Reliability and Performance Models for Grid Computing

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1 Relablty and Performance Models for Grd Computng Yuan-Shun Da,2, Jack Dongarra,3,4 Department of Electrcal Engneerng and Computer Scence, Unversty of Tennessee, Knoxvlle 2 Department of Industral and Informaton Engneerng, Unversty of Tennessee, Knoxvlle 3 Oak Rdge Natonal Laboratory 4 Unversty of Manchester Abstract Grd computng s a newly developed technology for complex systems wth large-scale resource sharng, wde-area communcaton, and mult-nsttutonal collaboraton. It s hard to analyze and model the Grd relablty because of ts largeness, complexty and stffness. Therefore, ths chapter ntroduces the Grd computng technology, presents dfferent types of falures n grd system, models the grd relablty wth star structure and tree structure, and fnally studes optmzaton problems for grd task parttonng and allocaton. The chapter then presents models for star-topology consderng data dependence and tree-structure consderng falure correlaton. Evaluaton tools and algorthms are developed, evolved from Unversal generatng functon and Graph Theory. Then, the falure correlaton and data dependence are consdered n the model. Numercal examples are llustrated to show the modelng and analyss. Keywords: Relablty, Performance, Grd computng, Modelng, Graph theory, Bayesan approach.

2 . Introducton Grd computng (Foster & Kesselman, 2003) s a newly developed technology for complex systems wth large-scale resource sharng, wde-area communcaton, and mult-nsttutonal collaboraton etc, see e.g. Kumar (2000), Das et al. (200), Foster et al. (200, 2002) and Berman et al. (2003). Many experts beleve that the grd technologes wll offer a second chance to fulfll the promses of the Internet. The real and specfc problem that underles the Grd concept s coordnated resource sharng and problem solvng n dynamc, mult-nsttutonal vrtual organzatons (Foster et al., 200). The sharng that we are concerned wth s not prmarly fle exchange but rather drect access to computers, software, data, and other resources. Ths s requred by a range of collaboratve problem-solvng and resource-brokerng strateges emergng n ndustry, scence, and engneerng. Ths sharng s hghly controlled by the resource management system (Lvny & Raman, 998), wth resource provders and consumers defnng what s shared, who s allowed to share, and the condtons under whch the sharng occurs. Recently, the Open Grd Servce Archtecture (Foster et al., 2002) enables the ntegraton of servces and resources across dstrbuted, heterogeneous, dynamc, vrtual organzatons. A grd servce s desred to complete a set of programs under the crcumstances of grd computng. The programs may requre usng remote resources that are dstrbuted. However, the programs ntally do not know the ste nformaton of those remote resources n such a large-scale computng envronment, so the resource management system (the bran of the grd) plays an mportant role n managng the pool of shared resources, n matchng the programs to ther requested resources, and n controllng them to reach and use the resources through wde-area network. The structure and functons of the resource management system (RMS) n the grd have been ntroduced n detals by Lvny & Raman (998), Cao et al. (2002), Krauter et al. (2002) and Nabrzysk et al. (2003). Brefly stated, the programs n a grd servce send ther requests for resources to the RMS. The RMS adds these requests nto the request queue (Lvny & Raman, 998). Then, the requests are watng n the queue for the matchng servce of the RMS for a perod of tme (called watng tme), see e.g. Abramson et al. (2002). In the matchng servce, the RMS matches the requests to the shared resources n the grd (Dng et 2

3 al., 2002) and then bulds the connecton between the programs and ther requred resources. Thereafter, the programs can obtan access to the remote resources and exchange nformaton wth them through the channels. The grd securty mechansm then operates to control the resource access through the Certfcaton, Authorzaton and Authentcaton, whch consttute varous logcal connectons that causes dynamcty n the network topology. Although the developmental tools and nfrastructures for the grd have been wdely studed (Foster & Kesselman, 2003), grd relablty analyss and evaluaton are not easy because of ts complexty, largeness and stffness. The grd computng contans dfferent types of falures that can make a servce unrelable, such as blockng falures, tme-out falures, matchng falures, network falures, program falures and resource falures. Ths chapter thoroughly analyzes these falures. Usually the grd performance measure s defned as the task executon tme (servce tme). Ths ndex can be sgnfcantly mproved by usng the RMS that dvdes a task nto a set of subtasks whch can be executed n parallel by multple onlne resources. Many complcated and tme-consumng tasks that could not be mplemented before are workng well under the grd envronment now. It s observed n many grd projects that the servce tme experenced by the users s a random varable. Fndng the dstrbuton of ths varable s mportant for evaluatng the grd performance and mprovng the RMS functonng. The servce tme s affected by many factors. Frst, varous avalable resources usually have dfferent task processng speeds onlne. Thus, the task executon tme can vary dependng on whch resource s assgned to execute the task/subtasks. Second, some resources can fal when runnng the subtasks, so the executon tme s also affected by the resource relablty. Smlarly, the communcaton lnks n grd servce can be dsconnected durng the data transmsson. Thus, the communcaton relablty nfluences the servce tme as well as data transmsson speed through the communcaton channels. Moreover, the servce requested by a user may be delayed due to the queue of earler requests submtted from others. Fnally, the data dependence mposes constrants on the sequence of the subtasks' executon, whch has sgnfcant nfluence on the servce tme. Ths chapter frst ntroduces the grd computng system and servce, and analyzes varous 3

4 falures n grd system. Both relablty and performance are analyzed n accordance wth the performablty concept. Then the chapter presents models for star- and tree-topology grds respectvely. The relablty and performance evaluaton tools and algorthms are developed based on the unversal generatng functon, graph theory, and Bayesan approach. Both falure correlaton and data dependence are consdered n the models. 2. Grd Servce Relablty and Performance 2.. Descrpton of the grd computng Today, the Grd computng systems are large and complex, such as the IP-Grd (Indana-Purdue Grd) that s a statewde grd ( IP-Grd s also a part of the TeraGrd that s a natonwde grd n the USA ( The largeness and complexty of the grd challenge the exstng models and tools to analyze, evaluate, predct and optmze the relablty and performance of grd systems. The global grd system s generally depcted by the Fg.. Varous organzatons (Foster et al., 200), ntegrate/share ther resources on the global grd. Any program runnng on the grd can use those resources f t can be successfully connected to them and s authorzed to access them. The stes that contan the resources or run the programs are lnked by the global network as shown n the left part of Fg.. 4

