Delay-Optimized File Retrieval under LT-Based Cloud Storage

Transcription

1 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 1 Delay-Optimized File Retieval unde LT-Based Cloud Stoage Haifeng Lu,Chuan Heng Foh, Senio Membe, IEEE, Yonggang Wen, Membe, IEEE, and Jianfei Cai, Senio Membe, IEEE Abstact Fountain-code based cloud stoage system povides eliable online stoage solution though placing unlabeled content blocks into multiple stoage nodes. Luby Tansfom (LT) code is one of the popula fountain codes fo stoage systems due to its efficient ecovey. Howeve, to ensue high success decoding of fountain codes based stoage, etieval of additional fagments is equied, and this equiement could intoduce additional delay. In this pape, we show that multiple stage etieval of fagments is effective to educe the fileetieval delay. We fist develop a delay model fo vaious multiple stage etieval schemes applicable to ou consideed system. With the developed model, we study optimal etieval schemes given equiements on success decodability. Ou numeical esults suggest a fundamental tadeoff between the file-etieval delay and the taget pobability of successful file decoding, and that the file-etieval delay can be significantly educed by optimally scheduling packet equests in a multi-stage fashion. Index Tems Delay-optimal etieval, LT codes, Cloud stoage. 1 INTRODUCTION Cloud stoage systems povide a scalable online stoage solution to end uses who equie flexible amount of stoage space but do not wish to own and maintain stoage infastuctue [1], []. Compaed with taditional data stoage, cloud stoage has seveal advantages. Fo example, end uses can access thei data anywhee though Intenet without botheing about caying physical stoage media. Also, diffeent uses can collaboatively contibute to the data stoed in cloud stoage with pemission fom the data owne. Due to its high populaity in industy, cloud stoage has been a hot topic in cloud computing community []. Geneally, cloud stoage systems ely on thousands of stoage nodes. A content is often fagmented and distibuted into a set of stoage nodes. To offe high eliability and availability of stoage sevices, edundancy of contents may be employed. Fagments of Haifeng Lu is with Alibaba Inc., China ( haifeng.lhf@alibabainc.com). Chuan Heng Foh is with the Institute fo Communication Systems Reseach, Univesity of Suey, Guildfod, Suey, GU XH, UK ( c.foh@suey.ac.uk). Yonggang Wen and Jianfei Cai ae with the School of Compute Engineeing, Nanyang Technological Univesity, 998, Singapoe ( ygwen@ntu.edu.sg, asjfcai@ntu.edu.sg). contents may be simply eplicated and stoed in diffeent stoage nodes to achieve edundancy. Othe than naive eplication, Lin et al. [4] poposed two QoSawae data eplication algoithms to educe stoage cost while maintaining QoS fo the applications. In [5], Weathespoon et al. analyzed two schemes fo edundancy: eplication and easue coding, concluding that easue-coded systems can povide highe availability with lowe bandwidth and less stoage space. Since then, thee ae plenty of woks on designing easue codes fo stoage systems, and they mainly focus on the eliability and availability of stoage systems [], [], [8], [9]. Recently, motivated by the geat success of ateless codes o fountain codes, which have vey low decoding complexity and can geneate infinite numbe of encoded packets, some woks [10], [11] have applied the popula ateless code o LT code, into cloud stoage systems and have achieved pomising pefomance. The main advantage fo a ateless code based cloud stoage system is that it significantly simplifies the challenging content placement and content ecovey poblems that need to be addessed in easue code based systems. This is because a ateless code based system can potentially geneate infinite numbe of encoded packets to be placed acoss the stoage system and to eplace those unavailable packets due to node failue. Howeve, the disadvantage of a ateless code based cloud stoage system is that it incus longe data etieval delay, since it equies moe coded packets due to its uncetainty in decoding diffeent fom its easue code countepat which is deteministic in decoding. Theefoe, in this pape we focus on designing an optimized etieval method fo an LT code based cloud system. As stated in [1], taffic hot spots emege at the coe of a cloud stoage system which is esponsible fo communications between the system and end uses. We call this coe as a potal. In this eseach, we focus on addessing the poblem how to minimize the etieval delay geneated by the potal in an LT-based cloud stoage system. In paticula, unlike the Maximum Distance Sepa-

2 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING able (MDS) codes such as Reed-Soloman code [1], which need deteministic numbe of edundant encoded packets fo decoding, a stoage collecto opeating LT decode equies an uncetain numbe of encoded fagments fom the stoage system fo successful decoding. These uncetain numbe of encoded packets add delay in stoage etieval. Thus, a tadeoff between the successful decoding pobability and stoage etieval delay exists. In this pape, we study such a tadeoff and popose a multiple-stage stoage etieval scheme whee we show that stoage etieval delay can be educed without sacificing the pefomance on decoding pobability. We futhe demonstate the optimal etieval setup fo the case of a two-stage etieval. The contibutions of this pape ae twofold. Fistly, we develop a model descibing the multi-stage equest scheme fo an LT code based stoage system. Secondly, we solve the optimization poblem fo the optimal two-stage equest scheme and povide a closed-fom expession fo analysis. The