# Supplemental File of S 2 -RAID: Parallel RAID Architecture for Fast Data Recovery

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1 JOURNAL OF L A T E X CLASS FILES, VOL. 6, NO. 1, JANUARY 27 1 Supplemental File of S 2 -RAID: Parallel RAID Architecture for Fast Data Recovery Jiguang Wan, Jibin Wang, Changsheng Xie, and Qing Yang, Fellow, IEEE APPENDIX A DATA LAYOUT OF S 2 -RAID A.1 Analysis of Conflict-Free Parallelism In this section, we show mathematically that S 2 -RAID can reconstruct data on a failed disk by reading each stripe on good disks in parallel without conflict. Let μ i,, μ i,1,, and μ i,g 1 be column sets representing elements in the 1 st, 2 nd,, and G th columns of the sub-matrix m i respectively. Let U i be a vector set representing all column vectors of m i. That is, U = (μ, μ,1 μ,g 1 ), U 1 =(μ 1, μ 1,1 μ 1,G 1 ),, and U G 1 =(μ G 1, μ G 1,1 μ G 1,G 1 ). Clearly, to realize the conflict-free parallel reconstruction the following two conditions must be satisfied simultaneously, { μi,j μ j,i = {e},μ i,j U i,μ j,i U j and i j μ i,j μ i,k =,μ i,j,μ i,k U i and j k, where i, j, k G 1 and {e} is an identity set with non-null element. In other words, there is only one common element when intersection column sets from different groups and no intersection among column sets in a group. This is the key to the S 2 -RAID layout. Now consider column vectors of matrix matrix m i and m j. We use C i,p and C j,q to represent (p +1) th column vector of the sub-matrix m i and the (q +1) th column vector of the sub-matrix m j respectively. Suppose the vectors C i,p and C j,q are known and given by C i,p = a,p a 1,p a 2,p a K 1,p and C j,q = b,q b 1,q b 2,q b K 1,q, where a i,j and b i,j denote elements at (j +1) th column (i +1) th row of the C i,p and C j,q, respectively. From Equation (1) we know that the elements of the C j,q can be obtained from sub-matrix m i though cyclic shift operations, we have b,q = a,q mod G b 1,q = a 1,[q+(j i)] mod G b 2,q = a 2,[q+2 (j i)] mod G, b K 1,q = a K 1,[q+(K 1) (j i)] mod G then C j,q can be represented by b,q a,q mod G b 1,q a 1,[q+(j i)] mod G C j,q = b 2,q = a 2,[q+2 (j i)] mod G b K 1,q a K 1,[q+(K 1) (j i)] mod G, 1 K G. What we want to show next is that the column subscripts of a,q mod G, a 1,[q+(j i) modg], a 2,[q+2 (j i)] mod G,, and a K 1,[q+(K 1) (j i)] mod G are different from each other. Suppose that there were two identical subscripts on row n1 and n2 of that column, the following equation would hold [q + n1 (j i)] mod G =[q + n2 (j i)] mod G, which implies [(n1 n2) (j i)] mod G =. (2) However, there are some limitations as follows: <n1,n2,i,j<g n1 n2,i j, G is a prime number which mean the true of the following inequalities { (n1 n2) 1, (j i) 1 (n1 n2)<g, (j i)<g. Finally, we can deduce the conclusion, (n1 n2) (j i) mod G. Then, Equation (2) can never hold. Therefore, their column subscripts are all different from each other indicating that they span all columns of matrix m i and there is one and only one of the same element in any column of m i. Consequently, the intersection between C i,p and C j,q is always one. That is, the system can carry out conflict-free parallel reconstruction.

