Measuring Fragmenaion of Two-Dimensional Resources Applied o Advance Reservaion Grid Scheduling Julius Gehr, Jörg Schneider Technische Universiae Berlin {jules,komm}@cs.u-berlin.de Absrac Whenever a resource allocaion fails alhough enough free capaciy being available, fragmenaion is easily spoed as cause. Bu how he fragmenaion in a sysem requiring coninuous allocaions like ime schedules or memory can be quanified is hardly analyzed. A Grid environmen using advance reservaion even combines wo-dimensions: ime and resource dimension. In his paper a new way o measure he fragmenaion of a sysem in one dimension is proposed. This measure is hen exended o incorporae also he second dimension. Exensive simulaions showed ha he proposed fragmenaion measure is a good indicaor of he sae of he sysem. 1 Inroducion Fragmenaion is a well known effec in resource allocaion. A reques will be rejeced due o fragmenaion, if i canno be allocaed o a coninuous range of resources while he overall remaining capaciy being sufficien o handle i. Therefore, i is raher easy o idenify fragmenaion as an individual reason for he rejecion of a single allocaion. On he oher hand, describing he general sae of an allocaion sysem by quanifying fragmenaion is a complex ask. Noneheless, such informaion could help o compare he effecs of a scheduling decision. The fragmenaion measure could be used as a forecas how likely fuure allocaions may succeed. Advance reservaion of resources has been idenified as a basic requiremen o guarany qualiy of service in disribued resource managemen sysems like he Grid [1]. Especially, complex Grid jobs composed of inerdependen sub-jobs so-called Grid workflows require he Grid resource managemen sysem o negoiae all sar imes for all sub-jobs o guaranee he successful execuion of he whole job unil he given deadline. Using advance reservaion for muli-uni resources, e.g., parallel compuer wih a number of processors or sorage sysems inroduces a new dimension for fragmenaion. I is no longer sufficien o have a coninuous ime span of a free resource, bu here mus also be a sufficien amoun of free unis on he same resource during he whole ime span. In his work we propose a fragmenaion measure for resource schedules, i.e., he fragmenaion of he ime dimension. This measure is furher exended o model resource allocaion schedules of muli-uni resources. The fragmenaion measure will be derived from former fragmenaion measures and furher discussed using exemplary schedules. To evaluae he resuling fragmenaion measure, he correlaion beween he measured fragmenaion of a schedule and he fuure rejecion rae will be analyzed. In he nex secions he specific applicaion domain Grid workflow scheduling is presened and he various reasons for rejecing an allocaion reques in his seing are discussed. 2 Two-dimensional Resource Allocaion in he Grid Nowadays, mos scieniss have already access o muliple high performance compuers. Sill i is a raher inconvenien ask o selec he righ one o execue a specific ask. The idea of Grid compuing is o provide a single enry poin (Grid broker) for all hese resources, which according o some allocaion sraegy akes care of selecing he resource, reserving i and monioring he execuion. By iner-organizaional cooperaion, such a Grid can grow furher and hereby incorporaing addiionally resource ypes as nework, sorage or even scienific insrumens. As he Grid couples pre-managed and raher big resources, he user ypically doesn book he whole resource bu a small pariion. Usually due o he disribued naure of he Grid, his pariion has o be wihin he same physical resource. Therefore, he Grid broker has o find for each allocaion a single resource wih sufficien free resources. Grid workflows (see Figure 1) are complex Grid jobs consising of a number of iner-dependen asks. In order o guaranee he successful execuion in he specified order,