5 Applcaton Programs Program Layer P, R, P, Resource access clamng Resource descrptons Resource requests Resource offers Inter-request RM Global RM Access Control Shared Resources Resource stes Matches Matches Request queue Request Layer Management Layer Network Layer Resource Layer R, Fve Layers Notatons: Grd Resource Management Large-Scale Grd Network P=Program R=Resource RM=Resource Management RMS=Resource Management System P, Fg.. Grd Computng System The dstrbuton of the servce tasks/subtasks among the remote resources are controlled by the Resource Management System (RMS) that s the bran of the grd computng, see e.g. Lvny & Raman (998). The RMS has fve layers n general, as shown n Fg. : program layer, request layer, management layer, network layer and resource layer. ) Program layer: The program layer represents the programs of the customer s applcatons. The programs descrbe ther requred resources and constrant requrements (such as deadlne, budget, functon etc). These resource descrptons are translated to the resource requests and sent to the next request layer. 2) Request layer: The request layer provdes the abstracton of program requrements as a queue of resource requests. The prmary goals of ths layer are to mantan ths queue n a persstent and fault-tolerant manner and to nteract wth the next management layer by njectng resource requests for matchng, clamng matched resources of the requests. 3) Management layer: The management layer may be thought of as the global resource allocaton layer. It has the functon of automatcally detectng new resources, montorng the resource pool, removng faled/unavalable resources, and most mportantly matchng the resource requests of a servce to the regstered/detected resources. If resource requests are matched wth the regstered resources n the grd, ths layer sends the matched tags to the next network layer. 5

6 4) Network layer: The network layer dynamcally bulds connecton between the programs and resources when recevng the matched tags and controls them to exchange nformaton through communcaton channels n a secure way. 5) Resource layer: The resource layer represents the shared resources from dfferent resource provders ncludng the usage polces (such as servce charge, relablty, servng tme etc.) 2.2. Falure analyss of grd servce Even though all onlne nodes or resources are lnked through the Internet wth one another, not all resources or communcaton channels are actually used for a specfc servce. Therefore, accordng to ths observaton, we can make tractable models and analyses of grd computng va a vrtual structure for a certan servce. The grd servce s defned as follows: Grd servce s a servce offered under the grd computng envronment, whch can be requested by dfferent users through the RMS, whch ncludes a set of subtasks that are allocated to specfc resources va the RMS for executon, and whch returns the result to the user after the RMS ntegrates the outputs from dfferent subtasks. The above fve layers coordnate together to acheve a grd servce. At the Program layer, the subtasks (programs) composng the entre grd servce task ntally send ther requests for remote resources to the RMS. The Request layer adds these requests n the request queue. Then, the Management layer tres to fnd the stes of the resources that match the requests. After all the requests of those programs n the grd servce are matched, the Network layer bulds the connectons among those programs and the matched resources. It s possble to dentfy varous types of falures on respectve layers: Program layer: Software falures can occur durng the subtask (program) executon; see e.g. Xe (99) and Pham (2000). Request layer: When the programs requests reach the request layer, two types of falures may occur: blockng falure and tme-out falure. Usually, the request queue has a lmtaton on the maxmal number of watng requests (Lvny & Raman, 998). If the queue s full when a new request arrves, the request blockng falure occurs. The grd servce usually has ts due tme set by customers or servce 6

7 montors. If the watng tme for the requests n the queue exceeds the due tme, the tme-out falure occurs, see e.g. Abramson et al. (2002). Management layer: At ths layer, matchng falure may occur f the requests fal to match wth the correct resources, see e.g. Xe et al. (2004, pp ). Errors, such as ncorrectly translatng the requests, regsterng a wrong resource, gnorng resource dsconnecton, msunderstandng the users' requrements, can cause these matchng falures. Network layer: When the subtasks (programs) are executed on remote resources, the communcaton channels may be dsconnected ether physcally or logcally, whch causes the network falure, especally for those long tme transmssons of large dataset, see e.g. Da et al. (2002). Resource layer: The resources shared on the grd can be of software, hardware or frmware type. The correspondng software, hardware or combned faults can cause resource unavalablty Grd Servce Relablty and Performance Most prevous research on dstrbuted computng studed performance and relablty separately. However, performance and relablty are closely related and affect each other, n partcular under the grd computng envronment. For example, whle a task s fully parallelzed nto m subtasks executed by m resources, the performance s hgh but the relablty mght be low because the falure of any resource prevents the entre task from completon. Ths causes the RMS to restart the task, whch reversely ncreases ts executon tme (.e. reduces performance). Therefore, t s worth to assgn some subtasks to several resources to provde executon redundancy. However, excessve redundancy, even though mprovng the relablty, can decrease the performance by not fully parallelzng the task. Thus, the performance and relablty affect each other and should be consdered together n the grd servce modelng and analyss. In order to study performance and relablty nteractons, one also has to take nto account the effect of servce performance (executon tme) upon the relablty of the grd elements. The conventonal models, e.g. Kumar et al. (986), Chen & Huang (992), Chen et al. (997), and Ln et al., (200), are based on the assumpton that the operatonal probabltes of nodes or lnks are constant, whch gnores the lnks' bandwdth, 7