2 JOURNAL OF L A T E X CLASS FILES, VOL. 6, NO. 1, JANUARY 27 2 A.2 Design of S 2 -RAID1 m primary = P, P,1, P,K 1 where P,j ( j K 1) is a G elements vector representing subraids of partition j on disks of group (also called primary group). Then the layout matrix of secondary group can be created by shift operations, P, SH r (P, ) m secondary = P,1 = SHr 1 (P,1 ), (3) SHr K 1 (P,K 1 ) P,K 1 Fig. 11. An overview of S 2 -RAID1 data layout. Fig. 11 shows one example of S 2 -RAID1 design with 8 disks and subraid size of 8. Disks D, D1, D2, and D3 are data disks and D4, D5, D6, and D7 are mirror disks, respectively. Instead of directly mirroring data stripes in traditional RAID1, S 2 -RAID1 skews data partitions in mirror disks by shifting each subsequent partition by one position as shown in the figure. In this way, data reconstruction can be done in parallel in case of a disk failure. As an example, if disk D2 fails in shaded area as shown in Fig. 11 and the data partitions in D2 had been mirrored from D4 to D7 (see the partitions.2, 1.2, 2.2, and 3.2 ), then they can be read out in parallel. In other words, the data reconstruction can be done 4 (β =4) times faster than the original RAID1. The data read from the 4 mirror disks then can be written in parallel to spare disks. Note that the 4 spare disks labeled S through S3 shown in Fig. 11 can be either physical spare disks if they are available or available disk space on the 8 data/mirror disks, as the similar descriptions on S 2 - RAID5. Of course, we also give the general layout of S 2 -RAID1 in the next section. A.3 S 2 -RAID1 Expansion Strategy Similar to S 2 -RAID5, S 2 -RAID1 is also easily expandable. The condition for generalizing S 2 -RAID1 is that disk number R must be an even number greater than 4. We separate R disks into two types of groups evenly, primary and secondary, and the notation N is a constant (N =2). We then divide every disk into K(1<K G) partitions and select one partition from each disk to form primary and secondary groups to form subraids. The subraid size G is equal to R/2, and the degree of parallelism, β, is equivalent to the number of disk partitions at the best (β K). The compositional procedure of S 2 -RAID1 is shown below. First, we give the initial matrix table m primary, where P,j denotes the (j +1)th line of the matrix table in secondary group, equaling to the result of SHr(P b,j ). The function of SHr(P b,j ) is similar to the Equation (1). Given the condition R =8, we can obtain the following initial layout matrix m primary, m primary = Recall that notation S.L represents logic disk number L in subraid number S. The identifier., for example, represents the first logic partition of the subraid. Then the final mapping table M with R =8based on Equation (3) is: M = , where notation S.L is an image of S.L. Obviously, this mapping table can achieve parallel reconstruction at 4 times speedup without access conflicts. APPENDIX B PROTOTYPE IMPLEMENTATION For the purpose of proof-of-concept and performance evaluation, we have built two prototypes of S 2 -RAID and Parity Declustering RAID in the same environment. Our implementations are based on the source code of the Linux software RAID (MD) and the following two sections give the details of them. B.1 Software Structure of S 2 -RAID S 2 -RAID was implemented in the Linux operating system inside the kernel and embedded into the iscsi target [8]. In the iscsi target below, we realized the S 2 -RAID functionalities. Fig. 12 shows the software structure of

