Efficient Striping Techniques for Variable Bit Rate Continuous Media File Servers æ

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1 Effcent Strpng Technques for Varable Bt Rate Contnuous Meda Fle Servers æ Prashant J. Shenoy Harrck M. Vn Department of Computer Scence, Department of Computer Scences, Unversty of Massachusetts at Amherst Unversty of Texas at Austn Amherst, MA 13 Austn, TX Abstract The performance of strped dsk arrays s governed by two parameters: the strpe unt sze and the degree of strpng. In ths paper, we descrbe technques for determnng the strpe unt sze and degree of strpng for dsk arrays storng varable bt rate contnuous meda data. We present an analytcal model that uses the server confguraton and the workload characterstcs to predct the load on the most heavly loaded dsk n redundant and non-redundant arrays. We then use the model to determne the optmal strpe unt sze for dfferent workloads. We also use the model to study the effect of varous system parameters on the optmal strpe unt sze. To determne the degree of strpng, we frst demonstrate that strpng a contnuous meda stream across all dsks n the array causes the number of clents supported to ncrease sub-lnearly wth ncrease n the number of dsks. To maxmze the number of clents supported n large arrays, we propose a technque that parttons a dsk array and strpes each meda stream across a sngle partton. Snce load mbalance can occur n such parttoned arrays, we present an analytcal model to compute the mbalance across parttons n the array. We then use the model to determne a partton sze that mnmzes the load mbalance, and hence, maxmzes the number of clents supported by the array. Keywords: Contnuous meda fle servers, strpng technques, dsk arrays 1 Introducton 1.1 Motvaton Advances n computng and communcaton technologes over the past few years have trggered the development of a wde range of nformaton servces (e.g., electronc newspapers, dstance learnng and self-paced educaton, vdeo mal, etc.). All of these servces nvolve storng, accessng, and processng multple types of nformaton (e.g., text, audo, vdeo, magery, etc., - whch we collectvely refer to as multmeda). Realzng such servces wll requre the development of fle servers that can effcently handle multple data types. To do so, such fle servers wll be requred to employ effcent placement technques. To help formulate the problem of effcent placement, let us frst ntroduce some termnology. Dgtzaton of audo yelds a sequence of samples and that of vdeo yelds a sequence of frames. A contnuously recorded sequence of audo samples or vdeo frames s referred to as a meda stream. Due to the large storage and bandwdth requrements æ A prelmnary verson of ths paper appeared n the Proceedngs of the Seventh IEEE Internatonal Workshop on Network and Operatng System Support for Dgtal Audo and Vdeo (NOSSDAV 97), pages 25 36, St. Lous, MO, May

2 of such meda streams, multmeda fle servers are generally founded on dsk arrays. To effcently utlze a dsk array, such servers strpe (.e., nterleave) meda streams across dsks n the array. A strpng polcy s governed by two parameters: the strpe unt sze, whch denotes the maxmum amount of logcally contguous data stored on a sngle dsk; and the degree of strpng, whch refers to the number of dsks across whch a partcular meda stream s strped. Recently, technques for determnng the strpe unt sze and the degree of strpng for workloads consstng of textual and numerc data accesses have been proposed [3, 5, 14]. However, these technques are not drectly applcable to fle servers optmzed for storng audo or vdeo (referred to as contnuous meda) due to the followng fundamental characterstcs: æ Real-tme requrements of contnuous meda: Textual and numerc data accesses requre good response tmes but no absolute performance guarantees. In contrast, due to ts real-tme nature, contnuous meda accesses requre the fle server to provde bounds on response tmes. Hence, a strpe unt sze that mnmzes the average response tme s consdered optmal for textual and numerc data [3], whle a strpe unt sze that mnmzes the tal of the response tme dstrbuton (possbly at the expense of an ncreased average response tme) s more desrable for contnuous meda data. Ths fundamental dfference n the optmzaton crteron has a sgnfcant mpact on the selecton of strpe unt sze. To llustrate, consder Fgure 1(a), whch depcts the hstogram of the response tme observed for two dfferent strpe unt szes (obtaned usng a workload of 6 vdeo clents accessng an array of 16 dsks). It shows that strpe unt szes of 32KB and 64KB yeld average response tmes of 3ms and 32ms, respectvely. The fgure also shows that the hstogram for the 32KB strpe unt sze has a longer tal. If data accesses do not mpose any real-tme constrants, 32KB would be chosen as the approprate strpe unt sze. For accesses wth real-tme constrants, a strpe unt sze of 64KB would be more desrable. As shown n Fgure 1(b), the block sze that mnmzes the average response tme contnues to dffers from one that mnmzes the 99 th percentle of the response tme (.e., the tal of the hstogram) across a wde range of clent workloads. æ Perodc and sequental nature of contnuous meda: In general, textual and numerc data accesses consst of aperodc reads and wrtes, whle contnuous meda workloads consst of reads and wrtes that are perodc and sequental. Moreover, contnuous meda applcatons have a sgnfcantly larger data rate requrement as compared to textual applcatons. These dfferences n workload characterstcs affect the optmal strpe unt sze. Due to the perodc and sequental nature of contnuous meda, most multmeda fle servers employ a serverpush archtecture to servce contnuous meda requests. Such servers servce clents by perodcally accessng and transmttng data wthout an explct request for each access (n contrast to a clent-pull archtecture employed by conventonal fle servers that access data only n response to explct clent requests). The workload seen by dsks n a server-push archtecture s markedly dfferent from those seen by dsks n a clent-pull archtecture. Due to these dfferences, technques developed for conventonal clent-pull based servers are napplcable to server-push based servers. Due to these dfferences, novel technques that optmze the performance of a multmeda fle server for contnuous meda data must be developed. 1.2 Research Contrbutons of Ths Paper In ths paper, we propose technques for determnng the strpe unt sze and the degree of strpng for fle servers storng varable bt rate contnuous meda data. We consder a fle server that servces clents by proceedng n terms of perodc rounds and argue that, n such envronments, a strpe unt sze that mnmzes the servce tme (.e., the total tme spent n retrevng the data requested n a round) of the most heavly loaded dsk s optmal. To determne the optmal strpe unt sze, we develop an analytcal model that uses the server confguraton and a dstrbuton of 2