emainde of this pape is oganized as follows. In Section, we biefly eview the elated wok. In Section, we intoduce the system achitectue and pesent the fomal definition of the optimal etieval delay poblem. A detailed desciption of LT codes and its decodability is pesented in Section 4. Section 5 pesents a igoous analysis on the delay fo diffeent etieval schemes togethe with simulation validation. Section povides analysis on the optimal two-stage equest scheme. We then demonstate the tadeoff between the delay and the pobability of successful decoding in Section. We also show some simulation esults using diffeent taffic models in Section 8. Finally, some impotant conclusions ae dawn in Section 9. RELATED WORK Easue code based cloud stoage systems: An example of a successful deployment of commecial cloud stoage system using easue coding is Symfom [14], [15]. In Symfom, Reed-Solomon code [1] is used fo edundancy geneation. Its success shows easuecode-based cloud stoage systems ae pactical and compaable with taditional eplication-based cloud stoage systems. Anothe eseach effot focuses on the design of stoage system opeational optimization based on existing easue codes to achieve some specific optimal pefomance matices. These opeational designs may include stoage edundancy ovehead optimization, stoage allocation and epai, and othes [1], [1], [18], [19]. As afoementioned, compaed with easue code based cloud stoage systems, LT code based systems simplify content placement and ecovey poblems at the cost of longe etieval delay. LT code: Luby Tansfom (LT) code [0] enjoys elatively low computational complexity of O(kln(k/δ)) fo ecoveing k symbols with O( kln (k/δ)) additional packets. Hee, 1 δ is the pobability of successful ecovey of all k symbols. The popety of low decoding complexity makes LT code suitable fo end uses equipped with modeate CPU. Anothe popety of LT code is ateless which means the edundancy atio of data can be abitaily changed without additional design. This popety is natually fit with cloud stoage systems with thousands of stoage nodes stoing vaious type of data. The pomising pefomance of LT code based stoage systems has been shown in [10], [11]. Data etieval pefomance in stoage systems: In addition to the eliability and availability issues of a stoage system, anothe impotant poblem aised in a stoage system is data etieval pefomance. Taditional distibuted file systems like NFS [1] and Spite [] ae single seve based and achieve acceptable etieval pefomance though file caching. Howeve, the etieval pefomance is bounded by the speed of the single stoage seve. Convesely, paallel file systems, such as Vesta [], Galley [4], PVFS [5] and GPFS [], allow data speading acoss multiple stoage nodes and povide paallel access to it. The high tansfe ate is eached by vaious techniques such as I/O scheduling, content pefetching, etc. Moe ecently, high etieval pefomance is achieved by intoducing fast stoage devices like solid-state dives (SSD) [] and DRAM [8]. Unlike these existing wok dealing with the etieval pefomance poblem fom the application level, we conside this poblem fom the netwok level since in an LT code based stoage system the pobabilistic LT decoding could esult in lage etieval delay due to decoding failue. In [9], [0], [1], [], the authos discussed some mechanisms fo educing data fetching latency in distibuted netwoks. Those mechanisms ae designed fo geneal content without consideing any coding metics, and they could be applied togethe with ou method. SYSTEM MODEL AND PROBLEM STATE- MENT In this section, we fist pesent a schematic desciption of a cloud stoage achitectue, in which LT-encoded data packets ae spead out acoss a pool of stoage nodes fo high availability. Following that, we fomulate a delay-optimal file-etieval poblem, which aims to minimize the etieval delay by stategically scheduling packet etieval equests. The notations used in this section ae summaized in Table 1..1 System Achitectue In this pape, we focus on a distibuted cloud stoage system (e.g., a community-based cloud stoage system as in Tahoe-LAFS []). As illustated in Figue 1, the cloud stoage system manages a set of stoage

3 This aticle has been accepted fo publication in a futue issue of this jounal, but has not been fully edited. Content may change pio to final publication. Citation infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 1 TABLE 1 Notations Notation k n λa l, σ θ llt ti s ni Desciption No. of oiginal symbols. No. of LT encoded symbols fo cetain taget decoding pobability. Aival ate fo ambient taffic. Mean and vaiance of length of ambient taffic packets. Aival ate fo LT taffic. Length of LT encoded symbols. Aival time fo ith LT encoded symbol. No. of stages. No. of LT encoded symbols equested in stage i. t0 nodes though a communication netwok. The uses can access to the stoage sevice though a dedicated potal. One paticula stoage application of inteest in this eseach is the file stoage. We assume that the stoage sevice povides both pemium LT-based file stoage and egula file stoage. Fo the LT-based file stoage, the potal, upon eceiving a file fom a use equest, encodes the file with a chosen LT code into a set of LT fagments o packets1 and speads them acoss diffeent nodes fo eliability and high availability. Upon eceiving a fileetieval equest, the potal sends a few packet etieval equests to diffeent stoage nodes, and gathes sufficient encoded packets to be fowaded to the use. Fo the egula file stoage, the potal simply stoes the eceived use file into a chosen stoage node, with some potential eplication in anothe set of nodes. We assume that the