl of a on ptabyt torag ytm bad on th analyi of additional data from a ytm imilar to that prntd in [4] but with a diffrnt ur ba, and thu diffrnt charactritic. W prnt tratgi for improving rliability by toring additional copi of dduplicatd fil for a ubt of th ytm, and how that whil for om catgori, dduplication ha a ngativ impact on rliability, for othr th impact i poitiv, dmontrating prviou prdiction mad in [4]. A. Rlatd Work Th cot of dduplication in trm of prformanc [5], [6] i wll undrtood. Rliability tudi hav bn much fwr in numbr. Sinc traditional dduplication kp only

2 a ingl intanc of rdundant data, dduplication ha th potntial to magnify th ngativ impact of loing a data chunk [7], [8] Howvr, du to th mallr numbr of dik rquird to tor dduplicatd data, dduplication alo ha th potntial to improv rliability a wll. Adminitrator and ytm architct hav found undrtanding th data rliability of thir ytm undr dduplication to b important but xtrmly difficult [9]. Quantitativ modling of rliability in a dduplication ytm i challnging, vn without taking into account th ptabyt cal of torag ytm. Firt, thr ar diffrnt typ of fault in a torag ytm, including whol dik failur [10], latnt ctor rror (LSE) [11], [12], and undtctd dik rror [13], [14], [15]. Scond, th fault can propagat du to th haring of data chunk or chaining of fil in a dduplication ytm. In ordr to corrctly undrtand th impact of th fault and thir conqunc on th rliability of our torag ytm, w nd to accuratly modl both torag ytm fault and fault du to data dduplication. Third, it i important to not that many of th fault w wih to conidr ar rar compard to othr vnt in th ytm, uch a dik crubbing, dik rbuild, and I/O. Calculating th impact from rar vnt in a ytm can b computationally xpniv, motivating u to find fficint way of mauring thir ffct on th rliability mtric of intrt. Th complxity of thi problm ari from two diffrnt cau. Th firt i th tat-pac xploion problm which can mak numrical olution difficult. A cond iu com from th tiffn that rult from rar vnt. For numrical olution tiffn introduc numrical intability, making olution impractical. Whn imulating, tiffn incra th numbr of vnt w mut proc, cauing a rulting incra in imulation complxity. B. Organization Thi papr i organizd a follow: Sction II introduc th modling formalim ud for our framwork; Sction III dicu th modl w u for dik, rliability group, and data dduplication rlationhip; Sction V dicu data dpndnc rlationhip and introduc th notion of a modl dpndnc graph (MDG); Sction VI dicu mthod for automatically idntifying ky vnt in an MDG; Sction VII u th mthod dcribd in Sction V and VI to dcompo a modl, whil prrving rward variabl rlationhip; Sction VIII dicu olution mthod for th dcompod modl, and prov th prrvation of rward mtric by th olution mthod; and finally Sction IX prnt th application of th mthod to our ytm of intrt and dicu th rult, and Sction X conclud th papr. II. BACKGROUND W prnt our mthod in th contxt of a gnric modl pcification languag bad on th notation prntd in [16]. Thi i intndd a an altrnativ to prnting our rult in a pcific formalim, both to implify th dicuion of our tchniqu and to gnraliz our mthod. Dfinition 1. A modl i a 5-tupl (S, E,, Λ, ) S i a finit t of tat variabl { 1, 2,..., n } that tak valu in N. E i a finit t of vnt, { 1, 2,..., m } that may occur in th modl. : E N 1 N 2... N n {0, 1} i th vntnabling function pcification. Λ : E N 1 N 2... N n (0, ) i th tranition rat function pcification. : E N 1 N 2... N n N 1 N 2... N n i th tat variabl tranition function pcification. W rprnt thi formalim viually uing circl for tat variabl, box for vnt, and arc to rprnt th dpndnc of th function, Λ, and on tat variabl and vnt. In addition to pcifying a modl of ytm, on mut pcify th prformability, availability, or dpndability maur for a modl. Th maur ar pcifid in trm of rward variabl [17]. Rward variabl ar pcifid a a rward tructur [18] and a variabl typ. Dfinition 2. Givn modl M = (S, E,, Λ, ), w dfin two rward tructur: rat rward and impul rward. A rat rward i dfind a a function R : P(S, N) R, whr for q P(S, N), R(q) i th rward accumulatd whn for ach (, n) q th marking of i n. An impul rward i dfind a a function I : E R, whr for E, I() i th rward arnd upon compltion of. whr P(S, N) i th t of all partial function btwn S and N. Dfinition 3. Lt Θ M = {θ 0, θ 1,...} b a t of rward variabl, ach with rward tructur R or I aociatd with a modl M. Th typ of a rward variabl dtrmin how th rward tructur i valuatd, and can b dfind ovr an intrval of tim, an intant of tim, or in tady tat, a hown in [19], [17]. A. Intant-of-Tim Variabl W rfr to variabl which ar ud to maur th bhavior of a modl at a particular tim t a intant-of-tim variabl [20], [17]. Such a variabl, θ(t) i dfind a:

3 whr θ t = R(ν) I ν t + E I() I t (1) I ν t i an indicator random variabl which rprnt th intanc of a marking uch that for ach (, n) ν, th tat variabl ha a valu of n at tim t. I t i an indicator random variabl which rprnt th intanc of an vnt that ha fird mot rcntly at tim t. B. Intrval-of-Tim Variabl In ordr to calculat mtric which accumulat ovr om fixd intrval of tim, w u intrval-of-tim variabl. Such variabl accumulat rward during om intrval of tim, and tak on th valu of th total rward for th dfind priod [20], [17]. Givn uch a variabl, θ [t,t+l], w dfin it a: θ [t,t+l] = whr R(ν)J ν [t,t+l] + E I()N [t,t+l] (2) J[t,t+l] ν i a random variabl which rprnt th total tim th modl pnt in a marking uch that for ach (, n) ν, th tat variabl ha a valu of n during th priod [t, t + l]. It i an random variabl which rprnt th numbr of tim an vnt ha during th priod [t, t + l]. C. Tim-Avragd