3 JOURNAL OF L A T E X CLASS FILES, VOL. 6, NO. 1, JANUARY 27 3 TABLE 3 A 9-Disk Parity Declustering Layout Based on Balanced Incomplete Block Design. Stripe Number Tuple Stripe Number Tuple, 1, 2 6, 4, 8 1 3, 4, 5 7 1, 5, 6 2 6, 7, 8 8 2, 3, 7 3, 3, 6 9, 5, 7 4 1, 4, 7 1 1, 3, 8 5 2, 5, , 4, 6 Fig. 12. Software structure of S 2 -RAID implementation. our prototype. It mainly includes three modules, iscsi target module, S 2 -RAID function module and configuration module. The iscsi target module modifies the SCSI command handling and disk I/O process. The disk I/Os of the iscsi target call upon interfaces of the S 2 -RAID module. The S 2 -RAID module implements the basic functions of S 2 -RAID5 and S 2 -RAID1 including RAID rebuilder based on MD. MD itself provides RAID rebuilder that allows parallel reconstruction of multiple RAIDs for a disk failure provided that these RAIDs do not share physical disks. When multiple RAIDs share physical disks, MD s rebuilder reconstructs data sequentially with no parallelism. Considering the mapping of S 2 -RAID, we employ two small mapping tables to manage data layouts as shown in Fig. 1 (see the figure in the paper) and Fig. 11. And the mapping mode within each subraid is the same as traditional MD. In this way, upon a disk failure, we are able to use MD s rebuilder to reconstruct data in parallel. In addition, some extra information should be recorded including spare disk locations. While the addressing of S 2 -RAID can be done by computation as described in Fig. 2 in the paper. For S 2 -RAID1 shown in Fig. 11, we built 16 subraids with RAID1 level using 2 disk partitions, then the S 2 - RAID module forms a uniform addressing space with RAID level, as called S 2 -RAID1. From point of view of RAID reconstruction, there are 4 independent conventional RAID1s without sharing physical disks allowing parallel reconstruction in case of one disk failure. The S 2 -RAID module finally maps the multiple sub- RAIDs to one unified LUN. This LUN presents to the iscsi target module as one logical disk for read and write I/Os. The configuration module provides RAID setup and configuration functions using mdadm commands to realize different S 2 -RAID functions. It also allows users to configure iscsi target by means of iscsi configuration functions. B.2 Implementation of Parity Declustering This section gives the implementation details of Parity Declustering. Parity Declustering is one of the existing works that is closely related to S 2 -RAID5. In order to show the difference both of them, we have implemented Parity Declustering based on MD. Our implementation is based on a data construction algorithm, which is derived from a balanced incomplete block design [3]. Through the concept we know, a block design table is an arrangement of ν distinct objects into b tuples, each of which contains k objects, such that each object appears in exactly γ tuples, and any pair of objects in tuples just appears λ tuples. There are also two notations C and G, where C is the number of disks in the RAID and G is the span of the stripe units with which parity unit can protect some smaller number of data units instead of C 1 and C = ν, G = k. We have implemented a 9-disk layout of Parity Declustering, with the stripe width of 3. The detail of distributing parameters are ν =9, b =12, k =3, and γ =4as shown in table 3. APPENDIX C ADDITIONAL EVALUATION OF S 2 -RAID C.1 Details of Testbed Settings In this section, we introduce the details of software and hardware configurations used in our evaluations as listed in Table 4 and Table 5. TABLE 4 Hardware Details of The Storage Client and Server. Note that all hardwares in either client or server are the same except for the disk and HBA. OS Disks (server) Disk (client) Mainboard CPU NIC Memory HBA (server) Fedora 14(kernel ) 1 Seagate ST332418AS, 32GB, 72RPM 15 Seagate ST3541AS, 5GB, 72RPM 1 Seagate ST332418AS, 32GB, 72RPM Supermicro X8DT6 Intel(R) Xeon(R) CPU Intel 82574L 8GB DDR3 Symbios Logic SAS28 C.2 Benchmark Settings Besides the traces, we also choose a set of standard benchmarks as test cases that are widely used in the research community and industry. The first benchmark we selected is PostMark [4] that is widely used as file system benchmark tool written

5 JOURNAL OF L A T E X CLASS FILES, VOL. 6, NO. 1, JANUARY 27 5 the whole disk. There are 8 disks in total (including 1 spare disk) involved in reconstruction process for 8-disk S 2 -RAID1, with 3 disks and 1 spare disk in primary group playing the role of spare disks and 4 disks in secondary group for reconstruction. While traditional RAID1 is based on a one-to-one reconstruction mode, with only two disks taking reconstruction by data migration. Better disk utilizations and high degree of parallelism in S 2 -RAID1 during reconstruction process will also adversely affect frontend I/O performance while reconstruction is going on. The results of frontend performance while reconstruction is in progress in background are shown in Fig. 14(b). S 2 -RAID1 has an average 27.13% performance degradation in terms of user response time compared with RAID1. For multimedia applications such as MM, this performance degradation is more pronounced. The fact is that all disks in S 2 - RAID1 are used for either reconstruction or responding frontend I/O requests. It is understandable, why frontend performance suffers a little bit - the heavy workloads of each disk. REFERENCES [1] Shimin Chen, Anastasia Ailamaki, Manos Athanassoulis, Phillip B. Gibbons, Ryan Johnson, Ippokratis Pandis, and Radu Stoica. TPC- E vs. TPC-C: characterizing the new TPC-E benchmark via an I/O comparison study. SIGMOD Record, 39(3):5 1, February 211. [2] Transaction Processing Performance Council. TPC benchmark tm C standard specification, 25. [3] Marshall Hall. Combinatorial Theory (2nd Edition). Wiley Interscience, [4] Jeffrey Katcher. Postmark: a new file system benchmark. Technical Report TR322, Network Appliances, [5] Alexey Kopytov. SysBench, a system performance benchmark, Available: 24. [6] Diego R. Llanos. TPCC-UVa: an open-source TPC-C implementation for global performance measurement of computer systems. SIGMOD Record, 35(4):6 15, December 26. [7] Jin Ren and Qing Yang. A new buffer cache design exploiting both temporal and content localities. In Proceedings of the 3th IEEE International Conference on Distributed Computing Systems, ICDCS 1, pages , 21. [8] iscsi Enterprise Target. Available: 212.

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