numbe of subscibes fo the pemium LT-based stoage sevice is much smalle than that fo the egula stoage sevice. As such, it is with high pobability that, at one instant, only one use will etieve his/he LT-coded files and the majoity of the file-etieval equests ae fo the egula stoage sevice. In this pape, we will analyze such a 1. Hee we use the tem packet intechangeably with fagment as we assume that an IP packet caies an LT encoded fagment in the system. t1... ti-1... Receive LT 1st LT packet packet equest aival ti... tn time... i-1-th LT i-th LT packet n-th LT packet packet aival aival aival Fig.. Aival pocess of LT encoded packets seen at the potal fo a file-etieval equest. simplified use case. The insights obtained though the mathematical famewok and numeical analysis will shed lights on moe pactical application scenaios.. Fig. 1. System achitectue fo a cloud stoage system. i Taffic Model In this achitectue, the potal node obseves two types of packets. The fist one is packets fo the egula stoage sevice, consideed as a pocess of ambient taffic. The ambient taffic follows a paticula aival pocess with an aival ate of λa. Moeove, the length of ambient packets is modeled as a andom vaiable with mean l and vaiance σ. The second one is the LT-coded packets fo the pemium stoage sevice. We assume the aival delay of each LT-coded packet is identically distibuted, each of which is a cetain andom vaiable with a mean value of θ 1. The length of LT-coded packets is assumed to be a constant, denoted by llt. When the potal equests n LT-coded packets fom the stoage gid, the packet aival pocess is illustated in Figue. We assume that the packet equest is sent out at time 0 (i.e., t0 = 0), and denote the aival time of the i-th LT-coded packet as ti. We also denote the inte-aival time between the i 1-th packet and the i-th packet as τi = ti ti 1.. Poblem Statement In the consideed distibuted stoage system, the bottleneck lies on the potal, because a lage numbe of uses can issue thei file equests to the potal. In this pape, we focus on the file-etieval delay, defined as the duation between the time fo the potal eceiving an LT-coded file equest and the time when the last LT-coded packet is sent out by the potal. The fileetieval delay is a good indicato of use expeience. Theefoe, we aim to educe the file-etieval delay by stategically scheduling the LT-coded packet equests. The delay-optimal file-etieval poblem is stated as follows. We assume a pobabilistic file-etieval model. In ode to achieve the pobability of successful decoding of p, the potal needs to equest n LT encoded packets fom the stoage gid. Taditionally, the n packets ae equested in one shot. This scheme is efeed to as a one-stage equest scheme. In this pape, we popose to use a multiple-stage equest scheme to impove the file-etieval pefomance. Specifically, the use can divide the equest into s stages, each P consisting of equests fo n1, n,, ns packets ( i ni = n).

4 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 4 If the use successfully decodes the file afte stage m, it will stop equesting the est s i=m+1 n i packets. In ou poposed multi-stage equest scheme, the designing objective is to minimize the aveage fileetieval delay, fo a given numbe of stages. Mathematically, it can be fomulated as the following optimization poblem, min D s (n 1,n,,n s ) (1) s s.t. n i = n, () i=1 n i 0,1 i s. () Hee D s ( ) denotes the aveage etieval delay unde a given equest scheme. Intuitively, multiple stages of packet etieval incus additional delay in stoage etieval. Howeve, we will show that the multi-stage etieval scheme can outpefom the single-stage etieval scheme in tems of the aveage file-etieval delay. This delay eduction oiginates fom the following obsevation. In the multi-stage etieval scheme, just etieving an adequate numbe of packets may aleady give sufficiently high success decodability, eliminating the need fo futhe ounds of packet etieval. Only when the file cannot be decoded with the set of eceived packets, the additional packet equests ae sent in subsequent stages. In such a case, the single-stage etieval scheme is a special case of ou poposed multi-stage etieval scheme, when s = 1. Fom an optimization pespective, the pefomance of the multistage etieval scheme cannot be wose than that of the single-stage scheme. 4 LT CODES AND ITS DECODABILITY In this section, we pesent a detailed desciption about LT-codes, including its encoding and decoding pocesses, and the pobability of decoding files fo a given numbe of etieved packets. 4.1 LT Encode/Decode LT codes ae the fist class of digital fountain codes, with nealy optimal easue coecting capability. Its main chaacteistic lies in employing a simple algoithm based on the exclusive-o opeation to encode and decode the message. As such, it is paticulaly appealing to cloud-based file stoage. In the LT-based cloud stoage system, a file is fist boken into k souce symbols, and then these k souce symbols ae encoded into n packets and spead acoss a pool of stoage nodes. The numbe of encoded packets n can be abitay lage depending on the duplication atio of the system. In this subsection, we biefly explain its encoding and decoding pocesses. The encoding and decoding pocesses of LT codes can be illustated via an example, as shown in Figue. S1 S S S a P1 P P P4 P5 c e g S1 1 S S S1 1 S S b d f h S1 1 S S S1 1 S S Fig.. An example of LT encoding and decoding pocedues fo k = 4 and n = 5.