Intrval-of-Tim Variabl Th final typ of rward variabl w will conidr ar tim-avragd intrval-of-tim variabl. Th variabl quantify accumulatd rward avragd ovr om intrval of tim [20], [17]. Givn uch a variabl, θ [t,t+l] w dfin it a: θ [t,t+l] = θ [t,t+l] l III. SYSTEM MODELS A. Empirical Analyi of Dduplicatd Storag Sytm In ordr to build a modl of our dduplicatd torag ytm, w xamind dduplicatd data tord in an ntrpri backup/archiv torag ytm that utiliz variablchunk hahing [21], [22]. W gnratd a modl of th dduplication rfrnc and tord intanc rlationhip in th mannr dcribd by [4]. Th data conitd of roughly 200,000,000 dduplicatd fil, which wr ortd into twlv catgori uing th algorithm prntd in [4]. (3) B. Fault Modl Whn modling fault, w utiliz modl of traditional dik failur, LSE and undtctd dik rror (UDE). Traditional dik failur ar aumd to b non-tranint and unrpairabl without driv rplacmnt. LSE can b ithr tranint or prmannt [11]. It i important to not that vn in th ca of a tranint LSE, a prviou tudy of LSE ha indicatd that data tord in th ctor i irrvocably lot, vn whn th ctor can latr b rad or writtn to proprly [11]. In our ytm modl, w conidr LSE to b corrctabl ithr whn th dik i ubquntly rbuilt du to a traditional dik failur, or upon prformanc of a crub of th appropriat dik. UDE rprnt ilnt data corruption on th dik, which i undtctabl by normal man [23], [13], [14]. UDE which manift during writ ar pritnt rror that ar only dtctabl during a rad opration ubqunt to th faulty writ. W conidr UDE to b corrctabl whn th dik i rbuilt bcau of a traditional dik failur, upon prformanc of a crub of th appropriat dik, or whn th rror i ovrwrittn bfor bing rad, although thi typ of mitigation produc parity pollution [14]. W draw our modl for total dik failur, LSE, and UDE from tho prntd in [4]. In gnral, for a fault to manift a a data lo rror, w mut xprinc a ri of fault within a ingl RAID unit. How th fault manift a rror dpnd on th ordring of fault and rpair action in a tim lin of ytm vnt. UDE cau a diffrnt kind of rror, which i largly orthogonal to RAID, by ilntly corrupting data which can thn b rvd to th ur. C. Dik Modl In ordr to undrtand th ffct of fault in an xampl ytm, w utiliz a formal modl of dik in our undrlying torag ytm. Each dik in our ytm i modld a a t of block. Th tat of a block i modld uing th variabl block_tat. Thi variabl indicat whthr th block i in a non-faulty tat, or i faulty du to an LSE, UDE, or total dik failur; tho poibiliti ar rprntd a 0, 1, 2, or 3, rpctivly. Each block modl contain vnt which rprnt fault, fault propagation, fault mitigation, and rpair. A full rprntation for a givn block i hown in Figur 1. A dik i modld a th collction of all block modl that har a common dik failur vnt, and all intrcting condary LSE vnt. D. Rliability Group Modl In ordr to charactriz th intraction of fault in our modl, w maintain a tat-bad modl of portion of th phyical dik. Givn a t of dik that ar groupd into an intrdpndnt array (uch a th t of dik in a RAID5 configuration, or a pair of dik that ar mirrord), ach trip in th array maintain it tat uing a tat

4 advanc_crub crub_proc advanc_crub block_tat block_tat block_tat block_tat block_tat block_tat Scondary_LSE Initial_LSE crub block_tat UDE rbuild parity_tat writ_block Total Dik Failur ddup_fail block_tat Failur (a) Exampl modl rprnting dduplication rlationhip. ddup_fail Failur block_tat block_tat Failur (b) Exampl modl rprnting dduplication rlationhip with 2-copy dduplication. Figur 3: Rprnting dduplication in our modling formalim LSE_count Scondary_LSE... Figur 1: Block modl diagram... Scrub Ovrwrit Rpair Strip_DFA TF UDE LSE Rad Figur 2: Strip modl diagram... Failur machin appropriat to th numbr of tolratd fault th configuration can utain without data lo. In ordr to charactriz th intraction of fault in our modl, w maintain a tat-bad modl of portion of th phyical dik. Each trip-bad tat machin i implmntd by toring th trip tat in a tat variabl calld Strip_DFA (a hown in Figur 2), which tak on valu {0, 1, 2, 3, 4, 5, 6} to rprnt th tat for our DFA. Evnt in thi trip modl rprnt tat tranition vnt and hav nabling condition bad on th fault prnt in block in th givn trip on th ytm. Anothr tat variabl, Failur, i t to 1 whn Strip_DFA ha a valu in {5, 6}. Th DFA maintaind by trip within our modld ytm ar gnratd automatically uing knowldg of potntial fault intraction and paramtr that dfin th iz of th dik array array and th numbr of dik fault n tolratd fault that th array can tolrat without data lo, a dfind by th array RAID lvl [24]. 1) Dduplication Modl: W u mpirical data to gnrat an timat of th probability dnity function (pdf) for a random variabl rprnting th numbr of rfrnc for a chunk in th givn catgory c. Uing thi pdf, f c (x) w can gnrat ralization of th random variabl dcribd by f c (x), allowing u to ynthtically crat a dduplication ytm with th am tatitical proprti a our xampl ytm. W ncod tho rlationhip whn gnrating th modl a dpndnc rlationhip and corrlatd failur that occur whn an undrlying block ha faild, a hown in th xampl in Figur 3a. In th figur, whn th block_tat tat variabl hown at th bottom of th diagram ha faild, and th Failur tat variabl indicat th failur of th trip. An additional failur of dduplicatd rfrnc occur bcau of th lo of a rquird intanc. Multicopy dduplication modifi th undrlying modl a hown in Figur 3b. IV. DEPENDENCE In ordr to improv th fficincy of our olution, w attmpt to xploit dpndnc rlationhip prnt in our modl. Our hypothi i that failur crat important dpndnc rlationhip within th modl, cauing u to valuat othrwi indpndnt ubmodl a a largr modl. W hypothiz that rpair action brak th dpndnci, until th nxt failur occur. A. RAID-Inducd Dpndnc Undr normal oprating condition, th componnt of a torag ytm can b conidrd largly indpndnt. Givn th aumption of uniform fil placmnt on th ytm (a i typical for larg-cal gnral purpo machin), fil ar rad and writtn to driv in an indpndnt fahion. In uch ca th ytm might b modld mor tractably a a t of indpndnt ytm, with ach ytm olvd individually. Onc a failur ha occurrd, th ntir RAID group mut b