8 communcaton tme and resource processng tme. Such models are not sutable for precsely modelng the grd servce performance and relablty. Another mportant ssue that has much nfluence the performance and relablty s data dependence, that exsts when some subtasks use the results from some other subtasks. The servce performance and relablty s affected by data dependence because the subtasks cannot be executed totally n parallel. For nstance, the resources that are dle n watng for the nput to run the assgned subtasks are usually hot-standby because cold-start s tme consumng. As a result, these resources can fal n watng mode. The consderatons presented above lead the followng assumptons that lay n the base of grd servce relablty and performance model. Assumptons: ) The servce request reaches the RMS and s beng served mmedately. The RMS dvdes the entre servce task nto a set of subtasks. The data dependence may exst among the subtasks. The order s determned by precedence constrants and s controlled by the RMS. 2) Dfferent grd resources are regstered or automatcally detected by the RMS. In a grd servce, the structure of vrtual network (consstng of the RMS and resources nvolved n performng the servce) can form star topology wth the RMS n the center or, tree topology wth the RMS n the root node. 3) The resources are specalzed. Each resource can process one or multple subtask(s) when t s avalable. 4) Each resource has a gven constant processng speed when t s avalable and has a gven constant falure rate. Each communcaton channel has constant falure rate and a constant bandwdth (data transmsson speed). 5) The falure rates of the communcaton channels or resources are the same when they are dle or loaded (hot standby model). The falures of dfferent resources and communcaton lnks are ndependent. 6) If the falure of a resource or a communcaton channel occurs before the end of output data transmsson from the resource to the RMS, the subtask fals. 8

9 7) Dfferent resources start performng ther tasks mmedately after they get the nput data from the RMS through communcaton channels. If same subtask s processed by several resources (provdng executon redundancy), t s completed when the frst result s returned to the RMS. The entre task s completed when all of the subtasks are completed and ther results are returned to the RMS from the resources. 8) The data transmsson speed n any mult-channel lnk does not depend on the number of dfferent packages (correspondng to dfferent subtasks) sent n parallel. The data transmsson tme of each package depends on the amount of data n the package. If the data package s transmtted through several communcaton lnks, the lnk wth the lowest bandwdth lmts the data transmsson speed. 9) The RMS s fully relable, whch can be justfed to consder a relatvely short nterval of runnng a specfc servce. The mperfect RMS can also be easly ncluded as a module connected n seres to the whole grd servce system Grd Servce tme dstrbuton and relablty/performance measures The data dependence on task executon can be represented by m m matrx H such that h k = f subtask needs for ts executon output data from subtask k and h k = 0 otherwse (the subtasks can always be numbered such that k< for any h k = ). Therefore, f h k = executon of subtask cannot begn before completon of subtask k. For any subtask one can defne a set φ of ts mmedate predecessors: k φ f h k =. The data dependence can always be presented n such a manner that the last subtask m corresponds to fnal task processed by the RMS when t receves output data of all the subtasks completed by the grd resources. The task executon tme s defned as tme from the begnnng of nput data transmsson from the RMS to a resource to the end of output data transmsson from the resource to the RMS. 9

10 The amount of data that should be transmtted between the RMS and resource j that executes subtask s denoted by a. If data transmsson between the RMS and the resource j s accomplshed through lnks belongng to a set γ j, the data transmsson speed s s j = mn ( bx ) () γ Lx j where b x s the bandwdth of the lnk L x. Therefore, the random tme t j of subtask executon by resource j can take two possble values a t j = tj = τ j + sj ˆ (2) f the resource j and the communcaton path γ j do not fal untl the subtask completon and t j = otherwse. Here, τ j s the processng tme of the j-th resource. Subtask can be successfully completed by resource j f ths resource and communcaton path γ j do not fal before the end of subtask executon. Gven constant falure rates of resource j and lnks, one can obtan the condtonal probablty of subtask success as p j (ˆ t j ( λ j +π j)ˆ t j ) = e (3) where π j s the falure rate of the communcaton path between the RMS and the resource j, whch can be calculated as π = λ, λ x s the falure rate of the lnk j x x γ j L x. The exponental dstrbuton (3) s common n software or hardware components relablty that had been justfed n both theory and practce, see e.g. Xe et al. (2004). These gve the condtonal dstrbuton of the random subtask executon tme t j : Pr( t j = tˆj) = pj( tj) and Pr( tj = ) = pj ( tj ). Assume that each subtask s assgned by the RMS to resources composng set ω. The RMS can ntate executon of any subtask j (send the data to all the resources from ω ) only 0

11 after the completon of every subtask k φ. Therefore the random tme of the start of subtask executon T can be determned as T = max( T k ) (4) k φ where T k s random completon tme for subtask k. If φ =,.e. subtask does not need data produced by any other subtask, the subtask executon starts wthout delay: T = 0. If φ, T can have dfferent realzatons Tˆ l ( l N ). Havng the tmet when the executon of subtask starts and the tme t j of subtask executed by resource j, one obtans the completon tme for subtask on resource j as t T + t j = j. (5) In order to obtan the dstrbuton of random tme tj one has to take nto account that probablty of any realzaton of events: tj Tˆ l + tˆ j = s equal to the product of probabltes of three - executon of subtask starts at tme Tˆ l : q l =Pr( T = Tˆ l ); - resource j does not fal before start of executon of subtask : p j ( Tˆ l ); - resource j does not fal durng the executon of subtask : p j ( tˆ j ). Therefore, the condtonal dstrbuton of the random tme tj gven executon of subtask starts at tme Tˆ l ( T = Tˆ l ) takes the form Pr( t j Tˆ l + tˆ j = ) =p j ( Tˆ l )p j ( tˆ j ) = p j ( Tˆ l + tˆ j ) = e Pr( t j = )=- p j ( Tˆ l + j tˆ )=-e ( λ )( ˆ j + π j Tl + tˆj) ( λ )( ˆ j + π j Tl + tˆj)., (6) The random tme of subtask completon T s equal to the shortest tme when one of the resources from ω completes the subtask executon: T mn ( = tj ). (7) j ω