0 0 0 2 Figure 1: Simple workflow example wih sequence dependency. he Grid broker negoiaes fixed sar imes for each sub-job wih all booked resources. Using so-called advance reservaions herefore increases he search space for a se of valid allocaions. One dimension is he sar ime for each job and he oher is he resource having sufficien free capaciy. Scheduling Grid workflows has a high complexiy. Therefor, advance reservaion Grid broker use ofen a heurisic ha may rejec a workflow even if here is a valid consellaion for allocaing he Grid workflow. However, a workflow may only acceped if a valid consellaion has been definiely found. In he nex secion we furher discuss he various reasons for a reques rejecion. 3 Fragmenaion - Reason for Rejeced Allocaion Requess A complex Grid job like a Grid workflow can be rejeced by a number of reasons. In his secion we discuss all possible reasons for rejecion and how hey could be improved or a leas deeced beforehand. Assuming a new Grid workflow wf1 as depiced in Figure 1 is submied o he Grid broker managing a small Grid wih wo compue resources wih 1 respecively 2 CPUs. The Figures 2 and 3 show for each of he following idenified reasons exemplary schedule-siuaions. 1. Rejecion because of premaure abor of he scheduling uni. The simples cause of rejecion is ha he scheduling uni was no able o find a valid assignmen even hough one exised. As discussed before, his may happen due o sric run-ime requiremens of he heurisic scheduler. In his case, i is he faul of he scheduler and i could be solved by improving he scheduling algorihm or allowing for longer processing ime. 2. Rejecion because of high uilizaion. If here is no vacancy on he resources as in Figure 2a he scheduling uni will rejec he reques. There is nohing one could do abou i and he resource owners are ineresed in high uilizaion anyway since i maximizes heir earnings. 1 (a) Fully uilized resources. 2 3 (d) Fragmened free blocks. 4 5 1 (b) wf1 scheduled for he firs ime. 6 3 1 (c) wf1 and wf1 can be scheduled ogeher. 2 0 4 1 6 3 5 (e) Large free block. Figure 2: Causes of Rejecion 3. Rejecion because of workflow srucure. In Figure 2b wf1 was successfully scheduled for he firs ime, bu i is no possible o assign his workflow a second ime ono he same resources. However, a modified version wf1 which is wf1 bu wih a reversed sequence dependency could be assigned as shown in Figure 2c. Therefore, he inabiliy of assigning wf1 wice does no resul ou of uilizaion since wf1 demands he same quaniy of resources as wf1. However, rejecion because of he inner srucure of workflows canno be measured in a way ha a cerain schedule-siuaion is more or less prone o srucural problems. Therefore, his problem canno be idenified beforehand. 4. Rejecion because of fragmenaion. If free pars of he resources are shaered in space and ime rejecion because of fragmenaion will appear. In he siuaion depiced in Figure 2d, i is eviden ha wf1 canno be scheduled. However, if opimizaion could compac he reservaions in order o form a large free block as in Figure 2e, wf1 will hen be admied. Since Figure 2d and Figure 2e have he same uilizaion i becomes clear ha fragmenaion is a cause of rejecion in is own righ. 5. Rejecion because of unfavorable previous decisions For an example on his cause a simple workflow consising of only one sub-job is considered. If his workflow was assigned as in Figure 3a, he rejecion of wf1 neiher comes from fragmenaion nor from
0 (a) Blocked firs slos. 0 (b) Unblocked firs slos. Figure 3: Rejecion because of previous decision. uilizaion since he uilizaion and fragmenaion are boh very low. The assignmen made in Figure 3a urns ou o be unfavorable, he workflow wf1 could have been admied if he previous assignmen had been as in Figure 3b. This example may sugges ha i is beer o schedule all workflows as lae as possible. This however does no solve he problem, since wf1 may also have demanded slo 1 and 2. The problem raher origins from he inabiliy o foresee he fuure. Usually, he scheduler considers only he requesed Grid workflow. So by changing he reservaions of already admied workflows, he problem could be parially solved a he expense of longer run imes. Since all oher causes of rejecion can be handled or are ineviable wihou changing he Grid seup, we furher analyze he fragmenaion as a way o describe he sae of he Grid. 4 Relaed Work Using advance reservaion is an imporan allocaion sraegy, widely used, e.g., in Grid ool kis such as Globus [3] or VRM [1], as hey provide simple means for planning of resources and in paricular co-allocaions of differen resources. Besides flexibiliy and easy suppor for co-allocaions, e.g., while processing complex workflows, advance