5 conidrd dpndnt a furthr failur dirctly impact th intgrity of fil in th RAID group. Thi dpndnc can b rmovd onc uccfully rcovr action hav rpaird all failur within th RAID group, allowing th dik in th group to onc again b conidrd indpndnt. B. Dduplication Inducd Dpndnc An additional form of dpndnc in torag ytm ar tho caud by dduplication. Whn a failur occur for a chunk which tor an intanc of a dduplicatd rourc in th torag ytm, a dpndnc i cratd for all rfrnc to that chunk, and th othr dik in th faild intanc RAID group. Should th RAID uffr additional failur which mak th intanc unrcovrabl, all rfrnc to th intanc will alo b unrcovrabl. A bfor, rcovry of th faild intanc and rpair liminat thi dpndnc. C. Important Evnt For both of th major ourc of dpndnc, an important point i that th vnt that cau and rmov dpndnc ar rar vnt. Fault of intrt occur rarly in th ytm, bcau of th rat ud by th modl that rprnt thm. Rpair action, whil having rat that ar rlativly fat compard to that of failur, can only occur whn a fault ha changd ytm tat. Thu, thy ar rar du to thir nabling condition, which ar rarly mt. In latr ction, w will analyz rar vnt along with dpndnc rlationhip to form a tratgy for olving our ytm. V. UNDERSTANDING DEPENDENCE RELATIONSHIPS W concrn ourlv with four typ of dpndnc rlationhip: Rat dpndnc: Th two ubmodl can b aid to hav rat dpndnc if th tranition rat function Λ of an vnt in on ubmodl i dfind in trm of th tat variabl of th othr ubmodl. Extrnal dpndnc: Whn an vnt in on ubmodl ha an vnt-nabling function,, dfind in trm of th tat or tat variabl of anothr ubmodl, w ay th ubmodl fatur xtrnal dpndnc. -dpndnc: Whn th firing of an vnt chang th valu of tat variabl in two or mor othrwi indpndnt ubmodl, w ay that thy fatur - dpndnc. Rward dpndnc: Whn a rward variabl θ i Θ M xit uch that it rward tructur i dfind in trm of th tat variabl of two ubmodl, or in trm of th vnt of two ubmodl, w ay thy fatur rward dpndnc. Dpndnci btwn ubmodl rult from dirct dpndnci btwn vnt and tat, or from indirct dpndnci rulting from a qunc of dirct dpndnci. A. Modl Dpndncy Graph W dfin a way to codify th rlationhip by contructing a modl dpndncy graph (MDG). W will u an MDG in conjunction with rar vnt found in th modl via th mthod dicud in Sction VI a input to an algorithm w introduc in Sction VII, to provid a propod dcompoition of M. Dfinition 4. Th MDG of a modl M i dfind a an undirctd labld graph, G M = (V, A, L), whr V i a t of vrtic compod of thr ubt V = V S V E V Θ, A i a t of arc conncting two vrtic uch that on vrtx i alway an lmnt of th ubt V S and on vrtx i alway an lmnt of th ubt V E, or on vrtx i of th ubt V Θ whil th othr i of th t {V E V S }, and L i a t of labl applid to lmnt of A from th t {, Λ,, R}. Lt V S dnot th ubt of vrtic rprnting th tat variabl S M; V E dnot th ubt of vrtic rprnting th vnt E M, and V Θ dnot th ubt of vrtic rprnting rward variabl from Θ M. W contruct G M uing th modl pcification from Dfinition 1 of a modl M, from Sction II. G M ha a nod for vry tat variabl in S and vnt in E, and rward variabl in Θ M, with arc conncting an arbitrary tat variabl i to an arbitrary vnt j, iff Th nabling condition of j dpnd on th valu of i. Thi indicat an xtrnal dpndnc and i markd with th labl. Th rat of th vnt j dpnd on th valu of i. Thi rprnt a rat dpndnc and i markd with th labl Λ. Th firing of j chang th valu of i. Thi rprnt a dpndnc and i markd with th labl. An arc labld R connct a nod rprnting an lmnt θ i Θ M to a nod rprnting an lmnt a j {S E} iff a j S and θ i i a rat rward dfind in trm of a j. a j E and θ i i an impul rward dfind in trm of a j. W rprnt an MDG graphically a hown in Figur 4. Stat variabl ar rprntd by circl, vnt by quar, and rward variabl by diamond. Arc in an MDG rprnt dpndnci. A hown in Figur 4 rat dpndnci ar labld Λ (a); xtrnal dpndnci ar labld with (b); -dpndnci ar labld with (c); and rat dpndnci, whthr impul or rat rward, ar labld with R (d,). VI. FAILURE, RECOVERY, AND MITIGATION EVENTS In ordr to find a way to dcompo a torag modl, w could rquir th ur to pcify th failur and rpair action in th modl, a with th dcompoition mthod

6 a) b) c) d) ) Λ R r 0 0 R r 0 2 Rat dpndnc Extrnal dpndnc dpndnc }Rward dpndnc Modl Rward Variabl Modl Dpndnc Graph Idntify Rar Evnt Dcompoition Dcompod Submodl Ξ im Sim Solv for nxt Ξ num Figur 5: Ovrviw of olution Mthod τ τ τ Num τ Figur 4: Two xampl of nar-indpndnt ubmodl. prntd by [25]. Idally, howvr, w wih to b abl to idntify th vnt without ur input. Th charactritic that t th vnt apart from othr in th modl i that thy ar rar. A. Idntifying Rar Evnt In ordr to find th vnt which rprnt whol dik fault, LSE and UDE, w nd to idntify vnt which ar locally rar. An vnt i, Λ( i, q) may b dfind uch that it rat i much l than that of othr vnt in th modl, i.. Λ( i, q) < µ max q. In th ca w can claify th local rat of i to b rar. In th ca of an vnt with a tat-dpndnt rat (i.., whr Λ( i, q) vari for diffrnt q), it may b uful to crat two virtual vnt, i,1 and i,2, with th firt virtual vnt rplacing i for valu of Λ( i, q) that contitut non-rar vnt, and i,2 rplacing i for valu that qualify a rprnting rar vnt. For our dduplication ytm, th locally rar rat which hav Λ( i, q) < µ max q, play a part