12 Accordng to the defnton of the last subtask m, the tme of ts begnnng corresponds to the servce completon tme, because the tme of the task proceeds wth RMS s neglected. Thus, the random servce tme Θ s equal to T m. Havng the dstrbuton (pmf) of the random value Θ T m n the form q Pr( ˆ ml = Tm = Tml ) for l N m, one can evaluate the relablty and performance ndces of the grd servce. In order to estmate both the servce relablty and ts performance, dfferent measures can be used dependng on the applcaton. In applcatons where the executon tme of each task (servce tme) s of crtcal mportance, the system relablty R(Θ*) s defned (accordng to performablty concept n Meyer (980), Grass et al. (988) and Ta et al. (993)) as a probablty that the correct output s produced n tme less than Θ*. Ths ndex can be obtaned as N m R( Θ*) = q ( Tˆ ml ml < Θ*). (8) l= When no lmtatons are mposed on the servce tme, the servce relablty s defned as the probablty that t produces correct outputs wthout respect to the servce tme, whch can be referred to as R( ). The condtonal expected servce tme W s consdered to be a measure of ts performance, whch determnes the expected servce tme gven that the servce does not fal,.e. W N m = Tˆ ml qml / R( ). (9) l= 3. Star Topology Grd Archtecture A grd servce s desred to execute a certan task under the control of the RMS. When the RMS receves a servce request from a user, the task can be dvded nto a set of subtasks that 2

13 are executed n parallel. The RMS assgns those subtasks to avalable resources for executon. After the resources complete the assgned subtasks, they return the results back to the RMS and then the RMS ntegrates the receved results nto entre task output whch s requested by the user. The above grd servce process can be approxmated by a structure wth star topology, as depcted by Fg. 2, where the RMS s drectly connected wth any resource through respectve communcaton channels. The star topology s feasble when the resources are totally separated so that ther communcaton channels are ndependent. Under ths assumpton the grd servce relablty and performance can be derved by usng the unversal generatng functon technque. Program Resource Resource Program Resource Resource Management System (RMS) Resource Program Resource Program Star Topology Fg. 2. Grd system wth star archtecture. 3.. Unversal Generatng Functon The unversal generatng functon (u-functon) technque was ntroduced n (Ushakov, 987) and proved to be very effectve for the relablty evaluaton of dfferent types of mult-state systems. 3

14 The u-functon representng the pmf of a dscrete random varable Y s defned as a polynomal K y u( z) = α k z k, (0) k = where the varable Y has K possble values and α k s the probablty that Y s equal to y k. To obtan the u-functon representng the pmf of a functon of two ndependent random varables ϕ(y, Y j ), composton operators are ntroduced. These operators determne the u-functon for ϕ(y, Y j ) usng smple algebrac operatons on the ndvdual u-functons of the varables. All of the composton operators take the form U(z) = K u( z) u j( z) = ϕ k = α K j K K j y y k jh k z α jhz = ϕ h= k = h = α k α ϕ( yk, y jh ) jhz The u-functon U(z) represents all of the possble mutually exclusve combnatons of realzatons of the varables by relatng the probabltes of each combnaton to the value of functon ϕ(y, Y j ) for ths combnaton. In the case of grd system, the u-functon u j (z) can defne pmf of executon tme for subtask assgned to resource j. Ths u-functon takes the form () u tˆ z = p t z j j ( ) j (ˆ j ) + ( pj (ˆ tj )) z (2) where tˆ j and p j ( tˆ j ) are determned accordng to Eqs. (2) and (3) respectvely. The pmf of the random start tme T for subtask can be represented by u-functon U (z) takng the form where q Pr( ˆ l = T = Tl ). L = l l = Tˆ l U ( z) q z, (3) For any realzaton Tˆ l of T the condtonal dstrbuton of completon tme tj for subtask executed by resource j gven T ˆ = Tl accordng to (6) can be represented by the u-functon 4

15 + u Tˆ t j ( z, Tˆ l ) p j ( Tˆ l + tˆ j ) z ˆ l j = + ( p ( Tˆ + tˆ )) z j l j. (4) The total completon tme of subtask assgned to a par of resources j and d s equal to the mnmum of completon tmes for these resources accordng to Eq. (7). To obtan the u-functon representng the pmf of ths tme, gven T ˆ = Tl, composton operator wth ϕ(y j, Y d ) = mn(y j,y d ) should be used: ˆ ˆ ˆ ˆ Tˆ ˆ ˆ l + tj ˆ ˆ u ( z, Tl ) = uj ( z, Tl ) ud ( z, Tl ) = [ p j ( Tl + tj ) z + ( p j ( Tl + tj )) z ] mn [ ( ˆ ˆ Tˆ + ˆ ) ( ( ˆ ˆ p T + t z l td d l d + pd Tl + td )) z ] mn ˆ ˆ ( ˆ ˆ ) ( ˆ T + mn(ˆ,ˆ ) ˆ l tj td ) ( ˆ ˆ )( ( ˆ ˆ ˆ T + = p T + t p T + t z + p T + t p T + t )) z l td j l j d l d d l d j l j ˆ ˆ ( ˆ ˆ )( ( ˆ T + t ˆ l j )) ( ( ˆ ˆ ))( ( ˆ ˆ + p j Tl + tj pd Tl + td z + p j Tl + tj pd Tl + td )) z. The u-functon u (, ˆ z T l ) representng the condtonal pmf of completon tme T for subtask assgned to all of the resources from set ω ={j,,j } can be obtaned as u ( z, ˆ ) can be obtaned recursvely: T l (5) u ( z, Tˆ ) = u ( z, Tˆ ) u ( z, Tˆ )... u ( z, Tˆ ). (6) l j u (, ˆ ) (, ˆ z Tl = uj z Tl ), l j mn 2 l j l mn mn u ( z, Tˆ ) = u ( z, Tˆ ) u ( z, Tˆ ) for e = j 2,, j. (7) l l mn Havng the probabltes of the mutually exclusve realzatons of start tme T, q = Pr( T = Tˆ ) and u-functons u ( z, ˆ ) representng correspondng condtonal l l T l dstrbutons of task completon tme, we can now obtan the u-functon representng the e l uncondtonal pmf of completon tme T as U ( z) N = ql l = u ( z, Tˆ Havng u-functons Uk ( z ) representng pmf of the completon tme T k l ). (8) for any subtask k φ = k,..., k }, one can obtan the u-functons (z) representng pmf of subtask start { tme T accordng to (4) as U 5