(a) LT encoding: encode 4 souce symbols,s 1 -S 4, into 5 packets,p 1 -P 5. (b)-(h) LT decoding to ecove the 4 souce symbols. An LT encode follows a thee-step pocess to geneate one encoded packet. Fist, a degee d [1, k] is andomly chosen fom a degee distibution of Ω d. Second, d distinct souce symbols ae chosen, unifomly at andom, fom the set of souce symbols. Finally, the encoded packet is equal to the bitwise sum, modulo, of those d chosen souce symbols. The infomation about the set of chosen souce symbols is called coding vecto and it is ecoded in the head of the encoded packet. The same pocess epeats until n encoded packets ae geneated. In Figue, Panel (a) demonstates an example of LT encoding pocedue with 4 binay souce symbols (i.e. k = 4), denoted by S 1,S,,S 4, and 5 LT encoded packets (i.e. n = 5), denoted by P 1,P,,P 5 with value The LT decoding pocess employs a simple message passing algoithm [4], which iteatively decodes souce symbols fom degee-1 encoded packets. Conside the same example illustated in Figue. At the fist iteation, the only encoded packets with degee equal to 1 is P (efe to Panel (a)). Thus, the souce symbol S which is connected to P is set to 1 (efe to Panel (b)). Afte S is ecoveed, the encoded packets which have connections with S ae extacted by the value of S and the connections ae emoved (efe to Panel (c)). Afte this step, a new symbol S 4 can be ecoveed and the same decoding pocedue is executed iteatively until no moe degee-1 encoded packets can be found. If thee ae still some encoded packets which ae not ecoveed, a decoding failue is epoted. In ode to ecove the est of souce symbols, one can equest moe encoded packets. Clealy, the LT decoding pocess cannot guaantee to ecove

5 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 5 Successful pobability k=100 k=00 k=00 k=400 k=1000 simulation fitting n/k Fig. 4. Fitting esults vesus simulation esults. all the souce symbols. In this example, if P 5 is missing, the decoding pocess will be teminated at Panel (b) since no moe degee-1 encoded packets can be found afte ecoveing S. 4. LT Decodability As shown in the example in Section 4.1, the LT decoding pocess is pobabilistic. Specifically, given a set of packets with a full-ank coding matix, the decoding pocess could still fail with a cetain pobability. Moeove, the successful pobability of decoding inceases as the numbe of eceived packets n inceases. In this case, LT decodability, denoted by f k (n), is defined as the pobability of successful decoding k souce symbols fom a set of n encoded packets (n k). Existing eseach on chaacteizing the LT decodability has not esulted in a closed-fom solution yet. In [5], the authos poposed a dynamic pogamming algoithm to calculate the LT decodability, with a computing complexity of O(n log n). Anothe algoithm in [] educes the computing complexity to O(k logk). Howeve, none of these pevious eseach effots esulted in a closed-fom solution to the LT decodability, while elying on numeical evaluation using dynamic pogamming. TABLE Fitting esults fo α and β k α β SSE In this pape, we deive an empiical model fo the LT decodability, by leveaging the numeical evaluation techniques fom []. Note that, in an LT-based cloud stoage system, the decodability unde 50% is of less engineeing inteest. As a esult, we focus on the decodability ove 50% in the est of this pape. We use a two-step pocedue to deive the empiical model fo the LT decodability, as follows: Fistly, we plot the LT decodability as a function of the numbe of encoded packet, by using the numeical technique suggested in []. Secondly, we use a geneic function of f k (n) = 1 e α(n k β) to cuve fit the numeical esults, whee α and β ae two fitting paametes elated to k. We illustate this appoach in Figue 4. Fist, we obtain the simulated LT decodability fo the cases of k = 100, k = 00, k = 00, k = 400 and k = 1000 as illustated with the solid cuves. Second, using MATLAB s Cuve Fitting Tool with a specific fitting function of f k (n) = 1 e α(n k β), we obtain the fitting esults of α and β togethe with the sum of squae eos (SSE) in Table. Finally, we compae the LT decodability between the simulation esults and the fitting esults (the dotted cuves) fo k = 100, 00, 00, 400, and 1000, in Figue 4. It can be obseved that the empiical model captues the simulation esults with a decent accuacy. In the est of this pape, we will use this empiical model of the LT decodability as a basis fo ou optimization famewok. 5 FILE-RETRIEVAL DELAY ANALYSIS In this section, we chaacteize the aveage fileetieval delay fo diffeent equest schemes in the pesence of ambient taffic. To validate ou analytical esults, we un compute simulation witten in Java using NetBeans.1. All the simulation esults ae obtained by aveaging 1000 simulation instances. In simulations, we set that the length of packets in the ambient taffic follows an exponential distibution. Although it is a simplified model, the insights obtained with this simple model can be applied to guide pactical system design. Table lists the default values of some paametes in the simulato. TABLE Default values in Notation Physical Meaning Value l Length of packet fo ambient taffic 104 Bytes l LT Length of LT-coded packet 104 Bytes Tansmission ate 800 kbps 5.1 Delay Analysis fo One Stage Request Scheme We fist investigate the file-etieval delay D 1 (n) fo n LT packets in the one-stage equest case. This delay consists of two pats as shown in Figue 5(a). The fist pat aises fom the taffic aiving befoe the LT equest and the second pat aises fom the taffic aiving afte the LT equest. In this analysis, we assume Poisson aival pocess fo ambient taffic and LT-coded packets. The fist pat only contains ambient taffic and it is teated as a classical M/G/1 queue. Moe infomation on M/G/1 queue can be found in [].

