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distributed storage systems ( dss ) are widely used today for storing data reliably over long periods of time using a distributed collection of storage nodes , which may be individually unreliable .
application scenarios include large data centers such as total recall @xcite , oceanstore @xcite and peer - to - peer storage systems such as wuala @xcite , that use nodes across the internet for distributed file storage .
one of the challenges for dss is the _ repair problem _ : if a node storing a coded piece fails or leaves the system , in order to maintain the same level of reliability , we need to create a new encoded piece and store it at a new node with the minimum repair bandwidth . to solve this problem , dimakis _ et al .
_ introduced a generic framework based on _ regenerating codes _ ( rc ) in @xcite .
rc use ideas from network coding to define a new family of erasure codes that can achieve different trade - offs in the optimization of storage capacity and communication costs .
the optimal tradeoff curve for achievable codes has two extremal points which are of particular interest : the minimum storage regenerating ( msr ) codes with minimum possible storage size for a given repair capability , and the minimum bandwidth regenerating ( mbr ) codes with minimum possible repair bandwidth .
consider a minimum storage system , where a source file of size @xmath3 units is split into @xmath4 parts , defined over a finite field @xmath5 and stored across @xmath6 nodes in the dss . while the economy in storage is highly desirable
, issues may arise when the system tries to repair failure at the optimal repair bandwidth . specifically ,
if @xmath7 or @xmath3 grows to be arbitrarily large , then the system may become inefficient and impractical due to the high computational complexity or the fast growing storage consumption .
another challenge for dss is data availability , which is of critical importance to a peer - to - peer ( p2p ) storage / backup system that relies on a swarm of distributed and independent nodes for file storage . as the nodes not only differ in their storage capacity and traffic bandwidth , but they may not be online or available at all times . hence , there is a pressing need to increase the data availability , such that infomation is available with a probability approaching 1 .
clearly , p2p enrironments are heterogeneous by nature , and code design for such systems must explicitly account for this heterogeneity .
the primary interest of this paper is to study a _
non - homogeneous dss _ , where one _ super node _ has a larger storage size and higher reliability and availability than the other storage nodes .
we study a class of high - rate @xmath8 mds storage codes , and show that with mds code such non - homogeneous dss can achieve the optimal bound in @xcite when repairing single or double - node failures , but require smaller @xmath3 and @xmath7 than the traditional homogeneous model in @xcite . another proposed scheme based on non - mds codes
is shown to repair 1-node failure below the optimal repair bandwidth bound in @xcite .
moreover , we show that our proposed non - homogeneous dss schemes can achieve a higher data availability than the traditional homogeneous dss scheme .
this paper is organized as follows .
section ii shows the definition of non - homogeneous dss .
section iii shows three proposed schemes of exact repair with @xmath8 storing codes in non - homogeneous dss .
section iv shows the numerical results of our schemes and the comparison with previous methods .
finally , the paper is concluded in section v.    ]
in this section , we present a brief review of the traditional homogeneous dss proposed in @xcite .
then , a new model of non - homogeneous dss is proposed to realize the practical dss .
we follow the definition of traditional homogeneous dss using @xmath9 regenerating codes over finite field @xmath5 .
this network has @xmath6 storage nodes and every @xmath4 nodes suffice to reconstruct all the data .
the size of the file to be stored is @xmath3 units and partitioned into @xmath4 equal parts @xmath10 where @xmath11 . after encoding them into @xmath6 coded parts using an @xmath12 maximum distance separable ( mds ) code , we store them at @xmath6 nodes .
we define here the mds property of a storage code using the notion of data collectors as presented in @xcite .
a storage code where each node contains @xmath13 worth of storage , has the mds property if a data collector can reconstruct the all the @xmath3 units by connecting to any @xmath4 out of @xmath6 storage nodes .
when a node fails , the data stored therein is recovered by downloading @xmath14 packets each from any @xmath15 of the remaining @xmath16 nodes ; the total repair bandwidth is then @xmath17 as shown in fig .
[ fig : traditionalrepair ] .
it has been shown in @xcite that there exists an optimal tradeoff between the storage per node , @xmath18 , and the bandwidth to repair one node , @xmath19 . in this paper , we focus on the extreme point where the smallest @xmath20 corresponds to a _ minimum - storage regenerating _ ( _ msr _ ) code .
@xmath21 to minimize @xmath22 , let @xmath23 and we get @xmath24 . in the case of high - rate codes @xmath25 ,
a lower bound for repair bandwidth @xmath26 of 1-node failure was shown as @xcite : @xmath27      a non - homogeneous dss with the parameter @xmath28 is a distributed storage systems with @xmath29 nodes based on @xmath12 storing codes and the amount of data stored and downloaded from any nodes are variable .
node @xmath30 in the network stores @xmath31 units .
when node @xmath30 fails then it is repaired by downloading @xmath32 packets from node @xmath33 , @xmath34.@xmath35    it is clear that we must have @xmath36 for all @xmath37 since a node can not transmit more information than it is storing . when @xmath38 for all @xmath39 , we obtain the traditional homogeneous dss . when @xmath40 , there are more redundant blocks than the storage nodes .
the storage process has to decide which node(s ) to store more blocks .    in this paper
, we present the idea of non - homogeneous dss using the following setting : there is one big node , called the _ super node _ , which has a larger storage capacity and higher reliability and availability than the other nodes .
such scenario is possible in practical system , e.g. in a peer - to - peer backup system , the super node could be the service provider that has higher availability and provides higher storage capacity than other peers .
consider a system with one super node and three other storage nodes non - homogeneous dss based on a @xmath41 mds code , which can be denoted as @xmath42 .
assume a file of size @xmath43 , then this file is divided into @xmath44 parts , each part containing @xmath45 packets . after encoding them into @xmath46 encoded parts or @xmath47 packets , we store the first @xmath48 packets in the super node , and each of the remaining three nodes stores @xmath49 packets as shown in fig . [
fig : examplenon - homogeneousdss ] .
mds codes and @xmath50 storage nodes.(node @xmath51 is the super node.)[fig : examplenon - homogeneousdss ] ]
in this paper , we limit our study to high - rate @xmath52 exact - repair storing codes . this homogeneous problem has been considered in @xcite .
we propose three efficient dss schemes using mds and non - mds storage codes in such @xmath53 non - homogeneous dss , which are denoted as scheme a , b , and c in table .