3 Probablty (a) 16 dsks, 6 clents strpe unt sze = 32kB strpe unt sze = 64kB Optmal strpe unt sze (kb) (b) 16 dsks Average response tme 99th percentle of the response tme Response tme (ms) Number of clents Fgure 1: Effect of dfferent metrcs on the strpe unt sze. the number of blocks accessed by a clent n a round to predct the servce tme of the most heavly loaded dsk n both redundant and non-redundant arrays. By determnng the servce tme of the most heavly loaded dsk across a range of block szes, a strpe unt sze that mnmzes the servce tme can be chosen. We valdate the accuracy of our model through extensve trace-drven smulatons. We demonstrate that, contrary to conventonal wsdom, a large strpe unt sze does not necessarly yeld good server performance. Instead, such a strpe unt sze can adversely affect the qualty of servce guarantees provded to clents, thereby reducng the number of clents supported by the server. We also use the model to: (1) evaluate the effect of varous system parameters (such as the number of clents, number of dsks, etc.) on the strpe unt sze, and (2) derve technques for selectng an optmal strpe unt sze for varous desgn scenaros. We then use the model to determne the optmal degree of strpng for varable bt rate meda streams. We demonstrate that strpng a meda stream across the entre array causes the number of clents supported to ncrease sub-lnearly wth ncrease n number of dsks. To maxmze the number of clents supported n large arrays, we propose a technque that parttons a dsk array and strpes each meda stream across a sngle partton. Snce load mbalances can occur n such parttoned arrays, we present a model to compute the mbalance across parttons. We then use the model to determne a partton sze that mnmzes the load mbalance, and hence, maxmzes the number of clents supported by the array. The rest of ths paper s organzed as follows. In Secton 2, we address the ssue of determnng an optmal strpe unt sze. Secton 3 descrbes technques for determnng the degree of strpng. Secton 4 descrbes related work, and fnally, Secton 5 summarzes our results. 2 Determnng the Strpe Unt Sze Consder a multmeda server that nterleaves meda streams across dsks by storng successve blocks of a stream on consecutve dsks n a round-robn manner. The unt of nterleavng, referred to as a meda block or a strpe unt, denotes the maxmum amount of logcally contguous data stored on a sngle dsk. 1 Due to the perodc nature of meda playback, the server servces multple clents by proceedng n perodc rounds. Durng each round, the server retreves a fxed number of meda unts (e.g., vdeo frames or audo samples) for each clent. To ensure contnuous playback, the number of meda unts accessed for a clent must be suffcent to sustan ts playback rate, and the servce tme (.e., the total tme spent n retrevng meda unts durng a round) must not exceed the duraton of a round. If each meda stream s compressed usng a varable bt rate (VBR) compresson algorthm, then the szes of successve meda unts wthn a stream wll vary. Although each clent accesses a fxed number of meda unts n each round, due to varable meda unt szes, the number of blocks requested by the clent can vary from one round 1 We shall use the terms meda block and strpe unt nterchangeably n ths paper. 3

4 to another. The server can servce such clents ether by retrevng a varable number of blocks across rounds, or by retrevng a fxed number of blocks across rounds and employng prefetchng and bufferng schemes to smooth out the varatons. Dependng on the amount of varaton n the stream bt rate, the latter approach can substantally ncrease the ntaton latency (snce suffcent amount of data must be prefetched before the clent can ntate playback). Our experments wth MPEG-1 vdeo clents ndcate that, for a round duraton of 1s, accessng data at the average bt rate can cause the ntaton latency to be more than 2s. 2 In contrast, snce no smoothng s performed when a varable number of blocks are accessed, the clents can ntate playback wthout any delay. However, accessng a varable number of blocks can cause load mbalances across dsks n the array and can reduce the number of clents supported by the server. A key challenge s to devse strpng technques that reduce the load mbalance so as to maxmze the number of clents supported. In ths paper, we assume that the server servces clents by accessng a varable number of blocks across rounds, and determne the strpe unt sze and the degree of strpng that acheves ths objectve. In servers that access a varable number of blocks, the set of dsks accessed by dfferent clents durng a round are dfferent, and hence, the total number of blocks accessed can vary from one dsk to another. Snce some dsks are more heavly loaded than others, the servce tme of some of these dsks may occasonally exceed the round duraton, causng playback dscontnutes at clent stes. To mnmze the frequency of such playback dscontnutes, the server must mnmze the servce tme of the most heavly loaded dsk n the array. The servce tme of the most heavly loaded dsk depends on the meda block sze. To observe ths, consder a small meda block sze. Such a block sze ncreases the number of blocks accessed from the array durng a round, thereby dstrbutng the load across dsks and reducng the load mbalance. However, t also ncreases the overhead due to seek and rotatonal latency, thereby ncreasng the servce tme of the most heavly loaded dsk. In contrast, a large block sze reduces the overhead of seek and rotatonal latency, but ncreases the load mbalance, and hence, the servce tme of the most heavly loaded dsk. The server must select a meda block sze that balances these tradeoffs and mnmzes the servce tme of the most heavly loaded dsk n the array. In what follows, we present an analytcal model that uses the characterstcs of the workload and the confguraton of the server to predct the servce tme of the most heavly loaded dsk n non-redundant and redundant dsk arrays. By computng the servce tme of the most heavly loaded dsk across a range of block szes, a meda block sze that mnmzes the servce tme can be chosen. 2.1 Analytcal Models for Determnng the Load on the Array A Model for Non-redundant Arrays Consder a multmeda server that nterleaves meda streams across a dsk array. Gven the confguraton of the server (e.g., number of dsks, ther physcal characterstcs, the round duraton, etc.) and the clent characterstcs (e.g., number of clents, trace of the meda unt szes for each clent, playback rate, etc.), the servce tme of the most heavly loaded dsk n redundant and non-redundant dsk arrays can be computed as follows: 1. Compute the dstrbuton of the number of blocks accessed from a dsk by each clent durng a round usng a trace of meda unt szes. 2. Compute the dstrbuton of the total number of blocks accessed from a dsk by summng the number of blocks requested by each clent from that dsk. 3. Compute the dstrbuton of the number of blocks accessed from the most heavly loaded dsk. 2 The latency s smaller f the clent retreves the fle as a pecewse constant bt rate (CBR) stream [18], rather than a pure CBR stream. However, pecewse CBR retrevals result n a varable load on dsks, snce bt-rate of clents changes over tme and dfferent clents retreve data at dfferent rates at any nstant. Our models are vald for VBR as well as smoothed pecewse CBR retrevals. 4

5 Frst of the requested blocks Frst of the requested blocks R R R R R R R R R R R R R R R R R Dsks j D 2 D 1 D Dsks j D 2 D 1 D Case 1 : Clent requests 5 blocks Case 2 : Clent requests (D+2) blocks R = requested block Fgure 2: Dfferent scenaros n whch clent accesses a block from dsk j. 4. Gven the dstrbuton of the number of blocks accessed from the most heavly loaded dsk, compute the servce tme dstrbuton for the dsk usng a dsk model. To derve the model for non-redundant arrays, consder a server that nterleaves meda streams across an array of D dsks. Let n clents access the server, each retrevng a meda stream, 3 and let B denote the meda block sze. Snce the server accesses a fxed number of meda unts for each clent durng a round, the dstrbuton of the number of blocks accessed by the clent durng a round can be determned from a trace of the meda unt szes. Let b k, obtaned from ths dstrbuton, denote the probablty that clent accesses k blocks from the array n a round, and let p k j denote the probablty that clent accesses k blocks from dsk j n a round. To compute p1 j, observe that clent wll access exactly one block from dsk j n a round f: (1) t requests m blocks (1 ç m ç D) from the array and the frst of these blocks s stored ether on dsk j or any of the prevous m, 1 dsks; or (2) t requests D + m blocks (1 ç méd) from the array and the frst of these block s stored any dsk other than dsk j or any of the prevous m, 1 dsks. Fgure 2 llustrates these cases. Due to the VBR nature of meda streams, the number of blocks accessed by a clent vares from one round to another. Hence, after a small number of rounds, the frst block s equally lkely to be accessed from any of the dsks n the array. Consequently, p 1 j = D X b m æ m D,1 X D + b D+m æ D, m D Generalzng, clent wll access k blocks (k = 1; 2; 3:::) from dsk j f: (1) t requests èk, 1è æ D + m blocks (1 ç m ç D) from the array and the frst of these blocks s stored on dsk j or any of the prevous m, 1 dsks; or (2) t requests k æ D + m blocks (1 ç méd) from the array and the frst of these blocks s stored on any dsk other than dsk j or any of the prevous m, 1 dsks. Hence, p k j = D X b èk,1èæd+m X æ m D + D,1 b kæd+m æ D, m D Lastly, the probablty that clent does not access dsk j s p j =1, P 1 k=1 p k j. Let X j be a random varable denotng the number of blocks accessed by clent from dsk j durng a round. Then, P èx j = kè =p k j (3) Then, the total number of blocks accessed from dsk j durng a round, N j, can be computed as N j = nx =1 (1) (2) X j (4) 3 Snce contnuous meda requests are domnated by read requests, we confne our focus to read requests. 5