reservaions also have oher advanages such as an increased admission probabiliy when reserving sufficienly early, and reliable planning for users and operaors. In conras o he synchronous usage of several differen resources, where also queueing approaches are conceivable, advance reservaions have a paricular advanage when ime-dependen co-allocaion is necessary. Suppor for advance reservaions has been inegraed ino several managemen sysems for disribued and parallel compuing [10]. In [1], advance reservaions have been idenified as essenial for a number of higher level services, such as SLAs. I remains o be examined how fragmenaion can be measured in he special domain of grid resources. The neares soluion would be o reuse ideas of how o measure fragmenaion in oher domains, e.g., in file sysems and fragmenaion in main memory. A deailed performance comparison of he 4.4BSD Logsrucured File Sysem and he 4.4BSD Fas File Sysem (FFS) is presened in [9]. The following hree facors were found o be relevan for conribuing o free space fragmenaion in a FFS: high file urnover, high uilizaion and file sysem age. This was furher analyzed by comparing FFS file sysems of file servers under real workloads wih he performance of empy FFS file sysems over a period of nine monhs. I was implied ha he observed performance degradaion is due o he fragmenaion of he file sysem. However, he acual fragmenaion of he free space blocks in he file sysems as well as he non-coniguiy of files were no quanified. The file sysem approaches OBFS, XFS and yfs were presened in [12, 11, 14]. Archiecures and deails regarding he implemenaion e.g.: usage of logs or B* or B+ rees were discussed and performance sudies were done, however wihou acually quanifying fragmenaion. Sears e.al. [8] deal wih he problem of wheher a binary large objec (BLOB) should be sored in he file sysem or in a daabase. The size of a BLOB was considered as a main facor concerning his quesion and i was sudied how he performance of read and wrie operaions in file sysems and daabases depends on he size of he BLOB. Furhermore, performance degradaion due o fragmenaion was examined by using fragmens per objec as a measure. This approach will be discussed laer. In [4, 13] he characerisics of dynamic memory allocaors were sudied. The sudied allocaors manage he general purpose heap, which can be requesed or freed by a program a any ime. They have o deal wih problems such as finding a free block for saisfying a malloc() reques, choosing one block ou of many possible ones, spliing a block which is larger han he requesed one, coalescing wo or more adjacen freed blocks, demanding more memory from he operaing sysem wih e.g. sbrk() o serve a malloc() reques. I is oulined ha previous sudies wrongfully use synheic races and herefore disregard imporan regulariies in real program s memory usage paerns. This sudy uses a lo of common programs gcc, perl, ec. and wo values were measured over he ime: Firs, live memory as requesed by he program. Second, he memory used by he allocaor o fulfill he requess. Four mehods of how o measure fragmenaion were inroduced, e.g.: The amoun of memory used by he allocaor divided by he amoun of memory requesed by he program, averaged
over all poins in ime. I was saed ha he quesion of wheher a high degree of fragmenaion is presen a a cerain siuaion or no is dependen on he sizes of he following malloc()s. This is due o he fac ha a program which requess only very small porions of memory can be saisfied wih small splinered pieces of memory and herefore he fragmenaion has no impac. Alhough being appropriae for his applicaion, hese kind of measures are raher measuring he wasage of memory which indirecly also serves as a measure of fragmenaion. Furhermore, he domain of memory managemen does no map ono he domain of grid resources very well because here is no mach for sbrk() (which incremens he daa segmen size) wihin he grid domain. In addiion he main memory can be considered as homogeneous which is no rue for grid resources. For he main memory, apar from effecs of localiy, i does no maer wheher objec1 is in cell1 and objec2 is in cell2 or vice versa. However, for grid resources i does maer wheher reservaion1 is in slo1 and reservaion2 is in slo2 since slo2 could be oo lae for