Double Disk Failures Research Articles

HVD code: a class of MDS array codes for tolerating triple disk failures

In this study, the authors propose a class of exclusive OR (XOR)-based maximum distance separable (MDS) array codes, referred to as horizontal-vertical-diagonal (HVD) codes, which can tolerate triple disk failures with optimal update complexity. The HVD code has a similar data/parity layout to the horizontal-vertical (HV) code and has many figures of merits as the HV code. Also, an efficient recovery algorithm is proposed, which can be implemented to accelerate the recovery process of double disk failures by making full use of all kinds of parities in the HVD code. The authors evaluate the encoding/decoding efficiency of the HVD code by comparing the number of XORs and the number of disk reads with other MDS array codes. Results show that the HVD code inherits most of the merits of the HV code and requires less number of disk reads when recovering single disk failure.

Optimized Recovery Algorithms for RDP<inline-formula> <tex-math notation="LaTeX">$(p, 3)$ </tex-math> </inline-formula> Code

This paper is concerned with the extended RDP codes, RDP(p, 3), which can tolerate up to three concurrent disk failures. For single-disk failure recovery, we derive a lower bound on disk reads. To approach the derived lower bound, we propose an algorithm that aims at maximizing the overlapping symbols among different types of parity set. For double-disk failure recovery, we propose an algorithm to accelerate the recovery process by shortening the recovery chains with the help of the extra parities. Numerical results for p ≤ 47 show that the proposed algorithm can further reduce by approximately 10% disk reads for single-disk recovery.

A New Construction of EVENODD Codes With Lower Computational Complexity

EVENODD codes are binary array codes for correcting double disk failures in RAID-6 with asymptotically optimal encoding and decoding complexities. However, the update complexity of EVENODD is sub-optimal. We propose a new construction of binary maximum distance separable array codes, namely EVENODD+, such that the encoding, decoding, and update complexities of EVENODD+ are less than those of EVENODD in general. Moreover, EVENODD+ achieves asymptotically optimal update complexity.

HV Code: An All-Around MDS Code for RAID-6 Storage Systems

The increasing expansion of data scale leads to the widespread deployment of storage systems with larger capacity and further induces the climbing probability of data loss or damage. The Maximum Distance Separable (MDS) code in RAID-6, which tolerates the concurrent failures of any two disks with minimal storage requirement, is one of the best candidates to enhance the data reliability. However, most of the existing works in this literature are more inclined to be specialized and cannot provide a satisfied performance under an all-round evaluation. Aiming at this problem, we propose an all-round MDS code named Horizontal-Vertical Code (HV Code) by taking advantage of horizontal parity and vertical parity. HV Code achieves the perfect I/O balancing and optimizes the operation of partial stripe writes, while preserving the optimal encoding/decoding/update efficiency. Moreover, it owns a shorter parity chain which grants it a more efficient recovery for single disk failure. HV Code also behaves well on degraded read operation and accelerates the reconstruction of double disk failures by executing four recovery chains in parallel. The performance evaluation demonstrates that HV Code well balances the I/O distribution. HV Code also eliminates up to 32.2 percent I/O operations for partial stripe writes in read-modify-write mode, and reduces up to 28.9 percent I/O operations for partial stripe writes in reconstruct-write mode. Moreover, HV Code reduces 5.4 $\sim$ 39.8 percent I/O operations per element for the single disk reconstruction, decreases 8.3 $\sim$ 39.0 percent I/O operations for degraded read operations, and shortens 47.4 $\sim$ 59.7 percent recovery time for double disk recovery.