6 Due to the VBR nature of vdeo streams, the number of blocks accessed by clents from the array are ndependent of each other. Thus, X 1j ;X 2j ; :::;X nj are ndependent random varables, and hence, the dstrbuton of N j can be obtaned by applyng the the z-transform 4 to (4). That s, where ZèN j è= ny =1 ZèX j è (5) ZèX j è=p j + zp1 j + z2 p 2 j + z3 p 3 j +æææ (6) Then, the number of blocks accessed from the most heavly loaded dsk 5 s gven by N max = maxèn 1 ;N 2 ;æææ;n D è (7) Due to the round robn nature of meda stream placement, the number of blocks accessed from a dsk s not ndependent of the load on ts neghborng dsks. Snce the precse dependence of these random varables on each other s dffcult to characterze, and snce the maxmum of D dependent random varables s dffcult to compute, as an approxmaton we assume that N j s are ndependent of each other. Later n ths secton, we demonstrate that ths approxmaton does not cause any naccuraces n the predctons of the model. Then, the dstrbuton of N max can be computed as F Nmax èxè =F N1 èxè æ F N2 èxèæææf ND èxè (8) where F Nj s the cumulatve probablty dstrbuton functon of the random varable N j [16]. Havng determned the dstrbuton of the number of blocks accessed from the most heavly loaded dsk, the servce tme of the dsk can then be computed by usng a dsk model. We use one such model that has been proposed n the lterature [14, 22] (see Appendx A for the complete dsk model). The servce tme to access N max blocks of sze B as predcted by the dsk model s: ç max = N max æ èt s + t r è+n max æ B æ t t (9) where t s and t r denote the seek tme and rotatonal latency ncurred whle accessng a block from dsk and t t denotes the transfer tme for a unt amount of data. Thus, gven the server confguraton and the workload characterstcs, the model computes the servce tme dstrbuton of the most heavly loaded dsk for a partcular block sze. Moreover, the model also yelds the dstrbuton of the number of blocks accessed from a dsk wth average load (.e., N j ). The servce tme of such a dsk can then be computed usng the dsk model A Model for Redundant Arrays Snce dsk arrays are hghly susceptble to dsk falures, multmeda servers employ redundances n data storage to guarantee hgh avalablty of data. Most redundant arrays are based on the Redundant Array of Independent Dsks (RAID) archtecture [6, 17]. RAID arrays compute redundant blocks (referred to as party) by takng an exclusve-or operaton over data blocks stored on G, 1 dsks, where G é 2, and store t on another dsk. The party block together wth all the data blocks over whch party s computed s referred to as a party group. In the presence of a dsk falure (also referred to as the degraded mode), the server reconstructs a block stored on the faled dsk by 4 The z-transform of a random varable U s the polynomal ZèUè =a + za 1 + z 2 a 2 + æææwhere the coeffcent of the th term n the polynomal represents the probablty that the random varable equals. Thats,PèU = è=a.ifu 1; U 2; :::; U n are n ndependent random varables, and Y = P n ZèUè. The dstrbuton of Y can then be computed usng a polynomal multplcaton of U, thenzèyè=q n =1 =1 the z-transforms of U 1; U 2; æææ; U n [16]. 5 Note that the dsk that s most heavly loaded wll vary from one round to another. Regardless of whch dsk s the most heavly loaded n a partcular round, N max represents ts load. 6

7 accessng the party block and data blocks of the party group stored on survvng dsks. A commonly used RAID archtecture s RAID-5 whch uses block-nterleaved party and unformly dstrbutes party blocks across dsks n the array. The multple RAID-5 archtecture s an extenson of the RAID-5 array n whch the array s parttoned nto clusters of dsks, wth each cluster ndependently computng party nformaton [6]. In the rest of ths secton, we assume a multple RAID-5 archtecture for our model. However, the basc approach used n our model s applcable to other RAID archtectures as well. Consder a multmeda server servcng n clents from a RAID-5 array consstng of D dsks. Let G denote the party group sze, where G ç D. Then the array contans P = D=G clusters. Let us assume that the server computes party blocks over a sequence of successve blocks from the same meda stream (.e., all data blocks of a party group are consecutve blocks of the same meda stream). Consequently, the server stores successve blocks of a meda stream on dsks storng data blocks of the party group and skps over dsks storng the party blocks. Snce each of the P clusters contans a dsk storng a party block, a request for more than D, P consecutve blocks causes a dsk to be reaccessed. Fault-free Case To compute the servce tme of the most heavly loaded dsk n the fault-free mode, let b k denote the probablty that clent accesses k blocks from the array durng a round, and let p k j denote the probablty that clent accesses k blocks from dsk j durng a round. To compute p 1 j, note that clent wll access dsk j only f dsk j stores a data block (.e., does not store a party block). Moreover, clent wll access a block from dsk j f: (1) t requests m blocks (1 ç m ç D, P ) from the array and the frst of these blocks s stored on dsk j or any of the prevous m,1 dsks storng data blocks; or (2) t requests D, P + m blocks (1 ç méd, P ) from the array and the frst of these blocks s stored on any dsk storng data blocks other than dsk j or any of the prevous m, 1 dsks. Snce party blocks are unformly dstrbuted across dsks, one out of every G blocks stored on a dsk s a party block. Hence, the probablty that dsk j stores a data block s è1, 1=Gè. Due to the VBR nature of meda streams, the frst block s equally lkely to be accessed from any of the D, P dsks storng data blocks. Hence, we get p =è1, 1 1 æ è D,P X j G è X D,P,1 b m m æ D, P + èd,p è+m b æ D, P, m D, P! (1) Generalzng, the probablty that clent accesses k blocks from dsk j s p k =è1, 1 æ è D,P X j G è X D,P,1 èk,1èæèd,p è+m m b æ D, P + kæèd,p è+m b æ D, P,! m D, P èk =1; 2; 3;:::è (11) Snce P = D G,, è1 1 G get D,P X p k j = D,P è can be rewrtten as D. Substtutng ths value n the above equaton and smplfyng, we èk,1èæèd,p è+m b X æ m D + D,P,1 kæèd,p è+m b æ D, P, m D èk =1; 2; 3;:::è (12) Let X j be the random varable representng the number of blocks accessed by clent from dsk j durng a round. Then P èx j = kè = p k j. Usng ths dstrbuton of X j, the dstrbutons of the number of blocks accessed and the servce tme of the most heavly loaded dsk n the fault-free state can be derved usng the method presented n Secton Falure Case To compute the servce tme of the most heavly loaded dsk n degraded mode, assume that dsk f n the array experences a falure, where 1 ç f ç D. Snce each cluster ndependently computes party, dsks that do not belong 7