in idntifying rar vnt in th ca of total dik failur, initial latntctor rror, and undtctd dik rror. Th vnt hav rat which ar rar compard to othr within th modl, bad imply on th valuation of thir rat function Λ( i, q). Th final, and potntially mot difficult to idntify, fahion in which vnt may b rar i whn thir nabling condition dfind by ar rar. Dpit th difficulty in finding uch vnt, thy ar important a thy rprnt rcovry/rpair action, among othr thing. It i important to idntify vnt rprnting rcovry, mitigation, and propagation a wll a fault. Th difficulty i that rcovry action hav high rat compard to mot failur. Howvr, inc thy ar not nabld unl a failur ha occurrd, thy ar dpndnt on a rar nabling condition. Latnt ctor rror alo partially fall into thi catgory. Whil an initial LSE i dfind by a locally rar rat, tudi [12] hav hown that thr i a priod aftrword during which thy bcom frqunt. Thi prcondition of a rcnt LSE crat a condition whr an othrwi common vnt, i rar du to th tat in which it i common bing rar. Whil calculation of all rar nabling condition i a difficult problm, w can u domain knowldg of torag ytm rliability modl to aid u in finding crtain cla Figur 6: Modl rpartition aftr ach rar vnt. of rar nabling condition. A w notd bfor, rcovry action ar by ncity paird with a prviou fault in th modl. Without a fault, th tat variabl in th modl can not hav valu uch that th rcovry action can fir, and o rcovry action dirctly dpnd on rar vnt. By analyzing th dpndnc rlationhip in our modl w can idntify th rar vnt givn th aumption that our modl bgin with no initial fault. B. Partitioning W idntify a crtain ubt E R E a rar vnt, givn om partitioning chm, a firt dicud in Sction VI-A. For th modl w hav tudid, it ha bn appropriat to aum a tatic partitioning paramtr µ max. W lct a valu for µ max bad on th fact that fault vnt ar many ordr of magnitud rarr than non-fault vnt. Two clutr of rat wr aily idntifid uing k-man clutring with two clutr. Givn a valu for µ max, w find tho vnt i E for which Λ( i, q) < µ max q and dfin th t of th vnt a E R. Thi partitioning idntifi tho rar vnt that ar rar du to a locally rar rat, or comptition. VII. DECOMPOSITION In thi ction w prnt an algorithm for dcompoing M, uing th MDG gnratd from M, G M. M will b dcompod into a t of n ubmodl Ξ = {ξ 0, ξ 1,..., ξ n } that can b conidrd indpndnt in th abnc of th firing of a rar vnt. W alo dicu how to rpartition M uing G M aftr a rar vnt ha fird, producing a nw t of indpndnt ubmodl, a illutratd in Figur 5. Whn viualizing our imulation a a tim-lin, a hown in Figur 6, w rprnt th firing of a rar vnt with th ymbol τ. Each tim a rar vnt fir, it rult in a rvaluation of our dcompoition of M, rprntd in Figur 5 a τ. Givn a modl and a t of rward variabl, w gnrat an MDG and a t of idntifid rar vnt, E R. Uing th MDG, G M, th t of rar vnt, E R, and th currnt valu of all tat variabl in th modl M, w produc a dcompoition of

7 R r Λ R (a) Compod Modl R r 0 G 0 2 Λ R 1 { c 2{ G 3 3 } G 0 (b) Dcompod Modl Figur 7: Exampl dcompoition of a modl dpndncy graph G M to G M. th modl M into a t of ubmodl Ξ R and Ξ!R. From Ξ R, Ξ!R, and th ubt Ξ ER Ξ of all ubmodl (in both Ξ R and Ξ!R ) that contain vnt in E R, w produc two nw t of ubmodl: Ξ Sim = Ξ R (Ξ!R Ξ ER ) (4) Ξ Num = Ξ!R \ (Ξ!R Ξ ER ). (5) Th ubmodl in th t Ξ Sim and Ξ Num will thn b pad to an appropriat olution mthod and olvd until a rar vnt fir, at which point w rpat th dcompoition tp bad on th nw tat of th modl. By dcompoing our modl in thi fahion, w hop to rmov from conidration vnt for which thr i no currnt dirct or indirct dpndncy from our rward variabl in th abnc of a rar vnt firing. In ordr to form thi dcompoition, howvr, w mut analyz th dpndnci in G M, and from th rult of that analyi form a dcompod modl dpndncy graph G M that can b ud to idntify a ubmodl dcompoition of M. W form a dcompod modl dpndncy graph G M for M by firt rmoving all -dpndnci that involv vnt in E R. For vry vrtx aociatd with a tat variabl who only -dpndnci involv vnt in E R, w rplac tho vrtic with nw vrtic from a t V C, which rprnt contant tat variabl who valu ar qual to thir initial condition. All vrtic that rprnt vnt with rat dpndnt on tat variabl that ar now rprntd by contant vrtic ar xamind. If uch vnt hav tranition rat function pcification uch that Λ(, q) = 0 for all q N 1 N 2... N n givn V C, or hav nabling function pcification uch that (, q) = 0 for all q givn V C, thy ar rmovd. All dpndnci of rmovd vnt ar alo rmovd. Th proc i rpatd, xamining all V S and V E itrativly until no nw vrtic ar rmovd. Thi proc for gnrating G M uing G M and E R i prntd in Algorithm 1. Th graph G M that rult from th application of Algorithm 1 to G M and E R i thn ud to dtrmin if a valid partition of th modl M xit for our tchniqu. If G M dfin multipl unconnctd ub-graph, G M = {g 0 g 1...