6 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING Classical M/G/1 queuing delay Ambient taffic 1... n 10 8 Eq. () Appoximation Classical M/G/1 queuing delay LT packets equest Fist equest (a) case Fist Stage 1... n 1 Ambient taffic (b) case Fig. 5. Components of delay. Second equest Second Stage 1... n Fig.. Compaison between D 1 (n) and D 1 (n). with delay denoted by W. The second pat can be futhe divided into two sub-pats. Fistly, it contains a constant which is the tansmission time of n LT packets, which can be deived as nllt. The est is the time to pocess the ambient taffics which aive at the potal duing the inte-aival time of each LT packet. This pat is denoted by T. Thus, the delay D 1 (n) can be expessed as to appoximate D 1 (n). Simila esults ae found with diffeent settings of λ a and θ. Fo bevity, those esults ae not epoted D 1 (n) = W + nl LT +T. (4) Taking an expectation on both sides, we obtain D 1 (n) = W +T + nl LT = λ a(σ +l ) (1 λal ) + λ al = Γ+ λ al n i=1 n i=1 1 iθ + nl LT τ i + nl LT Γ+ λ al θ (ln(n)+1)+ nl LT, (5) whee Γ is a constant when the paametes fo the taffic ae detemined. Thee ae thee tems in Eq. (5), Γ, the logaithm tem and the linea tem. In ode to ensue the potal has a stable state, λ a should be less than 100 packets pe second unde ou setting stated in Table. Unde this condition, the ode of magnitude of Γ is 10. The numbe of souce symbols k usually exceeds 100 which implies the numbe of equested packets n is also lage than 100. Thus, compaed to the linea tem which has the ode of magnitude equal to 10 o 10, Γ is negligible. If we assume that λ a and θ ae compaable (e.g. λa θ 1), the logaithm tem has the ode of magnitude equal to 10 which is also negligible. As a esult, the linea tem D 1 (n) = nllt can be used to appoximate D 1 (n). In Figue, we plot the values of D 1 (n) and D 1 (n) fo n vaying fom 100 to 800 when λ a = 0 and θ = 50. We can obseve that Γ and the logaithm tem contibute little to the etieval delay. So it is easonable to use D 1 (n) Fig.. Delay in one-stage equest. 1: λ a = 10, θ = 5; : λ a = 0, θ = 50; : λ a = 0, θ = 50. In Figue, we plot the numeical etieval delay D 1 (n), compaed with the simulation esults, as a function of the numbe of equest packets unde diffeent λ a and θ settings. Notice that the numeical esults match the simulation esults well, confiming that using D 1 (n) to appoximate Eq. (5) is applicable. 5. Delay Analysis fo Multiple Stage Request In this subsection, we fist investigate the file-etieval delay fo the two-stage equest scheme. The packet aival pocess is illustated in Figue 5(b). Suppose in the fist stage, the use equests n 1 LT encoded packets. The delay fo the fist stage is D 1 (n 1 ). Afte decoding these n 1 packets, if the use fails to decode the oiginal file, it continues to equest the n packets. Hee we assume that duing the decoding pocess of n 1 packets, the tansmission queue in the potal has aleady etuned to the steady state. This assumption is easonable since it does take some time fo the use to detemine if the n 1 LTencoded packets ae decodable. Thus, the delay fo

7 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING the second stage is identical to the fist stage except fo the numbe of encoded packets the use equests. As a esult, the oveall file-etieval delay fo the two stage equest case is given by D (n 1,n ) = f k (n 1 )D 1 (n 1 ) + (1 f k (n 1 ))(D 1 (n 1 )+D 1 (n )) = D 1 (n 1 )+(1 f k (n 1 ))D 1 (n ), () whee f k (n) is the pobability of successful decoding with n LT encoded packets. Using the fitting esults pesented in Section 4., we can calculate f k (n) appoximately. Taking expectation on both sides, we have D (n 1,n ) = D 1 (n 1 )+(1 f k (n 1 ))D 1 (n ). () We plot in Figs. 8-9 the aveage file-etieval delay fo the two-stage equest scheme, as a function of the numbe of packets equested in the fist stage, unde diffeent settings of k, n and taffic loads. Notice that the numeical esults match the simulation esults well, egadless of the taffic loads. Moeove, in all esults, the file-etieval delay fist deceases and then inceases as the numbe of packets equested in the fist stage inceases. This consistent tend indicates a potential fo file-etieval delay minimization by choosing an appopiate numbe of packets equested in the fist stage. We shall addess this optimization poblem in Section (a) λ a = 10,θ = (c) λ a = 0,θ = (b) λ a = 0,θ = (d) λ a = 0,θ = 50 Fig. 8. Delay in two-stage equest when k = 100 and n = 00 unde diffeent taffic loads. Both numeical and simulation esults confim that the minimal delay exists. The esult fo the two-stage equest scheme can be genealized into abitay t-stage equest. Specifically, the delay expession fo t-stage equest can be defined (a) λ a = 10,θ = No. of equest in fist stage (c) λ a = 0,θ = (b) λ a = 0,θ = 5 Simulaaton (d) λ a = 0,θ = 50 Fig. 9. Delay in two-stage equest when k = 400 and n = 00 unde diffeent taffic loads. Both numeical and simulation esults confim that the minimal delay exists. ecusively, D t (n 1,n,,n t ) = (1 f k (n 1 ))(D 1 (n 1 ) + D t 1 (n,,n t )) + f k (n 1 )D 1 (n 1 ), whee the initial values fo D 1, D ae defined in Eq. (4) and