8 Frst block s between dsks j and f R R R R R Dsks f j D 2 D 1 D Case 1 : Clent accesses dsk j but not dsk f Dsks R = requested block A = addtonal block A R R R R A A A A A f j D 2 D 1 D Case 2: Clent accesses dsk f but not dsk j (an addtonal block s accessed from dsk j) A A R R R R R A A A Dsks f j D 2 D 1 D Case 3 : Clent accesses both dsks j and f Fgure 3: Dfferent scenaros n whch clent accesses a block from dsk j n degraded mode. to the cluster contanng dsk f are unaffected by ths falure, and hence, for these dsks, the number of blocks accessed n a round s the same as that n the fault-free state. All dsks belongng to the cluster contanng dsk f, however, wll experence an ncrease n load whenever a clent accesses a block from dsk f. To compute the number of blocks accessed by clent from dsk j belongng to the cluster contanng dsk f,let æ denote the number of dsks storng data blocks contaned between dsks j andf (ncludng dsk j), and let æ denote the number of dsks storng data blocks not contaned between dsks j and f. Observe that, f no party block s stored on a dsk between dsks j and f,thenæ =j j, f j. Otherwseæ =j j, f j,1. In ether case, æ=d, P, æ. To compute p 1 j, note that clent wll access exactly one block from dsk j f t requests m blocks from the array and one of the followng three condtons hold: (1) a block s requested from dsk j but not from dsk f, or(2)a block s requested from dsk f but not from dsk j (and hence, a block must be accessed from dsk j to reconstruct the block on dsk f), or (3) a block s requested from both dsks j and f and both blocks belong to the same party group (and hence, no addtonal block needs to be accessed from dsk j). Fgure 3 llustrates these cases for an array wth G = D. To compute the probablty that clent accesses dsk j but not dsk f, let us frst consder the case when f éj. Clent wll access dsk j only f dsk j stores a data block of the party group. Moreover, clent wll access a block from dsk j but not dsk f f: (1) t requests m blocks (1 ç m ç æ) from the array and the frst of these blocks s stored on dsk j or any of the prevous m,1 dsks; or (2) t requests æ + m blocks (1 ç m ç æ,æ) from the array and the frst of these blocks s stored on dsk j or any of the prevous æ, 1 dsks; or (3) t requests æ+m blocks (1 ç m ç æ, 1) from the array and the frst of these blocks s stored on dsk j or any of the prevous æ, m, 1 dsks. A smlar argument holds for the case when f éj, except that we must consder the last block accessed by the clent nstead of the frst block. Snce the frst (last) block s equally lkely to be stored on any of the D, P dsks storng data blocks, and snce the probablty that dsk j stores a data block s è1, 1 G è,weget Substtutng D,P D p =è1, 1 G è æ è æx X æ,æ b m m æ D, P + b æ+m æ æ D, P + for, è1 1 è n the above equaton and smplfyng, we get G p = æx b m æ m æ,æ X D + b æ+m æ æ D + æx b æ+m æx b æ+m æ æ, m D æ æ, m D, P è (13) By symmetry, the probablty that clent accesses dsk f but not dsk j s the same as the probablty that t accesses dsk j but not dsk f. To compute the probablty that clent accesses a block from both dsks j and f, observe that the clent must request at least æ blocks from the array (see Fgure 3). Moreover, to be able to access dsk j and f both dsks j and f must store data blocks. Hence, the clent accesses blocks belongng to the same party group from dsks j and f f (1) t requests èm + æè blocks from the array, ( ç m ç æ) and the frst of these blocks s stored on a dsk not 8 (14)

9 contaned between dsks j and f; or (2) t accesses èd, P + mè blocks and the frst of these blocks s stored on a dsk such that only one block s accessed from dsks j and f. Snce two out of every G party groups wll store a party block on dsks j or f, the probablty that nether dsk j nor dsk f stores a party block s è1, 2 è. Hence, G the probablty of accessng blocks belongng to the same party group from dsks f and j s p =è1, 2 G è æ è æ X m= b m+æ æ m +1 D, P + æx D,P +m b æ, æ m +1 D, P! (15) Hence, summng the probablty of the three cases, we get p 1 j èæ; æè = 2 æ p + p,thats, p 1 j èæ; æè = 2 æ è æx b m è1, 2 G è æ è æ X m= æ m æ,æ X D + b æ+m æ æ D + æx b m+æ æ m +1 D, P + æx b æ+m æ æ, m D! + D,P +m b æ, æ m +1 D, P! (16) The value of p 1 j computed n the above equaton s a functon of parameters æ and æ. Dependng on whether or not a party block s stored on a dsk between dsks j and f, we have two cases. If a party block s stored on a dsk between dsks j and f, thenwegetæ 1 =j j, f j,1 and æ 1 = D, P, æ 1. Snce party blocks are unformly dstrbuted across dsks n the array, and the probablty that of ths case s s æ 1 G. If no party block s stored between dsks j and f,thenwegetæ 2 =j j, f j and æ 2 = D, P, æ 2, and the probablty of ths case s è1, æ 1 G è. Hence, the overall probablty that clent accesses one block from dsk j s p 1 j = æ 1 G æ p1 j èæ 1; æ 1 è+è1, æ 1 G è æ p1 j èæ 2; æ 2 è (17) Generalzng, the probablty that clent accesses k blocks from dsk j s where p k j èæ; æè = 2 æ è æx p k j = æ 1 G æ pk j èæ 1; æ 1 è+è1, æ 1 G è æ pk j èæ 2; æ 2 è (18) b m+æ è1, 2 G è æ è æ X m= æ m æ,æ X D + b æ+m+æ æ æ D + æx b m+æ+æ æ m +1 D, P + æx b æ+m+æ æ æ,! m + D D,P +m+æ b æ, æ m +1 D, P! (19) and æ =èk, 1è æ èd, P è. LetX j be the random varable representng the number of blocks accessed by clent from dsk j durng a round. Then P èx j = kè =p k j. Then, usng ths dstrbuton of X j, the dstrbuton of the number of blocks accessed and the servce tme of the most heavly loaded dsk n the degraded mode can be derved n a manner smlar to that n Secton Valdaton of the Models To valdate our models, we have bult an event-based, trace-drven dsk array smulator called DskSm. 6 We dgtzed a number of traces and used these traces to run smulatons over a wde range of system parameters (e.g., dfferent number of clents, dfferent number of dsks, dfferent round duratons, etc.). The characterstcs of the traces are 6 The source code for DskSm s publcly avalable from 9