16 U (z) can be obtaned recursvely: N k k max 2 max max k l l= Tˆ U ( z) = U ( z) U ( z)... U ( z) = q z l. (9) 0 U ( z) = z max, U ( z) = U ( z) U ( z) for e = k,, k. (20) e It can be seen that f φ = then 0 ( z) z U =. The fnal u-functon U m (z) represents the pmf of random task completon tme T m n the form N ˆ ( ) = m T U ml m z qml z. (2) l= Usng the operators defned above one can obtan the servce relablty and performance ndces by mplementng the followng algorthm:. Determne tˆ j for each subtask and resource j ω usng Eq. (2); Defne for each subtask ( m) U ( z) = U ( z) = z For all : If φ = 0 or f for any k φ Uk ( z ) z 0 (u-functons representng the completon tmes of all of the predecessors of subtask are obtaned) N 2.. Obtan = Tˆ U l ( z) ql z usng recursve procedure (20); l = 2.2. For l =,, N : For each j ω obtan u (, ˆ j z T l ) usng Eq. (4); Obtan u ( z ) usng recursve procedure (7); 6

17 2.3. Obtan U ( z ) usng Eq. (8). 3. If U m (z) = z 0 return to step Obtan relablty and performance ndces R(Θ*) and W usng equatons (8) and (9) Illustratve Example Ths example presents analytcal dervaton of the ndces R(Θ*) and W for smple grd servce that uses sx resources. Assume that the RMS dvdes the servce task nto three subtasks. The frst subtask s assgned to resources and 2, the second subtask s assgned to resources 3 and 4, the thrd subtask s assgned to resources 5 and 6: ω = {,2}, ω 2 = {3,4}, ω 3 = {5,6}. The falure rates of the resources and communcaton channels and subtask executon tmes are presented n Table. Table. No of subtask Parameters of grd system for analytcal example No of resource j λ j +π j (sec - ) tˆj (sec) p j tˆ ( j ) Subtasks and 3 get the nput data drectly from the RMS, subtask 2 needs the output of subtask, the servce task s completed when the RMS gets the outputs of both subtasks 2 and 3: φ = φ 3 =, φ 2 = { }, φ 4 = {2,3}. These subtask precedence constrants can be represented by the drected graph n Fg. 3. 7

18 2 4 3 Fg. 3. Subtask executon precedence constrants for analytcal example. Snce φ = φ 3 =, the only realzaton of start tmes T and T 3 s 0 and therefore, U (z)=u 2 (z)=z 0. Accordng to step 2 of the algorthm we can obtan the u-functons representng pmf of completon tmes t, t 2, t 35 and t 36. In order to determne the subtask executon tme dstrbutons for the ndvdual resources, defne the u-functons u j (z) accordng to Table and Eq. (9): u ( z,0) = exp( ) z 00 + [ exp( )] z = 0.779z z. In the smlar way we obtan u 2 ( z,0) = 0.968z z ; u 35 ( z,0) = 0.86z z ; u 36 ( z,0) = 0.98z z. The u-functon representng the pmf of the completon tme for subtask executed by both resources and 2 s U ( ) = u ( z,0) = u ( z,0) z u (,0) mn 2 z = (0.779z z ) (0.968z z ) mn =0.779z z z. The u-functon representng the pmf of the completon tme for subtask 3 executed by both resources 5 and 6 s U ( ) = u 3 ( z,0) = u 35 ( z) 3 z u ( ) mn 36 z = (0.86z z ) (0.98z z ) mn =0.86z z z. Executon of subtask 2 begns mmedately after completon of subtask. Therefore, U 2 (z) = U ( ) =0.779z z z z 8

19 (T 2 has three realzatons 00, 80 and ). The u-functons representng the condtonal pmf of the completon tmes for the subtask 2 executed by ndvdual resources are obtaned as follows u (00 250) (00 250) 23 ( z,00) = e z + [ e ] z =0.9z z ; u (80 250) (80 250) 23 ( z,80) = e z + [ e ] z =0.879z z ; u 23 ( z, ) = z ; u (00 300) (00 300) 24 ( z,00) = e z + [ e ] z =0.726z z ; u (80 300) (80 300) 24 ( z,80) = e z + [ e ] z =0.68z z ; u 24 ( z, ) = z. The u-functons representng the condtonal pmf of subtask 2 completon tme are: u = 2( z,00) u23( z,00) u24( z,00) = (0.9z z ) (0.726z z ) mn mn =0.9z z z ; u = 2( z,80) u23( z,80) u24( z,80) = (0.879z z ) (0.68z z ) mn mn =0.879z z z ; u 2( z, ) = u23( z, ) u24( z, ) = z. Accordng to Eq. (8) the uncondtonal pmf of subtask 2 completon tme s represented by the followng u-functon mn U ( z) = u2( z,00) + u2( z,80) + 0. z =0.779(0.9z z z )+0.24(0.879z z z )+0.007z =0.70z z z z z The servce task s completed when subtasks 2 and 3 return ther outputs to the RMS (whch corresponds to the begnnng of subtask 4). Therefore, the u-functon representng the pmf of the entre servce tme s obtaned as U4( z) = U2( z) U3( z) =(0.70z z z z z ) (0.86z z max max 0.0z )=0.603z z z z z. 9

20 The pmf of the servce tme s: Pr(T 4 = 350) = 0.603; Pr(T 4 = 400) = 0.049; Pr(T 4 = 430) = 0.283; Pr(T 4 = 480) = 0.07; Pr(T 4 = ) = From the obtaned pmf we can calculate the servce relablty usng Eq. (8): R(Θ*) = for 350< Θ* 400; R(Θ*) = for 400< Θ* 430; R(Θ*) = for 430< Θ* 480; R( ) = and the condtonal expected servce tme accordng to Eq. (9): W = ( ) / = sec. 4. Tree Topology Grd Archtecture In the star grd, the RMS s connected wth each resource by one drect communcaton channel (lnk). However, such approxmaton s not accurate enough even though t smplfes the analyss and computaton. For example, several resources located n a same local area network (LAN) can use the same gateway to communcate outsde the network. Therefore, all these resources are not connected wth the RMS through ndependent lnks. The resources are connected to the gateway, whch communcates wth the RMS through one common communcaton channel. Another example s a server that contans several resources (has several processors that can run dfferent applcatons smultaneously, or contans dfferent databases). Such a server communcates wth the RMS through the same lnks. These stuatons cannot be modeled usng only the star topology grd archtecture. In ths secton, we present a more reasonable vrtual structure whch has a tree topology. The root of the tree vrtual structure s the RMS, and the leaves are resources, whle the branches of the tree represent the communcaton channels lnkng the leaves and the root. Some channels are commonly used by multple resources. An example of the tree topology s gven n Fg. 3 n whch four resources (R, R2, R3, R4) are avalable for a servce. The tree structure models the common cause falures n shared communcaton channels. For example, n Fg. 3, the falure n channel L6 makes resources R, R2, and R3 unavalable. Ths type of common cause falure was gnored by the conventonal parallel computng models, and the above star-topology models. For small-area communcaton, such as a LAN 20