}, a valid partition xit. If it do not, Algorithm 1 Rmov rar-vnt-bad dpndnci from G M. G M = (V, A, L ) G M P E R whil P do Rmov all dg in A containing at lat on vrtx in P. Do not rmov vrtic with dg labld or Λ if th vrtx i in E R. P for all v i V S do if! v iv j A uch that v iv j ha labl thn V V \ v i Crat a nw contant vrtx v ci V C V V v ci Aociat a valu qual to th initial marking of i S aociatd with v i with v ci nd if nd for for all v j V E do if v i v iv j A labld uch that v i V C thn if! q conitnt with th contant marking aociatd with vrtic in V C and ( j, q) = 1 thn P P v j nd if nd if if v i v iv j A labld Λ uch that v i V C thn if! q conitnt with th contant marking aociatd with vrtic in V C and Λ( j, q) 0 thn P P v j nd if nd if nd for V V \ P nd whil our tchniqu i not applicabl. Th ub-graph of G M corrpond to th ubmodl in our partition Ξ. For a givn ub-graph, g i = (V i, A i ), for ach v j V i uch that v j V S, w add th corrponding tat variabl to ξ i. For ach v j V i uch that v j V E, w add th corrponding vnt to ξ i. In addition, for ach ξ i Ξ w rtrict th dfinition of ( j, q), Λ( j, q), and ( j, q) to j ξ i and q N 1, N 2,..., N Sξi uch that S ξi ξ i. An xampl dcompoition of a mall modl i i hown in Figur 7. A. Mitigation, Rcovry, and Propagation Evnt In Sction VI w mntiond that by uing mthod dicud in thi ction, w would b abl to find mitigation, rcovry and fault propagation vnt. Through xcution of Algorithm 1 on M, with tat variabl t to rprnt an initially fault-fr modl, rmoving tat variabl and vnt in th mannr dcribd by Algorithm 1, it i guarantd that rcovry, mitigation and propagation vnt will b rmovd. Thi i du to th fact that th fault aociatd with tho action hav yt to occur. Sinc rcovry, mitigation and propagation action hav a dirct corrpondnc with fault in th ytm (giving thm thir rar nabling condition), in th abnc of a fault, thir nabling condition cannot

8 b mt by th contant placholdr w u to rprnt th ffct of a fault vnt firing. Thu tho vnt will b rmovd during our dcompoition tp. Whn Algorithm 1 i ud, th t of all vnt addd to P i th t of fault vnt in E R plu any vnt that dpnd on E R ; that rprnt rcovry, mitigation, or fault propagation; and that ar addd to E R whn our modl i bing dcompod and olvd during imulation. B. Analyzing Rward Variabl Dpndnci Rward variabl dpndnci prvnt dcompoition of othrwi indpndnt ub-graph by maintaining connctivity bad on rward dpndnc and hlp u choo olution mthod for ubmodl in Ξ. Propoition 1. In th abnc of th firing of a rar vnt, th rward variabl θ i i indpndnt from a ubmodl ξ j if no dirct dpndnc xit in G M from θ i to a vrtx in g j. Proof: If a dirct dpndnc xitd btwn a rward variabl θ i and a tat or vnt in ξ j thn G M would hav an dg conncting θ i to a vrtx in g j, and a path would xit. If thr wr an indirct dpndncy btwn θ i and a vrtx in g j, thn a path would xit btwn a vrtx v k that ha a dirct dpndncy with θ i and a vrtx in g j. Thn v k would b a vrtx in g j, and θ i would hav an dg conncting dirctly to a vrtx in g j. Givn G M, w divid all ubmodl in Ξ dfind by th indpndnt ub-graph of G M into two t: tho upon which rward variabl do and do not dpnd in th abnc of rar vnt. Th t of ubmodl ar calld Ξ R and Ξ!R, rpctivly. VIII. SOLVING THE DECOMPOSED MODEL W prnt in thi ction an algorithm for hybrid imulation of dcompod modl, and a dicuion of complmntary olution mthod from th litratur. Our hybrid imulation algorithm wa dignd to hlp u tudy th dpndability charactritic of dduplicatd data torag ytm. A. Hybrid Simulation of Rar-Evnt Dcompod Sytm Our tudy of rar-vnt-bad dcompoition mthod wa motivatd by a dir to tudy th dpndability charactritic of torag ytm that utiliz data dduplication, in a fault nvironmnt charactrizd by rar vnt. In ordr to timat th valu of rward variabl dfind for modl of th ytm, w hav mployd our dcompoition mthod and a hybrid imulation algorithm. Whn olving our modl, w viw trajctori of modl xcution a a tim ri τ 0 τ 1 τ 2 τ 3... whr τ 0 rprnt our tart tim, and ach ubqunt τ i rprnt th firing of a rar vnt. Th t Ξ Sim contain all ubmodl that contain ithr a rar vnt or a rward Algorithm 2 Hybrid Simulation of M Givn M, Θ M, G M and initial valu for all tat variabl. whil Θ M not convrgd do Gnrat G M and Ξ from G M. Driv Ξ Sim and Ξ Num. Simulat Ξ Sim until th nxt vnt i in th t E R. Gnrat πξ i for ach ubmodl Ξ Num. Gnrat a random tat for ξ i Ξ Num trating πξ i a th pmf of th random variabl. Rcompo M. Simulat th nxt rar vnt in M. U currnt tat of M a th nxt initial tat. nd whil dpndncy. Th t Ξ Num contain all ubmodl that hav nithr rar vnt nor rward dpndnci. From Propoition 1, rward variabl olution do not dpnd on Ξ Num. Thu w nd only olv th tat occupancy probability for all ubmodl in Ξ Num at th tim of th nxt rar vnt firing. W do o by making th aumption that th ubmodl ntr tady tat btwn firing of rar vnt. Thi aumption m appropriat for two raon. Th firt i th high probability of a long intr-vnt tim btwn rar vnt. Th cond rlat to th fact that for th torag ytm in which w ar intrtd, ytm tnd to ntr into tady tat intantly in th abnc of rar vnt. Th crub proc, for xampl, i alway in tady tat; th am hold tru for many rcovry, propagation, or mitigation ubmodl. Simulation of th modl M i prformd uing Algorithm 2, a modifid vrion of a tandard dicrt vnt imulator. Th gnral improvmnt offrd by thi algorithm com from th rduction of vnt that mut b imulatd in ordr to timat th ffct of rar vnt in th ytm. Bucklw and Radk [26] giv a gnral rul of thumb that in ordr to timat th impact of an vnt with probability ρ, w mut proc approximatly 100/ρ imulation. Our mthod k to rduc th numbr of vnt that mut procd for ach computd trajctory by liminating tho vnt that cannot impact Θ M without th firing of a rar vnt. Th prformanc improvmnt offrd by thi algorithm vari with th modl and with th dgr of dpndnc of th tat variabl and vnt in th modl. For modl who rulting Ξ do not hav th propr tructur, our propod hybrid imulator may provid no improvmnt. Btwn firing of a rar vnt, our mthod will produc a pd-up proportional to th rat at which w rmov vnt from xplicit imulation. Thu givn E a th t of all vnt i Ξ Num j PM uch that j / Ξ, our improvmnt i proportional to i E λ(i,ψi) P i E λ(i,ψi), ψ i Ψ. Thi improvmnt i du to th rmoval of vnt that, whil nabld, cannot chang th tat of our modl in a way that influnc our rward variabl. For intanc, a writ proc may till b nabld, but in th abnc of a UDE th writ proc cannot rult in th propagation of a UDE to parity.