10 Servce tme (msec) (a) RAID-, 16 dsks, 6 clents, 3 frames/s Average loaded dsk (smulator) Average loaded dsk (model) Servce tme (msec) (b) RAID-, 16 dsks, 6 clents, 3 frames/s Most heavly loaded dsk (model) Most heavly loaded dsk (smulator) Block sze (kb) Block sze (kb) Fgure 4: Varaton n the servce tme of the average loaded dsk and the most heavly loaded dsk. Servce Tme (msec) RAID-, 16 dsks, 6 clents, 3 frames/s 7th percentle (model) 7th percentle (smulator) 8th percentle (model) 8th percentle (smulator) 95th percentle (model) 95th percentle (smulator) Block sze (kb) Fgure 5: Valdaton of the model for varous percentles of the servce tme. lsted n Table 1. For each combnaton of parameters, we conducted multple smulaton runs and computed the 95% confdence ntervals of the expected number of blocks accessed and the expected servce tme of the most heavly loaded dsk. To valdate the model for non-redundant arrays, we computed the expected number of blocks accessed and the expected servce tme of the most heavly loaded dsk for each workload. The values predcted by the model were found to be wthn the 95% confdence ntervals obtaned from smulatons. Fgures 4(a) and (b) plot these values for one such workload. Smlar results were obtaned for varous percentles of the servce tme of the most heavly loaded dsk (see Fgure 5). The model for redundant arrays was valdated smlarly [19]. Thus, the smulaton results valdate the predctons made by our analytcal models over a large parameter space. The servce tme graphs of the average loaded dsk and the most heavly loaded dsk n Fgure 4 lead us to the followng observatons: æ As shown n Fgure 4(a), the servce tme of the average loaded dsk decreases monotoncally wth ncreasng block sze. Ths s because ncreasng the block sze decreases the number of blocks accessed from the dsk, thereby reducng dsk seek and rotatonal latency overheads. æ The servce tme of the most heavly loaded dsk, on the other hand, decreases ntally and then starts ncreasng wth ncrease n block sze (see Fgure 4(b)). To explan ths behavor, let us frst ntroduce some termnology. Let cn max and bç max, respectvely, denote the expected number of blocks accessed from the most heavly loaded dsk and the expected servce tme of the most heavly loaded dsk durng a round, and let bç avg denote the expected servce tme of the average loaded dsk. Then, the mbalance n the servce tmes of the most heavly loaded dsk and the average loaded dsk I s (referred to as the load mbalance) s defned as I s = bç max, bç avg bç max 1

11 Table 1: Characterstcs of Vdeo Traces MPEG Encodng Length Frame Bt rate Fle Pattern (frames) rate Mb/s Fraser IèBBPè 3 BB Newscast IèBBPè 3 BB Flntstones IèBBPè 3 BB = 1, bç avg bç max (2) From (9), the porton of the servce tme spent n dsk seek and rotatonal latency s cn max æ èt s + t r è = bç max, cn max æ B æ t t. Hence, the overhead due to seek and rotatonal latency O can be defned as: O = bç max, cn max æ B æ t t bç max = 1, c N max æ B æ t t bç max (21) Assumng a fxed server confguraton and workload characterstcs, ncreasng the block sze decreases the number of blocks accessed from the array. The smaller the number of blocks beng accessed, the smaller s the probablty of achevng equtable dstrbuton of load across dsks (snce the array becomes sparsely loaded). Hence, ncreasng block sze yelds an ncrease n the load mbalance I s. On the other hand, ncreasng the block sze causes the seek and rotatonal latency overhead to decrease. Fgure 6 shows these varatons n I s and O. For each meda block sze, the servce tme of the most heavly loaded dsk s governed by the relatve values of I s and O. As shown n Fgure 6, at small block szes, the latency overhead domnates, and hence the servce tme decreases wth ncrease n block sze. At large block szes, the load mbalance domnates the latency overhead, and causes the servce tme to ncrease wth ncrease n block sze. Consequently, the servce tme of the most heavly loaded dsk decreases ntally and then starts ncreasng wth ncrease n block sze. From the above analyss, we conclude that mnmzng the servce tme of the average loaded dsk requres the server to choose a block sze that s as large as possble. In contrast, mnmzng the servce tme of the most heavly loaded dsk requres the server to choose a block sze that mnmzes the combned effects of I s and O. To maxmze the number of clents supported for best-effort workloads, the server must mnmze the servce tme of the average loaded dsk, whle for contnuous meda workloads, mnmzng the servce tme of the most heavly loaded dsk s more desrable. Hence, the optmal block sze obtaned for the two envronments can dffer sgnfcantly. The precse value of the optmal block sze for a contnuous meda workload depends on the qualty of servce requrements of clents and the values of varous system parameters (such as the number of clents, ther playback rate, the number of dsks, etc.). In what follows, we examne the effect of these factors on the optmal block sze. For each parameter, we also compute the range of block szes that yelds a servce tme wthn x% of the mnmum. The upper and lower bounds of ths set of block szes defne the x% optmal envelope for the workload [3, 22]. By choosng a block sze that s contaned wthn the x% optmal envelope of all values of the parameter, the server can ensure performance that s wthn x% of the optmal regardless of the workload. 11

12 Normalzed metrc RAID-, 16 dsks, 6 clents Load mbalance Latency overhead Block sze (kb) Fgure 6: Varaton n the load mbalance and the latency overhead. 2.3 Factors Affectng the Optmal Block Sze Effect of Qualty of Servce Observe that, the model yelds a dstrbuton of the servce tme of the most heavly loaded dsk n the array. To determne the optmal block sze, the server must frst choose a partcular percentle of the servce tme as the metrc and then compute the block sze that mnmzes that percentle. The choce of a partcular percentle depends on the QoS requrements of clents (where QoS s defned to be the fracton of request deadlnes that can be volated). For nstance, the server can choose the expected value of the servce tme (whch, n our experments, approxmately corresponds to the 7 th percentle of the servce tme dstrbuton) to determne the block sze. In such a scenaro, there s a 3% chance that the actual value of the servce tme durng a round wll exceed ts expected value, resultng n a large number of request deadlne volatons. If clents have strngent qualty of servce (QoS) requrements (.e., they can tolerate only rare volatons of request deadlnes), then the server must choose hgher percentles of the servce tme to provde the desred performance guarantees. For example, by choosng the 95 th percentle of servce tme dstrbuton of the most heavly loaded dsk, the server can ensure that the servce tme does not exceed ts estmated value n more than 5% of the rounds. Snce dfferent percentles of the servce tme yeld dfferent optmal block szes (see Fgure 7(a)), the server must carefully choose an approprate percentle of the servce tme as the metrc based on the QoS requrements of clents. Fgure 7(b) shows the varaton n optmal block sze and the 5% optmal envelope for dfferent percentles of the servce tme. Larger percentles of the servce tme correspond to more strngent QoS requrements. To provde strngent QoS, the server must mnmze the varaton n servce tmes of the most heavly loaded dsk across rounds. Ths can be acheved by selectng a block sze whch reduces the load mbalance. Snce the load mbalance decreases wth decrease n the block sze (Fgure 6), a small block sze yelds better performance for more strngent QoS requrements. Hence, the optmal block sze and the 5% optmal envelope decrease wth ncrease n percentle of the servce tme. Observe from Fgure 7(a) that, the servce tme of the most heavly loaded dsk ncreases slowly for block szes larger than the optmal block sze. Ths mght lead us to beleve that choosng a block sze that s larger than the optmal wll yeld near optmal performance, whle reducng dsk latency overheads. However, Fgure 7(b) demonstrates that choosng the largest possble block sze contaned n the optmal envelope for a partcular QoS degrades performance for more strngent QoS. For nstance, choosng the upper 5% optmal envelope of the 7 th percentle (.e., 256KB) as the block sze wll cause a loss n performance for the 95 th percentle (snce 256KB s not contaned n the 5% optmal envelope of the 95 th percentle). Ths argument also shows that ad-hoc technques that select a large block sze (e.g., selectng the track sze as the block sze) can sgnfcantly affect the server performance, and hence, the number of clents supported. To acheve good performance over a range of QoS requrements, a block sze that s contaned wthn the x% optmal envelope of a wde range of percentles must be chosen. 12