21 or a cluster, such assumpton that gnores the common cause falures on communcatons s acceptable because the communcaton tme s neglgble compared to the processng tme. However, for wde-area communcaton, such as the grd system, t s more lkely to have falure on communcaton channels. Therefore, the communcaton tme cannot be neglected. In many cases, the communcaton tme may domnate the processng tme due to the large amount of data transmtted. Therefore, the vrtual tree structure s an adequate model representng the functonng of grd servces. R J, 48s 0.007s - J2, 25s 0.008s - L R2 L2 5Kbps 0.005s - 6Kbps 0.003s - 5Kbps 0.00s - L5 L3 L6 J, 38s 0.004s- J2, 35.5s 0.003s- R4 R3 4Kbps 0.004s - 20Kbps 0.002s - 0Kbps 0.004s - L4 RMS Fg. 3: A vrtual tree structure of a grd servce. 4.. Algorthms for determnng the pmf of the task executon tme Wth the tree-structure, the smple u-functon technque s not applcable because t does not consder the falure correlatons. Thus, new algorthms are requred. Ths secton presents a novel algorthm to evaluate the performance and relablty for the tree-structured grd servce based on the graph theory and the Bayesan approach Mnmal Task Spannng Tree (MTST) The set of all nodes and lnks nvolved n performng a gven task form a task spannng tree. Ths task spannng tree can be consdered to be a combnaton of mnmal task spannng trees (MTST), where each MTST represents a mnmal possble combnaton of avalable elements 2

22 (resources and lnks) that guarantees the successful completon of the entre task. The falure of any element n a MTST leads to the entre task falure. For solvng the graph traversal problem, several classcal algorthms have been suggested, such as Depth-Frst search, Breadth-Frst search, etc. These algorthms can fnd all MTST n an arbtrary graph (Da et al., 2002). However, MTST n graphs wth a tree topology can be found n a much smpler way because each resource has a sngle path to the RMS, and the tree structure s acyclc. After the subtasks have been assgned to correspondng resources, t s easy to fnd all combnatons of resources such that each combnaton contans exactly m resources executng m dfferent subtasks that compose the entre task. Each combnaton determnes exactly one MTST consstng of lnks that belong to paths from the m resources to the RMS. The total number of MTST s equal to the total number of such combnatons N, where N ω (22) = m j = j (see Example 4.2.). Along wth the procedures of searchng all the MTST, one has to determne the correspondng runnng tme and communcaton tme for all the resources and lnks. For any subtask j, and any resource k assgned to execute ths subtask, one has the amount of nput and output data, the bandwdths of lnks, belongng to the correspondng paths γ k, and the resource processng tme. Wth these data, one can obtan the tme of subtask completon (see Example 4.2.2). Some elements of the same MTST can belong to several paths f they are nvolved n data transmsson to several resources. To track the element nvolvement n performng dfferent subtasks and to record the correspondng tmes n whch the element falure causes the falure of a subtask, we create the lsts of two-feld records for each subtask n each MTST. For any MTST S ( N), and any subtask j ( j m), ths lst contans the names of the elements nvolved n performng the subtask j, and the correspondng tme of subtask completon y j (see Example 4.2.3). Note that y j s the condtonal tme of subtask j completon gven only MTST s avalable. 22

23 Note that a MTST completes the entre task f all of ts elements do not fal by the maxmal tme needed to complete subtasks n performng whch they are nvolved. Therefore, when calculatng the element relablty n a gven MTST, one has to use the correspondng record wth maxmal tme pmf of the task executon tme Havng the MTST, and the tmes of ther elements nvolvement n performng dfferent subtasks, one can determne the pmf of the entre servce tme. Frst, we can obtan the condtonal tme of the entre task completon gven only MTST S s avalable as Y = max ( y ) for any N: (23) { } j j m For a set ψ of avalable MTST, the task completon tme s equal to the mnmal task completon tmes among the MTST. Y ψ = mn( Y{ } ) = mn max ( yj ). (24) ψ ψ j m Now, we can sort the MTST n an ncreasng order of ther condtonal task completon tmes Y {}, and dvde them nto dfferent groups contanng MTST wth dentcal condtonal completon tme. Suppose there are K such groups denoted by G,...,, G2 GK where K N, and any group G contans MTST wth dentcal condtonal task completon tmes Θ ( 0 Θ < Θ2 <... < ΘK ). Then, t can be seen that the probablty Q = Pr( Θ = Θ ) can be obtaned as Q = Pr( E,E,E 2,..., E) (25) where E s the event when at least one of MTST from the group G s avalable, and E s the event when none of MTST from the group G s avalable. Suppose the MTST n a group G are arbtrarly ordered, and F j (j=,2,, N ) represents an event when the j-th MTST n the group s avalable. Then, the event E can be 23