reservaion1. Analogous, i is rue for he oher dimension ha i does maer wheher reservaion1 is assigned o resource1 and reservaion2 is assigned o resource2 since resource2 may no be capable of handling reservaion1. Anoher measure of fragmenaion was suggesed in [7] which is also par of he memory allocaor domain: F = MaxHeapSize MaxLiveByes MaxLiveByes. As in he case before concerning grid resources, here exiss no mach for he heap as well. The proposed measure is he memory wasage. I is no appropriae for he geomeric aspec of how much he free blocks have been splinered. Finally, in [5] i was proposed o measure fragmenaion LargesF reeblock as F = 1 AllF reememory which will be discussed laer on. 5 Measuring Fragmenaion In he las secion, approaches already exising for measuring fragmenaion have been sudied and alhough being mosly no applicable in he domain of grid resources, he ineresing ones will be furher discussed. Laer on, a new measuremen of fragmenaion will be proposed. A drawback which is inheren o all approaches sudied so far is heir limiaion o one dimension. However, grid resources have wo dimensions: ime and resource capaciy, e.g., number of CPUs or bandwidh. A procedure which ransfers hese one-dimensional approaches ino wodimensional ones will be inroduced. Table 1 gives an overview on he symbols used in he following secion. Symbol Explanaion b i widh of he i-h booked-block f i widh of he i-h free-block n number of free-blocks m number of booked-blocks f f = {f i 1 i n} f f s average F fragmenaion wih F [0, 1] and 0 for he lowes fragmenaion U uilizaion generaed by a schedule N non-linear uilizaion R raing, used o compare differen schedules p conrols how fas he fragmenaion converges o 1 Table 1: Used symbols 5.1 Fragmenaion in a Single Dimension In order o deermine a measure of fragmenaion, examples wih disinc schedule-siuaions were used. A small subse of hese examples is depiced in Figure 4. These siuaions were raed by using visual judgmen. Siuaions wih one large coniguous chunk of free space and very few and very small addiional spliners were regarded as siuaions wih nearly zero fragmenaion. The esimaed fragmenaion go as worse as he free space go splinered. In he following, a number of formulas were examined wheher hey fulfill hese requiremens. The mos simple approach is he one from [8] which suggess o measure fragmenaion as: F = 1 1 n, wih n being he number of free-blocks. A siuaion as in Figure 4f will be raed wih F = 2 3 whereas i measures F = 0 for Figure 4e. Boh siuaions are nearly idenical bu raed subsanially differen by he proposal. The problem of his measure is ha he sizes of he free blocks are no aken ino accoun. This approach is no suiable because i is reasonable o neglec small free blocks. In conras o he las approach, in Jan Lindblad s proposal for measuring fragmenaion [5] he size of he larges free block is considered: F = 1 max{f} The sizes of he oher blocks are however ignored and i raes differen fragmenaion siuaions as equal, e.g.: in Figure 4b he free space wihou he larges block is shaered ino many small and unusable spliners. Whereas, if he free space is divided in blocks of he same size as he larges free block (as in Figure 4c) he resource would be f i
00 200 300 400 500 (a) Unsusable free blocks: U = 3, F = very high. 00 200 300 400 500 (b) Shifed reservaions o form a usable block: U = 3, F = high. 00 200 300 400 500 (c) Shifed reservaions o form many usable blocks: U =.63, F = lower. 00 200 300 400 500 (d) Three reservaions a inerval end: U = 1 2, F = 0. 00 200 300 400 500 00 200 300 400 500 (e) Three reservaions: U = 1 2, F = 0. (f) Three reservaions: U = 1 2, F 0. 