13 Servce Tme (ms) (a) RAID-, 16 dsks, 6 clents 7th percentle 8th percentle 95th percentle Optmal block sze (kb) (b) RAID-, 16 dsks, 6 clents Upper 5% envelope Lower 5% envelope Optmal block sze Optmal block sze (kb) Percentle of servce tme Fgure 7: Effect of Qualty of Servce Normalzed metrc (a) RAID-, 16 dsks Load mbalance, 2 clents Latency overhead, 2 clents Load mbalance, 1 clents Latency overhead, 1 clents Optmal block sze (kb) (b) RAID-, 16 dsks Upper 5% optmal envelope Lower 5% optmal envelope Optmal block sze Block sze (kb) Number of clents Fgure 8: Effect of number of clents on the optmal block sze Effect of system parameters The model can also be used to study the effect of varous system parameters on the optmal block sze. Snce the servce tme of the most heavly loaded dsk s mnmzed when the combned effects of I s and O are mnmzed, the effect of varyng a system parameter on the optmal block sze can be analyzed by studyng ts effect on I s and O. We can ntutvely understand the effect of a parameter on the optmal block sze by assumng that the pont of ntersecton of I s and O governs the mnma of the servce tme curve. Then, f a change n the value of the system parameter ncreases the number of blocks accessed from the array, t ncreases the probablty of achevng equtable load dstrbuton across dsks, and hence, reduces I s. Such a reducton causes the I s curve to shft downward. Ths shfts the pont of ntersecton of I s and O (and hence, the mnma of the servce tme curve) to the rght, thereby ncreasng the optmal block sze. On the other hand, f a change n the value of the parameter causes a decrease n the number of blocks per dsk, then the load mbalance ncreases. Such an ncrease causes the pont of ntersecton of the I s and O curves to shft to the left, thereby reducng the optmal block sze. To llustrate, consder the effect of varaton n the number of clents on the optmal block sze. For a fxed server confguraton, ncrease n the number of clents ncreases the number of blocks accessed from the dsk array, and thereby ncreases the probablty of achevng equtable dstrbuton of load across dsks. Ths reduces the load mbalance I s, causng the I s curve to shft downwards. In contrast, the latency overhead curve, whch s governed mostly by the physcal characterstcs of dsks, shfts only margnally. Ths shfts the pont of ntersecton of I s and O curves to the rght (see Fgure 8(a)). Hence, the optmal meda block sze ncreases wth ncrease n the number of clents accessng the server (see Fgure 8(b)). The 5% optmal envelope also ncreases wth ncrease n number of clents for smlar reasons. We have determned the effect of varous system parameters, such as the number of dsks, ther physcal characterstcs, the playback rate of clents, the round duraton, etc., on the optmal block sze. The effect of all of these 13

14 Table 2: Effect of varous parameters on the block sze Parameter Number of clents Playback rate Qualty of Servce (QoS) Number of dsks Round duraton Dsk zones Party Group Sze Effect of ncrease n parameter on optmal block sze Block sze ncreases Block sze ncreases Block sze decreases Block sze decreases Block sze ncreases Block sze ncreases from nner zones to outer zones Block sze ncreases 3 25 (a) RAID-, 6 clents Upper 5% envelope Lower 5% envelope Optmal block sze Optmal block sze (kb) Number of dsks Fgure 9: Effect of the number of dsks on the optmal block sze. parameters on the optmal block sze can be explaned usng arguments smlar to those presented above. In what follows, we dscuss our results n detal (Table 2 summarzes these results). Number of Dsks For a fxed number of clents, ncreasng the number of dsks n the system decreases the number of blocks accessed per dsk. Ths decreases the probablty of achevng equtable dstrbuton of load across dsks, and hence, ncreases the load mbalance I s. An ncrease n I s causes the I s curve to shft upwards and the pont of ntersecton of I s and O to shft to the left. Thus, the optmal block sze decreases wth an ncrease n the number of dsks (see Fgure 9). Playback Rate and Round Duraton Assumng a fxed round duraton (playback rate), ncreasng the playback rate (round duraton) causes a clent to request a proportonately larger amount of data per round to sustan contnuous playback. Ths causes a larger number of blocks to be accessed from the array, thereby spreadng the load across dsks and reducng the load mbalance. Consequently, the optmal block sze and the 5% optmal envelope ncrease wth ncrease n playback rate (round duraton). (see Fgures 1(a) and 1(b)). Dsk Characterstcs To evaluate the effect of varyng dsk characterstcs on the optmal block sze, we frst defne the work coeffcent of adsk[3]: 14

15 35 3 (a) RAID-, 16 dsks, 6 clents 35 3 (b) RAID-, 16 dsks, 6 clents Upper 5% envelope Lower 5% envelope Optmal block sze Optmal block sze (kb) Upper 5% envelope Lower 5% envelope Optmal Block Sze Optmal block sze (kb) Playback rate (frames/s) Round duraton (s) Fgure 1: Effect of the playback rate of clents and the round duraton on the optmal block sze. Table 3: Characterstcs of varous Seagate Dsks Model Abbrev- Capacty Average Avg Rotatonal Transfer Transfer Work aton MB seek (ms) latency (ms) rate (MB/s) tme (ms/kb) Coeffcent Medalst M x 1,3 Hawk H x 1,3 Barracuda1 B x 1,3 Barracuda2 B x 1,3 Elte9 E x 1,3 Defnton 1 The work coeffcent of a dsk s defned as W = tme to transfer unt amount of data average seek + average rotatonal latency The work coeffcent measures the relatve varaton n the latency overheads and transfer tmes of dsks. Table 3 shows the characterstcs of varous Seagate dsks and ther work coeffcents. Recall from (9) that bç max = cn max æ èt s + t r è+cn max æ B æ t t Hence, from the defnton of O,weget: O =1, c N max æ B æ t t bç max = èt s + t r è èt s + t r è+b æ t t = 1 1+B æ t t èt s+t rè = 1 1+B æ W Hence, for a partcular block sze, ncreasng W decreases O. Ths causes the pont of ntersecton of the I s and O curves to shft to the left. Ths ndcates that the optmal block sze vares nversely wth the work coeffcent. Fgure 11(a) and Table 3 demonstrate ths behavor for dfferent Seagate dsks. Zoned Dsks Our experments thus far assumed a sngle transfer rate for the entre dsk. However, modern dsks are parttoned nto zones, wth outer zones havng hgher recordng denstes and larger data transfer rates as compared to nner zones. Due to larger transfer rates (and hence, smaller transfer tmes), outer zones have a smaller work coeffcent. Consequently, the optmal block sze and the 5% optmal envelope for a zone ncreases as we proceed from nner zones to outer zones (see Fgure 11(b)). 15 (22)