Eq. (). In Figue 10 and 11, we plot the optimal delay compaison obtained by diffeent equest schemes fo diffeent numbe of oiginal symbols unde diffeent taffic loads. Both figues show that the multiple stage equest schemes outpefom taditional onestage equest. Moe specifically, the two-stage equest scheme can educe etieval delay by 15% compaed with the one-stage equest. Whilst, the thee-stage equest scheme futhe educes the delay 4% only. It is obvious that to find an optimal equest in thee-stage equest is much moe difficult than that in two-stage equest and the impovement of thee-stage equest is not so significant. Consequently, we will only focus on finding an optimal two-stage equest scheme in the est of this pape. OPTIMAL SCHEDULING FOR PACKET RE- TRIEVAL In this section, we will investigate the optimal scheduling of packet equests to minimize the fileetieval delay fo two-stage equest scheme. The esults in Figue 8 and Figue 9 suggest an optimal two-stage equest scheme fom Eq. (). Anothe obsevation is that fo a ational taget successful pobability (lage than 90%), the optimal numbe of LT-coded packets equested in the fist stage fo a two-stage equest scheme yields f k (n 1 ) > 50%. Thus,

8 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING Thee stage (a) λ a = 10,θ = 5 Thee stage (c) λ a = 0,θ = Thee stage (b) λ a = 0,θ = 5 Thee stage (d) λ a = 0,θ = 50 Fig. 10. Minimal delay compaison among one-stage equest, two-stage equest and thee-stage equest when k = Thee stage (a) λ a = 10,θ = 5 Thee stage (c) λ a = 0,θ = Thee stage (b) λ a = 0,θ = 5 Thee stage (d) λ a = 0,θ = 50 Fig. 11. Minimal delay compaison among one-stage equest, two-stage equest and thee-stage equest when k = 400. we can apply the fitting function fo f k (n) to Eq. () and simplify the optimization poblem pesented in Section. fo the two-stage equest. Fo the convenience of discussion, we fomalize the simplified optimization poblem as: min D (n 1 ) = D 1 (n 1 )+e α(n 1 k β) D 1 (n n 1 ) s.t. kβ n 1 n,n 1 N. (8) By solving the above optimization poblem, we can detemine the optimal numbe of packets equested in the fist stage fo a two-stage equest scheme. Instead of finding the exact solution, we elax the intege constaint, and assume n 1 is a positive eal numbe to solve the appoximate poblem without intege constaint. Note that the minimum D (n 1 ) without the intege constaint is a lowe bound of the minimum D (n 1 ) with intege constaint. We obseved in Section 5.1 that D 1 (n) can be appoximated as llt n. Thus, D (n 1 ) in Eq. (8) is convex. Now, taking diffeentiation with espect to n 1 on both sides of Eq. (8) and set D n 1 = 0, we get the equation e α k (n1 kβ) = α k (n n 1) + 1. The solution to this equation is n 1 = n+ k α k α W(eα(n k β)+1 ). (9) Hee, W(x) is Lambet function [8] which is defined as x = W(x)e W(x). Note that W(x) cannot be expessed in tems of elementay functions, and thus calculating the eal value of W(x) is not an easy task. Thee exists liteatues on designing algoithm to compute W(x) [9], [40]. Howeve, in this pape, we ae moe inteested in the analytical bound of W(x). In [41], the authos poved the following bound of W(x), Lemma 1. Fo x > 1 we have W(x) logx (logx loglogx+1), (10) 1+logx with equality only fo x = e x W(x) 4 1 W(x) bound Fig. 1. Compaison between W(x) and its bound. Figue 1 shows the compaison between the tue value of W(x) and the lowe bound in Eq. (10). It is clea that this lowe bound is tight enough and we can use this bound to appoximate W(x). By applying Eq. (10) on Eq. (9), we have n 1 = n+ k α k α W(eα(n k β)+1 ) kβ + k (α( n k β)+1) log(α(n k β)+1) α α( n k β)+. (11) It is tivial that n 1 > kβ, and n 1 kβ + α k log(α(n k β)+1) kβ + kα(n k β) α = n. (1)

9 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 9 Eq. (1) shows n 1 n. Thus, we confim that n 1 is the optimal equest in the fist stage which minimizes the two-stage file-etieval delay. In Figue 1, we show Optimal fist stage equest Optimal fist stage equst (a) k = (c) k = 400 Optimal fist stage equest Optimal fist stage equest (b) k = (d) k = 1000 Fig. 1. Compaison of the optimal fist stage equest between simulation and appoximation unde diffeent k. the compaison of the optimal fist stage equest between simulation and appoximation unde diffeent k. In the simulation, the taffic paametes λ a and θ ae set to 0 packets pe second and 50 packets pe second espectively. The esults confim that the appoximation of the optimal fist stage equest obtained fom Eq. (11) is pactical. In Eq. (11), the optimal fiststage equest is unelated to taffic load (i.e. λ a, θ). When the numbe of oiginal symbols k is fixed (i.e. the paametes a and c ae known), the optimal fiststage equest is detemined. This is consistent with the esults shown in Figue 8 and 9. This phenomena can be explained by the following eason. Compaed with the LT encoded packets equested in each stage, the packets belonging to ambient taffic ae negligible. The file-etieval delay is mainly caused by the pocessing the equested LT encoded packets. Based on Eq. (11), if we incease the taget successful pobability (i.e. incease the numbe of equested packets n), the optimal numbe of packets