00 200 300 400 500 (g) High uilizaion, very very high fragmenaion. Figure 4: Reserved Timeslos for a single Resource much more uilizable. However his approach raes boh siuaions wih he same fragmenaion F = 5. This approach is also inappropriae. In Figure 4b and in Figure 4a he fragmenaion seems grave and he average free block size is small. In conras o Figure 4c and Figure 4e where he fragmenaion seems o be beer and he average free block size is larger. Therefore, he average free block size can be used o calculae fragmenaion as: F = 1 f This mehod works perfecly for Figure 4e and yields 0. Figure 4f conaining free blocks 4, 4, 242 is raed as 2 3. However, his approach is mahemaically idenical o he firs one since he arihmeic mean is f = 1 n n f i. The arihmeic mean may no be appropriae anyhow, since i is very fragile o ouliers. Therefore, i is reasonable o consider oher averages no so sensiive o ouliers. Anoher average, well known for is immuniy o ouliers, is he median which can be found by soring he values and choosing he one in he middle. In Figure 4f he observed values are (4, 4, 250) and herefore he median is 4. The median has idenified he 250 as being an oulier which migh be reasonable in some cases bu no in his one. Acually, boh values of 4 are ouliers whereas he large value is no. In order o boos he influence of he large values, hey can be raised o he power of a number greaer han 1. f i This leads o he new formula o measure fragmenaion: f p i F = 1 ( n ) p f i Several consequences arise for his kind of fragmenaion measuremen. Firs, i holds n f p i ( n f i) p. From his i convenienly follows ha F [0, 1). Furhermore, wo addiional saemens can be made abou his way of fragmenaion measuring. Firs, F = 0 if n = 1. Second, F 1 if n and blocks being sized equally. In he case of f 1 = f 2 =... = f n he fragmenaion formula can be simplified. f p i F = 1 ( n ) p = 1 ( n ) p f i f 1 = 1 nf p 1 (nf 1 ) p = 1 nf p 1 n p f p 1 = 1 1 n p 1 I converges as fas as large p ges. For p = 2 and equally sized fragmens he measure ransforms o F = 1 1 n which is he approach from [8]. E.g., for n = 1, 2, 3,..., 100 fragmenaion becomes F = 0, 1 2, 2 3,..., 99 100. This converging f p 1
Lising 1: 2D fragmenaion measure f l o a c a l c 2 D f r a g m e n a i o n ( ) { f l o a f r a g = 0. 0 ; i n num = 0 ; f l o a n e w f r a g = 0. 0 ; f o r ( i n demand = 1 o MAX DEMAND) { n e w f r a g = c a l c 1 D f r a g m e n a i o n ( a g o c c u p i e d v a c a n ( demand ) ) ; } i f ( n e w f r a g!= 0) { f r a g += n e w f r a g ; ++num ; } } reurn f r a g / num ; Figure 5: The simulaed Grid wih 8 compue resources. behavior for growing values of n is essenial for fragmenaion measures and naurally i is also fulfilled by F = 1 1 n. However, he insensiiviy o small ouliers has o be esed addiionally. Using he example from Figure 4f, for p = 2 he fragmenaion equals: F = 1 42 + 4 2 + 250 2 (4 + 4 + 250) 2 = 1 62532 66564 6 which is obviously beer han 2 3 yielded by F = 1 1 n. This approach has he necessary propery of being resisan o small negligible fragmens as long as one large fragmen exiss. As in he discussion before, in he following analysis p = 2 will be used. 5.2 Adding he Second Dimension Even hough a promising candidae has been found, i sill lacks he abiliy o measure fragmenaion in wo dimensions, i.e., for a ime schedule of muli-uni resources. I regards ime slos as occupied or vacan bu i does no deal wih he amoun of free resources (e.g. 1 cpu or 10 cpus) wihin he slos. However, his binary assumpion maches he poin of view of he scheduler while scheduling a new ask. All ime slos become eiher occupied or vacan for ha specific ask according o is demands on he resource. Thus, for a specific demand of resources he 1D measure can be applied. And in order ge a 2D measure he average over all possible demands of his resource can be calculaed. In Lising 1 such a calculaion is shown. This approach can be furher improved by using he relaive frequency of occurrences for every demand as average weighs. 6 Evaluaion As discussed in Secion 3, he rejecion raio should be highly influenced by he measured fragmenaion. To evaluae he proposed fragmenaion measure he rejecion rae was compared beween siuaions where he measured fragmenaion differed, while all oher parameers have been consan. Because, he acual disribuion of he job sizes, he job duraions ec. does no impac he general qualiy of he resuls, even when using simple models [6], he simulaions were made using a synheic workflow model. The modeled Grid (see Figure 5) consised of eigh compue resources wih 512, 256, 256, 128, 128, 96, 32, and 32 CPUs. As only he fragmenaion in he compue resources was analyzed, all compuers were conneced o a nework wihou any bandwidh or laency resricions. The simulaion was caried ou wih he simulaion mode of he VRM Grid broker [1] using he Grid workflow scheduler described in [2]. 6.1 Workload Consrucion A firs, more han 70,000 independen workloads have been creaed, each of hem conaining 10,000 workflows. Each workflow is assumed o be reserved in advance wih an inerarrival ime disribued normally wih parameers µ = 100 and σ = 1000. The decision agains he exponenial disribuion was made because he exponenial disribuion lacks he abiliy of separaely choosing variance and mean. In his simulaion seup, here is no way of conrolling he simulaions in such a way ha specific saes of uilizaion and fragmenaion can be achieved. However, a