9 Likwi whil it i important to rprnt th poition of th crub proc, it cannot rult in mitigation of a fault until a fault i prnt to mitigat. By rmoving tho vnt from our imulator, w improv th fficincy of our olution. B. Rquird Aumption A ky aumption of our olution mthod i that th torag ub-modl rach tady tat btwn rar vnt. For th torag ytm of intrt, th initial tranint priod i not part of th ytm liftim during production. W conidr th initial tat of any torag ytm to b faultfr and with uniform ditribution of fil acro th torag ytm itlf. Th tady tat of uch a ytm can b conidrd thi fault-fr condition, with th placmnt of fil dcribd by th obrvd mpirical ditribution ud to modl dduplication. Othr ub-modl that ar likly to b conidrd indpndnt, uch a th modl of th crub proc, ar charactrizd by priodic proc that ar ithr unprturbd by rar vnt in th ytm, or dormant in th abnc of fault. Thu, any ytm that i not currntly compod with anothr ytm that contain a failur can b aid to b in tady tat. C. Corrctn of Rward Variabl In thi ction, w how th thr typ of rward variabl dfind in Sction II. W rdfin th rward variabl for u with our dcompoition algorithm and hybrid olution mthod, dcribd prviouly. W dmontrat that th rulting rward variabl ar quivalnt to tho dfind in Sction II. Whn olving for intant-of-tim variabl, w u th am quation givn in Sction II: θ t = R(ν) I ν t + A I() I t (6) W valuat thi variabl in th am fahion for th ubmodl that contain th ncary tat variabl to tablih ach ν P(S, N) and ach A. Propoition 2. For a givn modl M, a dcompod MDG G M, and an intant-of-tim rward variabl θ t, olving for θ t uing Equation 6, and th appropriat ubmodl dcompoition propod by G M yild th am rult a th original modl M. Proof: From Propoition 1 w know that th rward variabl θ t i indpndnt from a ubmodl ξ j at tim t if no dirct dpndnc xit in G M from θ t to a vrtx in g j. Thu th olution at tim t for θ t for our dcompod ubmodl i th am a th olution for M. Bcau of our mthod of forming an MDG, rward dpndnci will xit btwn a variabl and all tat t d0 d0 d1 d1 t+l t τ τ τ Figur 8: Comparion of intant-of-tim rward variabl olution variabl and vnt, nuring that th ubmodl i dcompod in uch a way that th rulting ubmodl contain vrything ncary to valuat θ t. To olv for intrval-of-tim variabl uing our hybrid olution mthod, w mut provid a nw mthod of computation. Rcall from Sction II that an intrval-of-tim variabl θ [t,t+l] i dfind a follow: θ [t,t+l] = t+l R(ν)J ν [t,t+l] + A I()N [t,t+l] (7) W modify th computation of intrval-of-tim variabl to accommodat our olution tchniqu by uing multipl random variabl for J ν [t,t+l] and N [t,t+l] bad on th dcompoition and r-compoition of th undrlying modl, a dictatd by our olution tchniqu. A hown in Figur 8 w hav a t of n modl dcompoition that form intrval dfind by th tim d 0, d 1,..., d n 1 during th priod [t, t + l]. For th intrval, w crat n + 1 random variabl to rplac J ν [t,t+l] and N [t,t+l] : J ν [t,d 0], J ν [d 0,d 1],..., J ν [d n 1,t+l] N [t,d 0], N [d 0,d 1],..., N [d n 1,t+l] Each of th random variabl i quivalnt to tho from th prviou dfinition, but ovr a diffrnt intrval of tim. Th variabl ar ditinguihd by n+1 parat intrval in th t D = {[t, d 0 ], [d 0, d 1 ],..., [d n 1, t + l]} Bad on tho idntiti, w rdfin th mthod for calculating an intrval-of-tim variabl for our olution mthod a follow: Y [t,t+l] = d D R(ν)J ν d + A I()Nd (8) d D

10 Th diffrnc btwn that calculation and th on hown in Sction II ar illutratd in Figur 8. In th original modl, w u a ingl random variabl for ach rat and impul rward. Uing our mthod, howvr, w nd on for ach parat intrval in D. Th two mthod ar actually quivalnt, howvr, a th um of th nw indicator variabl yild th old indicator variabl. Th raon i that th dcompoition algorithm w hav prntd prrv rward dpndnci. Propoition 3. For a givn modl M, a t of dcompod MDG G ˆ M ovr th intrval [t, t+l], and an intrval-of-tim rward variabl Y [t,t+l], olving for Y [t,t+l] uing Equation 8 and th appropriat ubmodl dcompoition for ach intrval in D yild th am rult a th original modl M. Proof: Starting from th dfinition from Equation 8, w prov th quivalnc of Y [t,t+l] and θ [t,t+l] by contruction. Y [t,t+l] = = d D R(ν)J ν d + A I()Nd (9) d D R(ν) d D J ν d + A I() d D N d (10) Givn Propoition 1, which tat that all tat variabl and vnt rquird for calculating a rward variabl ar containd within th ubmodl containing th rward variabl itlf, w hav that: J ν [t,t+l] = d D J ν d (11) N [t,t+l] = d D N d (12) Th um of our nw indicator variabl ar th original indicator variabl from Sction II. Thu, Y [t,t+l] = I()N [t,t+l] (13) R(ν)J[t,t+l] ν + A = θ [t,t+l]. (14) Intrval-of-tim rward variabl calculatd with our mthod ar quivalnt to tho calculatd with typical dicrt vnt imulator. For tim-avragd intrval-of-tim variabl, w rdfin th variabl a W [t,t+l] = Y [t,t+l]. (15) l Th abov calculation i