24 expressed by N U j= E = F j, (26) and (25) takes the form N Pr( E, E, E 2,..., E) = Pr( U Fj, E, E 2,..., E). (27) Usng the Bayesan theorem on condtonal probablty, we obtan from (27) that j= N Q = ( F ) Pr( F, F,..., F, E, E, E F ) j= Pr L. (28) j ( j ) ( j 2) 2, The probablty Pr ( F j ) can be calculated as a product of the relabltes of all the elements belongng to the j-th MTST from group G. The probablty Pr(, F,..., F, E, E, E F ) followng two-step algorthm (see Example 4.2.4). F ( j ) ( j 2) 2L, j can be computed by the Step : Identfy falures of all the crtcal elements n a perod of tme (defned by the start and end tme), durng whch they lead to the falures of any MTST from groups G m for j m=,2, - (events E ), and any MTST S k from group G for k=,2,, j (events m F k ), but do not affect the MTST S j from group G. Step 2: Ggenerate all the possble combnatons of the dentfed crtcal elements that lead to the event F ( j ), F ( j 2),..., F, E, E2, L, E Fj usng a bnary search, and compute the probabltes of those combnatons. The sum of the probabltes obtaned s equal to Pr (, F,..., F, E, E, E F ) F ( j ) ( j 2) 2L, j. When calculatng the falure probabltes of MTSTs' elements, the maxmal tme from the correspondng records n a lst for the gven MTST should be used. The algorthm for obtanng the probabltes Pr{ E, E2, L E E } can be found n Da et al. (2002). Havng the condtonal task completon tmes Y{ } for dfferent MTST, and the correspondng probabltes Q, one obtans the task completon tme dstrbuton ( Θ, ), K, and can easly calculate the ndces (8) & (9) (see Example 4.2.5). Q 24

25 4.2. Illustratve Example Consder the vrtual grd presented n Fg. 3, and assume that the servce task s dvded nto two subtasks J assgned to resources R & R4, and J2 assgned to resources R2 & R3. J, and J2 requre 50Kbts, and 30Kbts of nput data, respectvely, to be sent from the RMS to the correspondng resource; and 00Kbts, and 60Kbts of output data respectvely to be sent from the resource back to the RMS. The subtask processng tmes for resources, bandwdth of lnks, and falure rates are presented n Fg. 3 next to the correspondng elements The servce MTST The entre graph consttutes the task spannng tree. There exst four possble combnatons of two resources executng both subtasks: {R, R2}, {R, R3}, {R4, R2}, {R4, R3}. The four MTST correspondng to these combnatons are: S : {R, R2, L, L2, L5, L6}; S 2 : {R, R3, L, L3, L5, L6}; S 3 : {R2, R4, L2, L5, L4, L6}; S 4 : {R3, R4, L3, L4, L6} Parameters of MTSTs' paths Havng the MTST, one can obtan the data transmsson speed for each path between the resource, and the RMS (as mnmal bandwdth of lnks belongng to the path); and calculate the data transmsson tmes, and the tmes of subtasks' completon. These parameters are presented n Table 2. For example, resource R (belongng to two MTST S & S 2 ) processes subtask J n 48 seconds. To complete the subtask, t should receve 50Kbts, and return to the RMS 00Kbts of data. The speed of data transmsson between the RMS and R s lmted by the bandwdth of lnk L, and s equal to 5 Kbps. Therefore, the data transmsson tme s 50/5=30 seconds, and the total tme of task completon by R s 30+48=78 seconds. Table 2: Parameters of the MTSTs' paths Elements, subtasks R, J R2, J2 R3, J2 R4, J Data transmsson speed (Kbps) Data transmsson tme (s) Processng tme (s) Tme to subtask completon (s) Lst of MTST elements 25

26 Now one can obtan the lsts of two-feld records for components of the MTST. S : path for J:(R,78); (L,78); (L5,78); (L6,78); path for J2: (R2,40); (L2,40); (L5,40); (L6,40). S 2 : path for J: (R,78), (L,78), (L5,78), (L6,78); path for J2: (R3,58), (L3,58), (L6,58). S 3 : path for J: (R4,53), (L4,53); path for J2: (R2,40), (L2,40), (L5,40), (L6,40). S 4 : path for J: (R4,53), (L4,53); path for J2: (R3,58), (L3,58), (L6,58) pmf of task completon tme The condtonal tmes of the entre task completon by dfferent MTST are Therefore, the MTST compose three groups: Y =78; Y 2 =78; Y 3 =53; Y 4 =58. G = {S 3 } wth Θ = 53; G 2 = {S 4 } wth Θ 2 = 58; and G 3 = {S, S 2 } wth Θ 3 = 78. Accordng to (25), we have for group G : Q =Pr(E )=Pr(S 3 ). The probablty that the MTST S 3 completes the entre task s equal to the product of the probabltes that R4, and L4 do not fal by 53 seconds; and R2, L2, L5, and L6 do not fal by 40 seconds. Pr(Θ=53)=Q =exp( )exp( )exp( ) exp( )exp( )exp( ) = Now we can calculate Q 2 as Q 2 = Pr(, E 2 ) E = Pr ( F ) ( E F ) = Pr( F ) ( F F ) = Pr( S ) ( S S ) 2 Pr 2 2 Pr because G 2, and G have only one MTST each. The probablty that the MTST S 4 completes the entre task PrS ( 4 ) s equal to the product of probabltes that R3, L3, and L6 do not fal by 58 seconds; and R4, and L4 do not fal by 53 seconds. ( ) 4 PrS = exp( ) exp( ) exp( ) exp( ) exp( ) To obtan Pr( S ) = Pr S 3 4, one frst should dentfy the crtcal elements accordng to the algorthm presented n the Da et al. (2002). These elements are R2, L2, and L5. Any falure occurrng n one of these elements by 40 seconds causes falure of S 3, but does not affect S 4. The probablty that at least one falure occurs n the set of crtcal elements s