[ table:(k+2,k)model ] . scheme a and c use mds codes while scheme b uses non - mds codes .
the new system consists of @xmath54 nodes which include @xmath4 nodes of storage size @xmath55 and one super node of size @xmath56 .    [ cols="^,^,^,^",options="header " , ]      in this subsection , we employ the framework proposed in @xcite to measure and compare the data availability between our proposed non - homogenous dss schemes and traditional homogenous dss schemes to show the efficiency of our proposed schemes .
let @xmath57 $ ] be the nodes online probability of @xmath29 nodes in the @xmath28 dss .
let the power set of @xmath29 , @xmath58 , denote the set of all possible combinations of online nodes .
let @xmath59 represents one of these possible combinations .
then , we will use @xmath60 to represent the event that combination @xmath61 occurs .
since node availabilities are independent , we have    @xmath62=\prod_{i\in a}p_{i}\prod_{j\in2^{h}\backslash a}\left(1-p_{j}\right)\ ] ]    let @xmath63 be the number of data blocks stored in storage node @xmath30 , for example @xmath64 , it means @xmath65 .
the data allocation of our schemes will be @xmath66 .
let @xmath67 be the subset containing those combinations of available nodes which together store @xmath4 different redundant blocks .
@xmath68 since the retrieval process needs to download @xmath4 different blocks out of the total @xmath6 redundant blocks , the probability of successful recovery for an allocation @xmath69 can be measured as @xmath70=\sum_{a\in l_{k}}pr\left[q_{a}\right]\\ = \sum_{a\in l_{k}}\left[\prod_{i\in a}p_{i}\prod_{j\in2^{h}\backslash a}\left(1-p_{j}\right)\right ] \end{array}\ ] ] to compare the data availability , we examine a scenario of node online probability where the online probability of super node is greater than the other node @xmath71 .
the data availability of homogeneous @xmath72 ( e.g. scheme in @xcite ) and non - homogeneous @xmath73 dss ( e.g. schemes a and c , since scheme b is based on non - mds code , it is excluded in this study as its availability is calculated in a different manner ) can be computed by the following equations :    @xmath74    @xmath75    let @xmath76 where @xmath77 . the condition @xmath78 will induce @xmath79)$ ] .
it can be seen that if @xmath80 , then @xmath81)\leq1\leq\chi$ ] .
therefore , @xmath78 for all @xmath80 .
we run the simulations for the case of @xmath82 and @xmath83 and obtain the result in fig .
[ fig : dataavailable ] .
it can be seen that for @xmath84 , data availability of non - homogeneous dss scheme outperforms the homogeneous dss scheme . for @xmath85
, the non - homogeneous schemes also have a big improvement when @xmath86 has a high online availability .
therefore , it can be seen that our proposed non - homogeneous dss schemes achieve a higher data availability than the traditional homogeneous dss .
the gap between the two becomes larger when the online availability of the super node increases , e.g. when @xmath86 is greater than 25% of @xmath87 , the data availability of the proposed non - homogenous over homogenous dss is increased by 10% .    ]
we proposed three distributed storage schemes for _ non - homogeneous dss _ with high rate @xmath8 codes .
two of the schemes make use of mds code , and can achieve optimal repair bandwidth of @xmath1 at smaller finite field @xmath7 and 75% smaller fragment @xmath3 than @xcite .
small @xmath3 and @xmath7 are desirable , because they reduce the update bandwidth and complexity .
another scheme based on non - mds code can achieve a smaller repair bandwidth than the optimal bandwidth based on mds code by @xmath2 for 1-node failure .
we further demonstrate that in such non - homogeneous dss , if we can ensure one super node with a higher online probability than the other nodes , we can achieve a higher data availability than the homogeneous dss .
this research is partly supported by the international design center ( grant no .
idg31100102 & idd11100101 ) .
li s work is supported in part by the national science foundation under grants no .
ccf-0829888 , cmmi-0928092 , and eager-1133027 .
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this paper describes a _ non - homogeneous _ distributed storage systems ( dss ) , where there is one super node which has a larger storage size and higher reliability and availability than the other storage nodes . 
 we propose three distributed storage schemes based on ( @xmath0 ) maximum distance separable ( mds ) codes and non - mds codes to show the efficiency of such non - homogeneous dss in terms of repair efficiency and data availability . 
 our schemes achieve optimal bandwidth @xmath1 when repairing 1-node failure , but require only one fourth of the minimum required file size and can operate with a smaller field size leading to significant complexity reduction than traditional homogeneous dss . moreover , with non - mds codes 
 , our scheme can achieve an even smaller repair bandwidth of @xmath2 . 
 finally , we show that our schemes can increase the data availability by 10% than the traditional homogeneous dss scheme .    
 exact - repair mds codes , non - homogeneous dss , minimum storage regenerating ( msr ) codes .