imilar to on givn in Sction II, but u th mthod for calculating intrval-of-tim variabl Spd-up 8 + p p p Tabl I: Spd-up ratio of th run-tim for variou RAID configuration. that tak into account th ubdiviion of th intrval [t, t+l] givn by D. Propoition 4. For a givn modl M, a t of dcompod MDG G ˆ M ovr th intrval [t, t + l], and a tim-avragd intrval-of-tim rward variabl θ [t,t+l], olving for W [t,t+l] uing Equation 15 and th appropriat ubmodl dcompoition for ach intrval in D yild th am rult a th original modl M. Proof: Givn quation 14, 15 i quivalnt to th dfinition prntd in Sction II. From Equation 14, w know that Y [t,t+l] = θ [t,t+l]. Subtituting θ [t,t+l] for Y [t,t+l] in Equation 15 yild Equation 3. In thi ction w hav dtaild a mthod to idntify all dpndnc rlationhip in a modl, M, uing an MDG, G M. W thn dtaild how to idntify rar fault in th modl. Uing th t of idntifid fault, E R, and th MDG, w howd how to nlarg th t E R to includ mitigation, rpair, and propagation action. W thn howd how to u E R with G M to form a t of dcompod ubmodl, Ξ, which w thn olvd uing Algorithm 2. IX. DISCUSSION W applid our mthod to a on ptabyt dduplicatd torag ytm to a th impact of dduplication on rliability. W modld our ytm undr thr diffrnt RAID configuration, 8+p, 8+2p, and 8+3p; on, two, and thr parity dik rpctivly. Th application of our framwork allowd u to automatically gnrat dtaild modl of our torag ytm from modl of individual componnt in th ytm itlf. Modl of dduplication rlationhip wr thn automatically gnratd from mpirical data. W compard th ffincy of our hybrid, dpndnc-bad, dicrt vnt imulator to an unmodifid dicrt vnt imulator, and achivd a ignificant pd-up a hown in Tabl I. In our prviou work, [4], w conductd a imilar tudy of larg-cal dduplicatd torag ytm, but howd only a dcra in rliability du to charactritic of th data tord in th tudid ytm. W prdictd that othr ytm may hav diffrnt rult givn othr dduplicatd diffrnt dduplication rlationhip. In th ytm tudid in thi papr, w confirm th prdiction mad in [4], onc again affirming th importanc of a dtaild undrtanding of th undrlying rlationhip in a dduplicatd torag ytm. A hown in Figur 9, whil th Archiv and Databa 1 fil catgori howd a dcra in rliability du to

11 !"##$%!"##$% &'( )*($ )++*($)*($ )+*($)*($ )*($)*($ &'( )*($ )++*($)*($ )+*($)*($ )*($)*($ (a) Corrupt data rvd pr yar du to UDE for th Databa 1 catgory. (b) Corrupt data rvd pr yar du to UDE for th Databa 2 catgory. Figur 10: Rat of corrupt data rvd for four xampl catgori, with no dduplication, 1 copy dduplication, and 1%, 10% and 50% 2 copy dduplication.! "# "$%"$%" "$&&%"$%" "$%"$%" Figur 9: Annual man fil lot du to failur of 8 + p. dduplication, th Cod and Databa 2 catgori howd an incra in rliability du to a mor vn ditribution of rfrnc, maning that no t of fil xitd which contributd diproportionatly to th impact from lo of dduplicatd rfrnc. Likwi, imilar rult wr hown for th impact of dduplication on corrupt data rvd du to UDE. Figur 10 how rat of corrupt data rvd pr yar for th catgori Databa 1 and Databa 2. In th ca of Databa 1, dduplication improv rliability, whr a for Databa 2, rliability dgrad. Rult wr imilar for othr fil catgori in our modld ytm. RAID implmntd a 8 + 2p and 8 + 3p provd highly fault-tolrant, with fw data lo vnt occurring, affirming that multi-copy dduplication i unncary for protction againt additional lo du to RAID failur in on ptabyt ytm. Such ytm, howvr, provid no additional protction againt UDE, howvr. For ytm in which dduplication caud a dcra in rliability, kping an additional copy for th mot rfrncd 1% of fil in ach catgory wa ufficint protction, though additional protction could b achivd (albit with diminihing rturn) by kping additional copi for largr portion of a givn catgory. From th rult, and th rult w providd in [4], it i clarly important to tak into account th actual dduplication charactritic of a givn ytm whn applying multi-copy dduplication tratgi. With our framwork it i poibl to modl and dvlop pr-catgory multi-copy dduplication chm to achiv th dird lvl of rliability, whil maintaining a high lvl of torag fficincy. Sytm dignr nd only ubmit to our framwork th ncary componnt lvl modl, RAID lvl, and th iz of th ytm, along with mpirical data about th footprint of th dduplicatd data thy wih to tor. Each catgory in th ytm can thn b analyzd at varying lvl of multicopy covrag. Ur can ntr a dird lvl of rliability for thir ytm, and uing our framwork dtrmin a lvl of multi-copy covrag which mt thir goal, whil obtaining mor torag fficincy can b prrvd than th naïv approach of aigning th am multi-copy chm to th ntir ytm. X. CONCLUSIONS Thi papr prnt a framwork for fficint olution of rliability modl of larg-cal torag ytm utilizing dduplication. Our framwork gnrat modl from componnt-bad tmplat, add dduplication rlationhip drivd from mpirical data, and idntifi dpndnc rlationhip in th gnratd modl. Th dpndnc rlationhip ar ud to improv th fficincy of modl olution whil laving th rward maur unaffctd. W dmontrat our mthod by olving a larg-cal torag ytm and achiv ignificant pd-up of roughly 20x whn compard to unmodifid dicrt-vnt imulation. Our rult how th importanc of dtaild modl of dduplication bad on fil catgori by howing om cat-