27 Pr ( S ) S 3 4 = exp( ) exp( ) exp( ) = Then, Pr(Θ =58) = Pr( 2E, ) E = ( ) PrS Pr( S ) 4 S 3 4 = = Now one can calculate Q 3 for the last group G 3 = {S, S 2 } correspondng to Θ 3 = 78 as E = Pr( F ) Pr( E, E F ) + Pr( F ) Pr( F, E E F ) Q 3 = Pr( 3, E2, E) , = Pr( S ) Pr( S, S S ) + Pr( S ) Pr( S, S S S ) , The probablty that the MTST S completes the entre task s equal to the product of the probabltes that R, L, L5, and L6 do not fal by 78 seconds; and R2, and L2 do not fal by 40 seconds. ( S ) Pr = exp( ) exp( ) exp( ) exp( ) exp( ) exp( ) = The probablty that the MTST S 2 completes the entre task s equal to the product of the probabltes that R, L, L5, and L6 do not fal by 78 seconds; and R3, and L3 do not fal by 58 seconds. ( S ) Pr 2 = exp( ) exp( ) exp( ) exp( ) exp( ) exp( ) = To obtan Pr( S S S ) 3, 4, one frst should dentfy the crtcal elements. Any falure of ether R4 or L4 n the tme nterval from 0 to 53 seconds causes falures of both S 3, and S 4 ; but does not affect S. Therefore, ( S S ) PrS 3, 4 = exp( ) exp( ) = The crtcal elements for calculatng Pr(, S S S ) 3, S are R2, and L2 n the nterval from 0 to 40 seconds; and R4, and L4 n the nterval from 0 to 53 seconds. The falure of both elements n any one of the followng four combnatons causes falures of S 3, S 4, and S, but does not affect S 2 :. R2 durng the frst 40 seconds, and R4 durng the frst 53 seconds; 2. R2 durng the frst 40 seconds, and L4 durng the frst 53 seconds; 3. L2 durng the frst 40 seconds, and R4 durng the frst 53 seconds; and

28 4. L2 durng the frst 40 seconds, and L4 durng the frst 53 seconds. Therefore, Pr 4 2 ( S, S3, S4S2 ) = = j= [ exp( λ j tj )] =0.230, where λ j s the falure rate of the j-th crtcal element n the -th combnaton (j=,2), (=,2,3,4); and t j s the duraton of the tme nterval for the correspondng crtcal element. Havng the values of Pr( S ), Pr( S ), Pr( S S S ), and Pr( S, S S S ) 2 3, 4 3, Pr(Θ =78)= Q 3 = = , one can calculate After obtanng Q, Q 2, and Q 3, one can evaluate the total task falure probablty as Pr(Θ = )=-Q -Q 2 -Q 3 = =0.3837, and obtan the pmf of servce tme presented n Table 3. Table 3: pmf of servce tme. Θ Q Θ Q Calculatng the relablty ndces. From Table 3, weone obtans the probablty that the servce does not fal as R ) = Q + Q + Q 0.664, ( 2 3 = the probablty that the servce tme s not greater than a pre-specfed value of θ*=60 seconds as 3 R ( θ*) = Q ( Θ < θ*) = = 0.528, = and the expected servce executon tme gven that the system does not fal as W 3 = Θ Q / R( ) = / = seconds. = 28

29 4.3. Parameterzaton and Montorng In order to obtan the relablty and performance ndces of the grd servce one has to know such model parameters as the falure rates of the vrtual lnks and the vrtual nodes, and bandwdth of the lnks. It s easy to estmate those parameters by mplementng the montorng technology. A montorng system (called Alertmon Network Montor, s beng appled n the IP-grd (Indana Purdue Grd) project ( to detect the component falures, to record servce behavor, to montor the network traffcs and to control the system confguratons. Wth ths montorng system, one can easly obtan the parameters requred by the grd servce relablty model by addng the followng functons n the montorng system: ) Montorng the falures of the components (vrtual lnks and nodes) n the grd servce, and recordng the total executon tme of those components. The falure rates of the components can be smply estmated by the number of falures over the total executon tme. 2) Montorng the real tme network traffc of the nvolved channels (vrtual lnks) n order to obtan the bandwdth of the lnks. To realze the above montorng functons, network sensors are requred. We presented a type of sensors attachng to the components, actng as neurons attachng to the skns. It means the components themselves or adjacent components play the roles of sensors at the same tme when they are workng. Only a lttle computatonal resource n the components s used for accumulatng falures/tme and for dvdng operatons, and only a lttle memory s requred for savng the data (accumulated number of falures, accumulated tme and current bandwdth). The vrtual nodes that have memory and computatonal functon can play the sensng role themselves; f some lnks have no CPU or memory then the adjacent processors or routers can perform ths data collectng operatons. Usng such self-sensng technque avods overloadng of the montorng center even n the grd system contanng numerous components. Agan, t does not affects the servce performance consderably snce only small part of computaton and storage resources s used for the montorng. In addton, such self-sensng technque can also be appled n montorng other measures. 29

30 When evaluatng the grd servce relablty, the RMS automatcally loads the requred parameters from correspondng sensors and calculates the servce relablty and performance accordng to the approaches presented n the prevous sectons. Ths strategy can also be used for mplementng the Autonomc Computng concept. 5. Conclusons Grd computng s a newly developed technology for complex systems wth large-scale resource sharng, wde-area communcaton, and mult-nsttutonal collaboraton. Although the developmental tools and technques for the grd have been wdely studed, grd relablty analyss and modelng are not easy because of ther complexty of combnng varous falures. Ths chapter ntroduced the grd computng technology and analyzed the grd servce relablty and performance under the context of performablty. The chapter then presented models for star-topology grd wth data dependence and tree-structure grd wth falure correlaton. Evaluaton tools and algorthms were presented based on the unversal generatng functon, graph theory, and Bayesan approach. Numercal examples are presented to llustrate the grd modelng and relablty/performance evaluaton procedures and approaches. Future research can extend the models for grd computng to other large-scale dstrbuted computng systems. After analyzng the detals and specfcty of correspondng systems, the approaches and models can be adapted to real condtons. The models are also applcable to wreless network that s more falure prone. Herarchcal models can also be analyzed n whch output of lower level models can be consdered as the nput of the hgher level models. Each level can make use of the proposed models and evaluaton tools. Acknowledgement: Ths work was supported n part by Natonal Scence Foundaton (NSF) under grant number References: Abramson, D., Buyya, R., Gddy, J. (2002), A computatonal economy for grd computng and ts mplementaton n the Nmrod-G resource broker, Future Generaton Computer Systems, vol. 8, no. 8, pp

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