equested in fist stage n 1 will be also inceased. The atio n 1 n is deceased when n is inceased. We plot the atio in Figue 14. This atio dopping phenomena can be undestood as follows. Fom the aspect of the decoding pobability, evey packet contibutes in the decoding pocess. Fom Figue 4, we see an unequal contibution of each packet to the decoding pobability. The contibution of an additional packet to the decoding pobability is high when the pobability stays at a lowe level. As the pobability pogesses highe, the contibution of each additional packet dops. In othe wods, to make a same gain in the decoding pobability, moe packets ae needed when the pobability is at a highe level. Notice fom Figue that the delay in the one-stage equest scheme inceases linealy with the numbe of equested packets. Consequently, tadeoff exists between decodability and delay. Instead of equesting all the packets equied by the taget decoding pobability at once, equesting a small potion of the numbe of packets needed in the fist stage will geneate a decoding pobability that contibutes most to the final decoding pobability; while the maginal contibution fom the second stage is elatively small. As the decodability inceases, the contibution fom the second stage becomes smalle. In othe wods, a pope numbe of packets equested in the fist stage will give us a elatively high decoding pobability. As a esult, setting a vey high decoding pobability is not ational Ratio (a) k = 100 Ratio (b) k = 400 Fig. 14. Optimal fist stage equest atio when k = 100 and k = 400. DELAY-DECODABILITY TRADEOFF We now investigate the fundamental tadeoff between the tageted decoding pobability and the fileetieval delay fo diffeent equest schemes. This delay-decodability tadeoff is impotant since it helps uses to make a ational equest scheme depending on the equiement of thei applications one stage equest two stage equest Taget successful pobability (a) k = one stage equest two stage equest Taget successful pobability (b) k = 400 Fig. 15. Tadeoff between decoding pobability and delay when k = 100 and k = 400. In Figue 15, we plot the delay-decodability tadeoff fo k = 100 and k = 400 espectively. The numeical esult is obtained by combining Eq. () and Eq. (11). Since taffic paametes do not have stong effect on the optimal etieval delay (efe to the figues shown in Section. 5), we choose λ a = 0,θ = 50 as an example. As shown in Fig. 15, the delay fo onestage equest inceases shaply when the decoding

10 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 10 pobability cosses 95%. In compaison, fo the twostage equest scheme, the delay is consistently lowe. This advantage eflects the gain fom being able to decode in the fist stage with an adequate instead of excessive numbe of LT-coded packets. Based on the epoted delay-decodability tadeoff, uses can choose an appopiate taget of decoding pobability, i.e. fo some delay sensitive applications like steam video, a lowe decoding pobability can be adopted such that the aveage etieval delay can be maintained unde a cetain theshold. equest scheme exhibits the same behavio even when the busty MMPP taffic model is used. With a diffeent decoding pobability, the best fist-stage equest can still be found. Only the aveage delay is affected since moe packets ae needed to achieve a highe decoding pobability. This obsevation confims the optimal equest in fist stage in Eq. (11) which is independent of the popety of ambient taffic. We believe that diffeent models of ambient taffic only affect on the oveall delay while using ou two-stage equest scheme can always give the minimal delay. 8 GENERAL TRAFFIC MODEL In Sections and, we assume that the aival pocess of ambient taffic follows Poisson pocess. In this section, we use a diffeent taffic model to examine how ou two-stage equest scheme pefoms. The Makov-modulated Poisson pocess (MMPP) [4] has been extensively used fo modeling pocesses whose aival ates vay andomly ove time [4]. We use MMPP to model the ambient taffic hee since it is moe geneal than simple Poisson pocess and can captue some featues to study bustiness of ambient taffic. In ou simulations, the MMPP taffic has aival ates of λ 1 and λ fo the two phases, and an exponentially distibuted ate switching duation with a mean value equal to 0.1s. We fix θ = 50 and vay the value of λ 1, λ and n. The simulation esults ae shown in Figue 1. 9 CONCLUSION In this pape, we have investigated the poblem of delay optimal file-etieval unde a distibuted cloud stoage system. The file is fist LT-encoded and spead out into a list of distibuted stoage nodes. Duing etieval, the use schedules the packet equest in a multi-stage manne, with an objective to minimize the aveage file-etieval delay. We developed an accuate model to chaacteize the aveage file-etieval delay fo diffeent equest stategies. Using this model, we deived an optimal two-stage equest scheme fo a given decoding pobability. Both simulation and numeical esults confim that this optimal scheme can educe the aveage delay damatically. Ou analysis offes a way fo stoage system opeatos to design an optimized stoage etieval scheme fo LT-based distibuted cloud stoage systems. Dealy (s) Dealy (s) (a) λ 1 = 10,λ = (c) λ 1 = 0,λ = 0 Dealy (s) Dealy (s) (b) λ 1 = 10,λ = (d) λ 1 = 0,λ = 0 Fig. 1. esult of MMPP model. (a)k = 100, n = 00. (b) k = 100, n = 50. (c) k = 400, n = 00. (d) k = 400, n = 800. Compaing with the simulation esults shown in Figue 8 and Figue 9, we can see that ou two-stage REFERENCES [1] Amazon S. [] EMC Atmos Online Stoage. [] L. Heilig and S. Voss, A scientometic analysis of cloud computing liteatue, IEEE Tansactions on Cloud Computing, vol., no., pp. 8, July 014. [4] J.