3 probabiliy 2 1 uilizaion fragmenaion (a) Spaial probabiliy densiy in dependence from fragmenaion and uilizaion. rejecion uilizaion fragmenaion (b) Rejecion rae in dependence from fragmenaion and uilizaion. Figure 6: Impac of fragmenaion and uilizaion. rejecion rae 0..0 rejecion rae 0 0.30 0 uilizaion: 0.1 0.15 0 0.10 0 0.30 fragmenaion uilizaion: 0.5 0.55 0 0.30 0 0.50 fragmenaion rejecion rae 2 6 0.10 rejecion rae 0 0.70 0 uilizaion: 0.3 0.35 0.10 0 0.30 0 fragmenaion uilizaion: 0.7 0.75 fragmenaion Figure 7: Impac of fragmenaion on he rejecion rae. high sandard deviaion had he consequence ha he simulaions covered a sufficienly large area in he uilizaionfragmenaion-domain. The book-ahead ime was normally disribued, oo, wih parameers µ = 150 and σ = 300. Each workflow consiss of a number of asks which are uniformly disribued over {4, 5,..., 9, 10}. Task duraion and requesed processors were also uniformly disribued over he inerval [250, 750] and {2, 4, 8, 16,..., 256}. The deadline of he workflow is he sum of all ask duraions and a normally disribued random value wih µ = 0 and σ = 300. The creaion uni of he synheic workflows ries o associae each ask wih zero o hree dependencies uniformly. These dependencies are composed of 66% sequenial dependencies and 33% synchronous dependencies. Creaing a new dependency fails if i creaes a loop wihin he workflow. A mos, 20 assignmen rials per dependency are done. 6.2 Workload Processing Each workload was processed by he simulaed grid and i has been raced if a disinc workflow could successfully be assigned o he grid. Before beginning a scheduling process, uilizaion and fragmenaion were also recorded. Each simulaion run has produced a lis of (rejecion, U, F ) samples wih rejecion {0, 1} and U [0, 1] and F [0, 1). Unforunaely, hese riples are no independen, e.g., if a workflow has been rejeced he successor is likely o be rejeced, oo. The closer heir arrival imes are he more likely is he rejecion of he successor since he probabiliy is high ha he reason remains which caused he firs rejecion. Furhermore, he farher he arrival imes are disribued over ime, he smaller is he auocorrelaion of he samples. In order o collec independen daa and o keep mahemaics simple, mos samples were discarded. For each independen run, he firs hundred samples were discarded which skips he iniial ransien phase. Afer each used sample, a number of samples were discarded in order o ge independen daa. These numbers were chosen randomly ou of {100, 101,..., 200}. Random values were used o circumven possible exising periodic behavior. The samples were divided ino differen caegories. An disinc sample (rejecion, U, F ) belongs o a disinc caegory S u,f wih u, f {0, 5, 0.1,..., 0.9, 0.95} if and only if u U < u + 5 f F < f + 5. The esimaed rejecion rae for a disinc caegory S u,f equals: i.rejecion i S u,f r u,f = S u,f 6.3 Simulaion Resuls Afer exensive simulaions have been done, he impac of he new fragmenaion measure on he rejecion rae was examined. Firs, i has o be menioned ha he sysem does no cover all possible S f,u caegories. In Figure 6a, he sae probabiliy of each caegory is depiced. I can be observed ha he uilizaion covers nearly he whole domain from 0% o 90% while he fragmenaion only covers he area from 0% only o 60%. Due o he sloed ime, free blocks have a leas he size of a slo and he number of free slos wihin an inerval is bounded. Hence, he fragmenaion is bounded, oo. Considering he slo size he search inerval size and he minimal ask size, he boundary caused by he sloed ime is approximaely 95%. Thus, here mus be anoher reason for he fragmenaion being bounded. The used scheduler[2] ends o schedule all asks as early as possible. This approach seems o avoid severe fragmenaion. Furher invesigaions is planned as fuure work. In Figure 6b, he rejecion rae is depiced. I can be clearly observed ha he rejecion rae grows wih higher uilizaion or wih higher fragmenaion. In order o clarify