16 25 2 (a) RAID-, 16 dsks, 6 clents Upper 5% envelope Lower 5% envelope Optmal block sze 3 25 (b) RAID-, 16 dsks, 6 clents Upper 5% envelope Lower 5% envelope Optmal block curve Optmal block sze (kb) Optmal block sze (kb) E9 M H B1 B2 Dsk Model Track transfer rate (MB/s) Fgure 11: Varaton n the optmal block sze wth dsk characterstcs. Fgure (a) compares the optmal block sze for dfferent Seagate dsks. Dsks used n the experment are Elte9, Medalst, Hawk, Barracuda1, and Barracuda2. Fgure (b) shows the varaton n the optmal block sze for dfferent transfer rates. Lower transfer rates represent nner zones. Servce tme (msec) (a) 32 dsks, 5 clents, normal mode 4 38 RAID-5 party group=4 RAID-5 party group=8 36 RAID-5 party group=16 34 RAID-5 party group=32 RAID Block sze (kb) Optmal block sze (kb) (b) RAID-5, 32 dsks, 5 clents, degraded mode Upper 5% envelope Lower 5% envelope Optmal block sze Party group sze Fgure 12: Effect of party group sze. Snce the optmal block sze vares across zones, a multmeda server can: () choose an optmal block sze for each zone, or () choose a sngle block sze for all zones. Recently several placement polces that employ dfferent block szes for dfferent zones have been proposed [2, 21]. Our models enable us to parameterze these polces by choosng an approprate block sze for each zone. Snce use of a sngle block sze for all zones can cause an ncrease n the servce tme of the most heavly loaded dsk, a multmeda server must choose a block sze that mnmzes ths ncrease across all zones. To do so, the server must select a block sze that s contaned wthn the x% optmal envelope of all zones. Ths ensures that the servce tme of the most heavly loaded dsk s always wthn x% of the mnmum. To llustrate, Fgure 11(b) shows that a block sze of 96KB s contaned wthn the 5% optmal envelope of all zones on the dsk. Observe that these polces form two ends of a spectrum. Whereas one yelds optmal performance, the other smplfes storage space management. The server can balance these tradeoffs by choosng an ntermedate polcy that groups consecutve zones and selects a sngle block sze for each group. Party Group Sze Snce non-redundant arrays do not mantan any party nformaton, the party group sze s a parameter that s relevant only to redundant dsk arrays. Fgure 12(a) depcts the servce tme of the most heavly loaded dsk n a RAID-5 array n the normal operatng mode. It demonstrates that, n the absence of a dsk falure, the servce tme of the most heavly loaded dsk n a RAID-5 array s almost dentcal to that obtaned for an equvalent RAID- array. Moreover, the servce tme of the most heavly loaded dsk s ndependent of the party group sze. Consequently, 16

17 the optmal block sze obtaned for a RAID-5 array n the normal operatng mode s ndependent of the party group sze and s dentcal to that obtaned for a RAID- array. Next consder the RAID-5 array wth a sngle dsk falure. Let G denote the party group sze. In such a scenaro, whenever a clent accesses a block stored on the faled dsk, the server must access the remanng blocks of the party groups stored on the survvng G, 1 dsks to reconstruct the requested block. Hence, wth ncrease n party group sze, the number of addtonal blocks that must be accessed to reconstruct a block on the faled dsk ncreases, ncreasng the load on survvng dsks. Ths results n an effectve ncrease n the playback rate of clents. As explaned n earler n ths secton, ncreasng the playback rate of clents causes an ncrease n the optmal meda block sze and the 5% optmal envelope. Hence, the optmal block sze and the optmal envelope n the degraded mode ncrease wth ncrease n the party group sze (see Fgure 12(b)). 2.4 Selectng an Optmal Block Sze Havng examned the effect of the server confguraton and the workload characterstcs on the block sze, we now present procedures for selectng an optmal block sze. The procedure for selectng an optmal block sze depends on the desgn goals for the multmeda server, whch n turn are dctated by the operatng envronment. To llustrate, for multmeda servers offerng commercal servces (e.g., vdeo-on-demand, onlne news, etc.), the prmary goal s to maxmze revenue by maxmzng the number of clents that can be supported by the server. In contrast, for multmeda servers whch servce clents wth heterogeneous QoS, the number of clents that can be supported depends on the exact workload mx (.e., the proporton of clents wth dfferent requrements). Snce the workload mx can vary over tme, the goal for such servers s to provde the best possble performance over a wde range of workloads. Dfferng desgn goals may requre the system desgner to choose completely dfferent meda block szes. To determne a block sze that maxmzes the number of clents supported, let us assume that all parameters determnng the server confguraton (.e., the number of dsks, ther physcal characterstcs, the round duraton, etc.) are known at desgn tme. Also, assume that the data rate of clents and ther QoS requrements are known. Then, a block sze that maxmzes the number of clents supported can be computed by the followng two step procedure: (1) For a gven number of clents, n, determne the servce tme of the most heavly loaded dsk for dfferent block szes and select the block sze that mnmzes the servce tme; (2) If the servce tme of the most heavly loaded dsk for ths block sze s less than the round duraton, then ncrement n and repeat step (1). The block sze that s obtaned when the servce tme of the most heavly loaded dsk equals the round duraton maxmzes the number of clents supported by the server. In general computng envronments, due to the heterogeneous nature of the workload, some of the workload characterstcs may be unknown at desgn tme (e.g., the number of clents accessng the server). In such a scenaro, a block sze that yelds good performance over a wde range of workloads must be chosen [3]. For every parameter that s unknown at desgn tme, the range over whch the parameter s lkely to vary must frst be estmated. The optmal block sze and the x% optmal envelope for each combnaton of these parameters s then computed usng the model. Let S 1 ;S 2 ;æææ denote sets, each contanng the x% optmal envelope for a partcular combnaton of these parameters. Then, the set of block szes that yelds servce tmes wthn x% of the mnmum over all possble combnatons of these parameters s S = S 1 ës 2 ë :::.IfS s empty, then the entre procedure must be repeated for a larger values of x untl a non-empty set of block szes s obtaned. Fgure 13 llustrates the process of computng a feasble soluton (.e., a non-empty set S) over a range of clent workloads. 3 Determnng the Degree of Strpng In addton to determnng the strpe unt sze, defnng a strpng polcy requres the determnaton of degree of strpng. A multmeda server can ether strpe a meda stream across all dsks n the array or across a subset of the dsks. Whereas the former polcy s referred to as wde strpng, the latter polcy s referred to as narrow strpng. 17

18 Infeasble Soluton Feasble Soluton Upper 7% envelope Block Sze (kb) Upper 5% envelope Lower 5% envelope Optmal block sze curve Block Sze (kb) Lower 7% envelope Optmal block sze curve Optmal block sze set S Number of Clents Number of Clents Fgure 13: Selectng a block sze that yelds near-optmal performance, regardless of the number of clents accessng the server. The shaded regon denotes the set of block szes S that yeld servce tmes wthn 7% of the mnmum for all workloads. To evaluate the relatve merts of these polces, consder a multmeda server that employs wde strpng to nterleave meda streams across dsks n the array. Let us assume that the performance of the server s measured n terms of the maxmum number of clents that t can support. In an deal scenaro, ncrease n the number of dsks n the system should result n a lnear ncrease n the number of clents that can be supported by the server. That s, the number of clents supported by a dsk array consstng of D dsks should be D tmes the number of clents that can be supported by a sngle dsk. However, as shown n Fgure 14(a), the number of clents supported by the server ncreases sub-lnearly wth ncrease n the number of dsks. Ths can be attrbuted to the followng two reasons: æ Real-tme requrements of clents: Due to the real-tme requrements of clents, the number of clents supported by the server s constraned by the most heavly loaded dsk. Specfcally, the number of clents accessng the server reaches ts maxmum value when the servce tme of the most heavly loaded dsk equals the round duraton. At ths pont, however, the servce tme of a dsk wth average load s smaller than the round duraton. The resultng load mbalance causes most of the dsks n the array to be under-utlzed. æ Reducton n optmal block sze: As explaned n Secton 2.3.2, an ncrease n the number of dsks n the system causes the load mbalance I s to ncrease. An ncrease n the number of dsks also ncreases the number of clents that can be supported by the server. Larger the number of clents accessng the server, the smaller the load mbalance I s. Thus, the combned effect of ncreasng the number of dsks and the number of clents accessng the server governs the actual value of I s. Fgure 14(b) plots the varaton n mbalance I s aganst the (number of dsks n the system, maxmum number of clents supported) pars. It llustrates that the ncrease n I s due to an ncrease n the number of dsks domnates the decrease n I s due to an ncrease n the number of clents, causng the actual mbalance to ncrease. Hence, a small block sze must be chosen to compensate for the ncreased mbalance, causng a decrease n the optmal block sze (see Fgure 14(c)). Snce a small block sze mposes a larger latency overhead, the overall throughput of the array decreases, causng a reducton n the number of clents that can be supported. To mnmze the mpact of these factors, a server can: (1) partton the dsk array nto mutually exclusve groups of dsks, and (2) strpe each meda stream only wthn a partton. Snce each partton acts as an ndependent dsk array and the number of dsks per partton s small, such an approach: (1) reduces the load mbalance wthn each partton, and (2) ncreases the optmal block sze for a partton (and thereby reduces the latency overhead). In such parttoned arrays, load mbalances can occur f clents are not equtably dstrbuted among all the parttons. Hence, the partton sze must be chosen so as to smultaneously mnmze the mpact of load mbalance across parttons and the load mbalance wthn a partton. In what follows, we frst present a model for determnng the load mbalance across parttons, and then descrbe a procedure for determnng the a partton sze that maxmzes the number of clents supported. 18