12 gori whr dduplication improv rliability, and om whr it dcra rliability. W how that vn for a imilar typ of catgory, th impact of dduplication may not b th am. Our rult mphaiz th nd to gnrat dtaild modl of dduplicatd ytm whn making dign dciion. Prviou work which uggtd broad application of multicopy dduplication, whil ffctiv for improving rliability, don t tak into account th diminihing rturn ralizd whn largr prcntag of th dduplicatd torag ytm tor multipl-copi of ach rfrncd chunk. Dtaild analyi ha provn difficult in th pat du to th complxiti involvd with olving larg modl containing rarvnt, but th framwork w prnt in thi papr ignificantly rduc th tim ndd to conduct uch tudi. Th limit of incrad rliability for ytm mploying th mthod ar th undduplicatd portion of fil. Givn that mot torag ytm ar ovrproviiond, or includ unud hot par to allow RAID rpair, w propo that thi undrutilizd pac might b ovrbookd to kp additional copi of undduplicatd portion of important fil, furthr improving rliability. REFERENCES [1] P. Lyman, H. R. Varian, K. Saringn, P. Charl, N. Good, L. L. Jordan, and J. Pal, How much information? [Onlin]. Availabl: [2] J. F. Gantz, C. Chut, A. Manfrdiz, S. Minton, D. Rinl, W. Schlichting, and A. Tonchva, Th divr and xploding digital univr: An updatd forcat of worldwid information growth through 2011, Whit Papr, IDC, March [3] J. F. Gantz and D. Rinl, Extracting valu from chao, Whit Papr, IDC, Jun [4] E. W. Rozir, W. H. Sandr, P. Zhou, N. Mandagr, S. M. Uttamchandani, and M. L. Yakuhv, Modling th fault tolranc conqunc of dduplication, in Rliabl Ditributd Sytm (SRDS), th IEEE Sympoium on, oct. 2011, pp [5] B. Zhu, K. Li, and H. Pattron, Avoiding th dik bottlnck in th data domain dduplication fil ytm, in USENIX FAST, 2008, pp [6] C. Unguranu, B. Atkin, A. Aranya, S. Gokhal, S. Rago, G. Calkowki, C. Dubnicki, and A. Bohra, HydraFS: A high-throughput fil ytm for th HYDRAtor contntaddrabl torag ytm, in FAST, 2010, pp [7] D. Bhagwat, K. Pollack, D. D. E. Long, T. Schwarz, E. L. Millr, and J.-F. Pri, Providing high rliability in a minimum rdundancy archival torag ytm, in IEEE MAS- COTS, 2006, pp [8] L. L. You, K. T. Pollack, and D. D. E. Long, Dp tor: An archival torag ytm architctur, in ICDE. IEEE, 2005, pp [9] L. Frman, How af i dduplication, NtApp, Tch. Rp., [Onlin]. Availabl: [10] B. Schrodr and G. A. Gibon, Dik failur in th ral world: what do an MTTF of 1,000,000 hour man to you? in FAST, 2007, p. 1. [11] L. N. Bairavaundaram, G. R. Goodon, S. Paupathy, and J. Schindlr, An analyi of latnt ctor rror in dik driv, SIGMETRICS 35, no. 1, pp , [12] B. Schrodr, S. Damoura, and P. Gill, Undrtanding latnt ctor rror and how to protct againt thm, in FAST, 2010, pp [13] A. Krioukov, L. N. Bairavaundaram, G. R. Goodon, K. Srinivaan, R. Thln, A. C. Arpaci-Duau, and R. H. Arpaci-Dua, Parity lot and parity rgaind, in FAST. USENIX, 2008, pp [14] J. L. Hafnr, V. Dnadhayalan, W. Blluomini, and K. Rao, Undtctd dik rror in RAID array, IBM J Rarch and Dvlopmnt 52, no. 4, pp , [15] E. W. D. Rozir, W. Blluomini, V. Dnadhayalan, J. Hafnr, K. K. Rao, and P. Zhou, Evaluating th impact of undtctd dik rror in RAID ytm, in DSN, 2009, pp [16] W. D. Oball II, Maur-Adaptiv Stat-Spac Contruction Mthod. U Arizona, [17] W. Sandr and J. Myr, A unifid approach for pcifying maur of prformanc, dpndability, and prformability, in DCCA 4. Springr, 1991, pp [18] R. A. Howard, Dynamic Probabilitic Sytm. Vol II: Smi- Markov and Dciion Proc. Nw York: Wily, [19] J. F. Myr, On valuating th prformability of dgradabl computing ytm, IEEE TC 29, pp , [20] W. H. Sandr and J. F. Myr, A unifid approach to pcifying maur of prformanc, dpndability, and prformability, Dpndabl Computing for Critical Application, vol. 4, pp , [21] M. O. Rabin, Fingrprinting by random polynomial, Tch. Rp., [22] A. Z. Brodr, Idntifying and filtring nar-duplicat documnt, in CPM. Springr, 2000, pp [23] L. N. Bairavaundaram, G. R. Goodon, B. Schrodr, A. C. Arpaci-Duau, and R. H. Arpaci-Dua, An analyi of data corruption in th torag tack, in FAST. USENIX, 2008, pp [24] D. A. Pattron, G. A. Gibon, and R. H. Katz, A ca for Rdundant Array of Inxpniv Dik (RAID), EECS Dpartmnt, UC Brkly, Tch. Rp. UCB/CSD , [Onlin]. Availabl: [25] G. Ciardo and K. S. Trivdi, A dcompoition approach for tochatic rward nt modl, Prformanc Eval. 18, no. 1, pp , [26] J. Bucklw and R. Radk, On th Mont Carlo imulation of digital communication ytm in Gauian noi, IEEE Tran. Comm. 51, no. 2, pp , 2003.