he impac of fragmenaion on he rejecion rae, in Figure 7 he rejecion rae is ploed in dependency on fragmenaion for some fixed values of uilizaion. A confidence inerval for he rejecion rae is depiced as doed line for a confidence of 99.993%. I can be clearly observed ha for very low uilizaion, fragmenaion remained quie low and no rejecion appeared. For more realisic scenarios, he uilizaion is beween medium and high. In he case of 50% uilizaion, he observed fragmenaion is beween 20% and 50%. The rejecion rae is beween 18% and 40% yielding a difference of more hen 20%. Thus he fragmenaion as measured by he new proposed measure has a huge impac on he rejecion rae. 7 Conclusion and Oulook In his paper a new fragmenaion measure for judging he sae of one-dimensional allocaion sysems like ime schedules was proposed. This measure was furher exended o also cope wih muli-uni resources over ime, hus handling boh dimensions wihin a single measure. Exensive simulaions showed, ha he proposed fragmenaion measure reflecs he sae of he allocaion sysem very well. Fuure work will deal wih furher analyzing he behavior of he new fragmenaion measure, i.e., wih oher values of p or in he conex of oher wo-dimensional resource allocaion sysems. The nex sep would be o enhance he scheduler o avoid allocaions leading o a high fragmenaion. References [6] V. Lo, J. Mache, and K. Windisch. A Comparaive Sudy of Real Workload Traces and Synheic Workload Models for Parallel Job Scheduling. In 4h Workshop on Job Scheduling Sraegies for Parallel Processing, Orlando, USA, volume 1459 of Lecure Noes in Compuer Science (LNCS), pages 25 46. Springer, 1998. [7] M. S. Neely. An analysis of he effecs of memory allocaion policy on sorage fragmenaion. Maser s hesis, Universiy of Texas a Ausin, 1996. [8] R. Sears, C. V. Ingen, and J. Gray. To BLOB or no o BLOB: Large objec sorage in a daabase or a filesysem? Technical Repor MSR-TR-2006-45, Microsof Research (MSR), Apr. 2006. [9] M. Selzer, K. A. Smih, H. Balakrishnan, J. Chang, S. Mc- Mains, and V. Padmanabhan. File sysem logging versus clusering: A performance comparison. In Proceedings of he USENIX 1995 Technical Conference, pages 249 264, New Orleans, LA, USA, Jan. 16 2995. [10] D. Snell, M. Clemen, D. Jackson, and C. Gregory. The Performance Impac of Advance Reservaion Mea-scheduling. In 6h Workshop on Job Scheduling Sraegies for Parallel Processing, Cancun, Mexiko, volume 1911 of Lecure Noes in Compuer Science (LNCS), pages 137 153. Springer, 2000. [11] A. Sweeney. Scalabiliy in he XFS file sysem. In USENIX Annual Technical Conference, pages 1 14, 1996. [12] F. Wang, S. A. Br, E. L. Miller, and D. D. E. Long. OBFS: A file sysem for objec-based sorage devices, May 04 2004. [13] P. R. Wilson, M. S. Johnsone, M. Neely, and D. Boles. Dynamic sorage allocaion: A survey and criical review. Lecure Noes in Compuer Science, 986:1??, 1995. [14] Z. Zhang and K. Ghose. yfs: A journaling file sysem design for handling large daa ses wih reduced seeking. In FAST. USENIX, 2003. [1] L.-O. Burchard, M. Hovesad, O. Kao, A. Keller, and B. Linner. The virual resource manager: An archiecure for SLA-aware resource managemen. In 4h Inl. IEEE/ACM Inl. Symposium on Cluser Compuing and he Grid (CCGrid) 2004, Chicago, USA, 2004. [2] J. Decker and J. Schneider. Heurisic scheduling of grid workflows supporing co-allocaion and advance reservaion. In B. Schulz, R. Buyya, P. Navaux, W. Cirne, and V. Rebello, ediors, 7h Inl. IEEE Inl. Symposium on Cluser Compuing and he Grid (CCGrid07), pages 335 342, Rio de Janeiro, Brazil, May 2007. IEEE CS Press. [3] I. Foser, C. Kesselman, C. Lee, R. Lindell, K. Nahrsed, and A. Roy. A Disribued Resource Managemen Archiecure ha Suppors Advance Reservaions and Co- Allocaion. In 7h Inernaional Workshop on Qualiy of Service (IWQoS), London, UK, pages 27 36, 1999. [4] M. S. Johnsone and P. R. Wilson. The memory fragmenaion problem: Solved? SPNOTICES: ACM SIGPLAN Noices, 34, 1999. [5] J. Lindblad. Handling memory fragmenaion. hp: //www.ednasia.com/aricle.asp?id=182 (visied December 1h 2006).