19 Number of clents supported (a) Wde Strpng Wde strpng (actual) Wde Strpng (deal) Load mbalance (b) Imbalance wthn a partton Load mbalance Optmal block sze (kb) (c) Varaton n optmal blocks sze Optmal block sze Number of dsks.325 (16,214)(24,39)(32,395) (48,58) (64,77) (Number of dsks, Maxmum number of clents) (16,214)(24,39)(32,395) (48,58) (64,77) (Number of dsks, Maxmum number of clents) Fgure 14: Loss n the number of clents supported n large dsk arrays and factors contrbutng to ths loss. 3.1 Modelng the Imbalance Across Parttons To compute the load mbalance across parttons, consder a dsk array consstng of D dsks that s parttoned nto groups of d dsks each. Let us assume that the server employs a placement polcy that assgns streams to parttons such that each partton s equally lkely to be accessed by a new request [8, 23]. That s, the probablty that a newly arrvng clent accesses a partton s q = d=d. Insuchascenaro, fn clents access the server, then the probablty that m clents access the j th partton s bnomally dstrbuted, and s gven as: è! P èy n j = mè = m æ q m æ è1, qè n,m (23) where Y j s a random varable representng the number of clents accessng the j th partton. Then the number of clents accessng the most heavly loaded partton s Y max = maxèy 1 ;Y 2 ;:::;Y D è (24) d Snce the load on a partton s ndependent of other parttons, Y 1 ;Y 2 ;:::;Y D d Hence, the dstrbuton of Y max can be computed as: are ndependent random varables. F Ymax èxè =F Y1 èxè æ F Y2 èxèæææf Y Dd èxè (25) where F Yj s the cumulatve probablty dstrbuton functon of the random varable Y j [16]. Gven the dstrbuton of Y and Y max, we can compute the expected number of requests on the average and the most heavly loaded parttons (denoted by by and by max, respectvely). Usng these values, we can defne the the load mbalance across parttons (denoted by I p )as: è I p = 1, Y b! Y b (26) max Thus, gven the number of dsks n the array and the partton sze, we can compute the load mbalance across parttons. 3.2 Determnng the Partton Sze For a fxed number of dsks, ncreasng the partton sze ncreases the load mbalance I s wthn a partton (Fgure 14(b)), whle decreasng the load mbalance I p across parttons (Fgure 15). Moreover, as shown n Fgure 14(c), ncreasng the partton sze results n a reducton n the optmal block sze (thereby ncreasng the seek and rotatonal latency overhead). Consequently, the server must determne the degree of strpng (.e., partton sze) that balances these tradeoffs. 19

20 Gven the models for predctng: (1) the load mbalance across parttons (Secton 3.1), (2) the load mbalance wthn a partton (Secton 2.1.1), a procedure for choosng a partton sze that maxmzes the number of clents supported by the server s as follows: Procedure ComputeParttonSze 1. Choose an ntal partton sze of d=1. 2. Usng the model presented n Secton 2.1.1, compute the maxmum number of clents, n, that can be supported by a sngle partton of sze d (.e., the number of clents at whch the servce tme of the most heavly loaded dsk equals the round duraton). 3. Assumng that n clents access the array, usng the model presented n Secton 3.1, compute the expected number of clents, by max, accessng the most heavly loaded partton. 4. If by max én, then ncrement n and repeat step (3). When by max = n,thenn denotes the maxmum number of clents that can be supported by the array wth a partton sze of d. 5. Increment the partton sze d, and repeat steps (2) thorough (4) untl no further mprovements n the number of clents s obtaned (.e., untl n starts decreasng wth ncrease n d). Ths yelds a partton sze that maxmzes the number of clents that can be supported. In the above procedure, note that the lmt on the number of clents that can be supported by the entre array s reached when the most heavly loaded partton reaches ts maxmum capacty. However, at ths pont, the number of clents accessng other parttons s less than ther maxmum capacty. Hence, the total number of clents that can be supported by the array does not ncrease lnearly wth number of parttons (.e., nén æ D d ). Fgure 16(a) llustrates the result of executng ths teratve procedure for an array of 12 dsks. Snce the number of clents that can be supported by the array s maxmzed at d =1, the array should be parttoned nto 12 parttons of 1 dsks each for optmal performance. Fgure 16(b) demonstrates the varaton n the optmal partton sze wth ncrease n the number of dsks n the array. Fnally, Fgure 16(c) llustrates the mprovement n the number of clents supported due to parttonng. For small dsk arrays, snce wde strpng s close to the deal case, the addtonal gans due to parttonng are small. For large dsk arrays, however, parttonng yelds a approxmately a 1% ncrease n the number of clents supported as compared to the wde strpng. Fgure 16(c) also demonstrates that parttonng coupled wth statc load balancng algorthms does not completely brdge the gap between the number of clents supported by the array n the deal case (.e., when the number of clents ncreases lnearly wth array sze) and that obtaned usng wde strpng. To further reduce the loss n the number of clents supported, the server must replcate streams across parttons and employ dynamc load balancng schemes. The mprovement n performance yelded by such a scheme s at the expense of hgher storage space requrement and more complex storage space management algorthms. Detaled cost-performance tradeoffs of such an approach s beyond the scope of ths paper. 4 Related Work Several research projects have developed smulaton and analytcal technques for optmzng the performance of strped dsk arrays for conventonal workloads [3, 4, 5, 14]. As demonstrated n Secton 1, due to the real-tme nature of contnuous meda accesses, these technques are not drectly applcable for optmzng performance n multmeda servers. The problem of determnng the optmal strpe unt sze for non-redundant arrays storng contnuous meda was studed n [22]. A model that predcts the servce tme of the most heavly loaded dsk for non-redundant arrays (henceforth referred to as the VRG model) was also proposed n the paper. The VRG model uses worst case assumptons about the number of blocks accessed by a clent durng a round to compute the servce tme of the most heavly loaded dsk. In contrast, our model uses actual dstrbutons of the number of blocks accessed by a clent 2

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