-W. Lin, C.-H. Chen, and J. Chang, Qos-awae data eplication fo data-intensive applications in cloud computing systems, IEEE Tansactions on Cloud Computing, vol. 1, no. 1, pp , Jan 01. [5] H. Weathespoon and J. Kubiatowicz, Easue coding vs. eplication: A quantitative compaison, Pee-to-Pee Systems, pp. 8, 00. [] J. Hafne, Weave codes: highly fault toleant easue codes fo stoage systems, in Poc. the 4th confeence on USENIX Confeence on File and Stoage Technologies. USENIX Association, 005, pp [] J. Plank and L. Xu, Optimizing cauchy Reed-Solomon codes fo fault-toleant netwok stoage applications, in IEEE Int. Symp. Netwok Computing and Applications. IEEE, 00, pp [8] C. Huang, M. Chen, and J. Li, Pyamid codes: Flexible schemes to tade space fo access efficiency in eliable data stoage systems, in IEEE Int. Symp. Netwok Computing and Applications. IEEE, 00, pp [9] F. Oggie and A. Datta, Self-epaiing homomophic codes fo distibuted stoage systems, in Poc. IEEE INFOCOM. IEEE, 010, pp [10] H. Xia and A. Chien, Robustoe: a distibuted stoage achitectue with obust and high pefomance, in Poceedings of the 00 ACM/IEEE confeence on Supecomputing. ACM, 00, p. 44. [11] N. Cao, S. Yu, Z. Yang, W. Lou, and Y. T. Hou, Lt codes-based secue and eliable cloud stoage sevice, in INFOCOM, 01 Poceedings IEEE. IEEE, 01.

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12 infomation: DOI /TCC , IEEE Tansactions on Cloud Computing IEEE TRANSACTIONS ON CLOUD COMPUTING 1 Haifeng Lu eceived his B.Eng degee in compute science and engineeing fom the Univesity of Science and Technology of China in 008. Since 009, he has been a PhD student at School of Compute Engineeing, Nanyang Technological Univesity. His eseach inteests include netwok coding, ateless coding, cloud computeing. Fom June 01 to July 014, he was a poject office at Rapid-Rich Object Seach (ROSE) Lab, NTU. Cuently, he woks at Alibaba Inc. as a senio secuity enginee. Yonggang Wen eceived the Ph.D. degee in electical engineeing and compute science fom the Massachusetts Institute of Technology, Cambidge, MA, USA, in 008. He is cuently an Assistant Pofesso with the School of Compute Engineeing, Nanyang Technological Univesity, Singapoe. Peviously, he was with Cisco, San Jose, CA, USA, as a Senio Softwae Enginee and a System Achitect fo content netwoking poducts. He has also been a Reseach Inten with Bell Laboatoies, Muay Hill, NJ, USA, Sycamoe Netwoks, Chelmsfod, MA, USA, and a Technical Adviso to the Chaiman at Linea A Netwoks, Inc., Milpitas, CA, USA. His cuent eseach inteests include cloud computing, mobile computing, multimedia netwoks, cybe secuity, and geen ICT. Chuan Heng Foh (S 00-M 0-SM 09) eceived his M.Sc. degee fom Monash Univesity, Austalia in 1999 and Ph.D. degee fom the Univesity of Melboune, Austalia in 00. Afte his PhD, he spent months as a Lectue at Monash Univesity in Austalia. In Decembe 00, he joined Nanyang Technological Univesity as an Assistant Pofesso until 01. He is now a Senio Lectue at The Univesity of Suey. His eseach inteests include potocol design and pefomance analysis of vaious compute netwoks, 5G netwoks, and data cente netwoks. He has authoed o coauthoed ove 100 efeeed papes in intenational jounals and confeences. He actively paticipates in IEEE confeence and wokshop oganization, including the Intenation Wokshop on Cloud Computing Systems, Netwoks, and Applications (CCSNA) whee he is a steeing membe. He is cuently an Associate Edito fo IEEE Access, IEEE Wieless Communications, and vaious othe Intenational Jounals. He is also the chai of the Special Inteest Goup on Geen Data Cente and Cloud Computing unde IEEE Technical Committee on Geen Communications and Computing (TCGCC). He is a senio membe of IEEE. Jianfei Cai (S 98-M 0-SM 0) eceived his PhD degee fom the Univesity of Missoui- Columbia. Cuently, he is the Head of Visual & Inteactive Computing Division at the School of Compute Engineeing, Nanyang Technological Univesity, Singapoe. His majo eseach inteests include visual infomation pocessing and multimedia netwoking. He has published ove 100 technical papes in intenational confeences and jounals. He has been actively paticipating in pogam committees of vaious confeences. He had seved as the leading Technical Pogam Chai fo IEEE Intenational Confeence on Multimedia & Expo (ICME) 01 and he cuently sits in the steeing committee of ICME. He was an invited speake fo the fist IEEE Signal Pocessing Society Summe School on D and high definition / high contast video pocess systems in 011. He is also an Associate Edito fo IEEE Tansactions on Cicuits and Systems fo Video Technology (T-CSVT), and a senio membe of IEEE.