An efficient data layout scheme for better I/O balancing in RAID-6 storage systems

Among redundant arrays of independent disks (RAID)-6 codes, maximum distance separable (MDS) based RAID-6 codes are popular because they have the optimal storage efficiency. Although vertical MDS codes exhibit better load balancing compared to horizontal MDS codes in partial stripes, an I/O unbalancing problem still exists in some vertical codes. To address this issue, we propose a novel efficient data layout, uniform P-code (UPC), to support highly balanced I/Os among P-coded disk arrays (i.e., PC). In UPC, the nonuniformly distributed information symbols in each parity chain of P-code are moved along their columns to other rows, thus enabling the parity chain to keep original parity relationships and tolerate double disk failures. The UPC scheme not only achieves optimal storage efficiency, computational complexity, and update complexity, but also supports better I/O balancing in the context of large-scale storage systems. We also conduct a performance study on reconstruction algorithms using an analytical model. Besides extensive theoretical analysis, comparative performance experiments are conducted by replaying real-world workloads under various configurations. Experimental results illustrate that our UPC scheme significantly outperforms the PC scheme in terms of average user response time. In particular, in the case of a 12-disk array, the UPC scheme can improve the access performance of the RAID-6 storage system by 29.9% compared to the PC scheme.

A New Non-MDS RAID-6 Code to Support Fast Reconstruction and Balanced I/Os

RAID-6 is widely applied to tolerate double concurrent disk failures in both disk arrays and storage clusters. Among numerous erasure codes developed to implement RAID-6, Maximum Distance Separable (MDS) Codes are highly popular. Owing to the limitation of parity generating schemes used in MDS codes, RAID-6-based storage systems suffer from unbalance I/Os and low reconstruction performance. Out of consideration for high performance and reliability, we propose a new class of XOR-based RAID-6 code (i.e. |$V^{2}$|-Code), which improves both load balancing and reconstruction performance of the MDS RAID-6 codes. |$V^{2}$|-Code, a very simple yet flexible Non-MDS vertical code, can be easily implemented and deployed in storage systems. |$V^{2}$|-Code's unique features include lowest density code, steady parity chain length and well-balanced computation. We perform theoretical analysis and empirical evaluation of the coding scheme by running a wide range of workload under various configurations. Experimental results show that |$V^{2}$|-Code outperforms four popular codes (i.e. EVENODD, RDP, X-Code and Code-M) in terms of load balancing and reconstruction time. In the single-disk-failure and double-disk-failure cases, |$V^{2}$|-Code can speed up the reconstruction time of X-Code by a factor of up to 3.31 and 1.79, respectively.

Rebuild processing in RAID5 with emphasis on the supplementary parity augmentation method[37

The rotated parity RAID5 disk array tolerates single disk failures by continuing operation by on-demand reconstruction of data blocks of the failed disk, until the systematic reconstruction of the contents of the failed disk is completed by the rebuild process on a spare disk. Supplementary Parity Augmentation (SPA), unlike the pyramid code, which has two parities covering half of the arrays disks each, extends RAID5's P parity with an additional S parity, which covers half of the disks. The extra load with respect to RAID5 of updating the S parity by one half of the disks is compensated by the more efficient on demand reconstructtion and rebuild processing when a disk fails. Although SPA has the same disk space redundancy level as RAID6, unlike RAID6 it can only deal with roughly half of all possible double disk failure cases for eight disks. For rebuild processing SPA reads half of the disks required by RAID5 and this leads to a higher Mean Time to Data Loss than RAID5, since fewer Latent Sector Errors are encountered. We review performance and reliability modeling of RAID5 arrays to provide insights into SPA's performance and reliability, which cannot be gained from numerical results alone. SPA is outperformed by the Intra-Disk Redundancy schemes combined with RAID5, which results in RAID6's reliability and RAID5 performance.

Extending and analysis of X-Code

X-Code is one of the most important redundant array of independent disk (RAID)-6 codes which are capable of tolerating double disk failures. However, the code length of X-Code is restricted to be a prime number, and such code length restriction of X-Code limits its usage in the real storage systems. Moreover, as a vertical RAID-6 code, X-Code can not be extended easily to an arbitrary code length like horizontal RAID-6 codes. In this paper, a novel and efficient code shortening algorithm for X-Code is proposed to extend X-Code to an arbitrary length. It can be further proved that the code shortening algorithm maintains the maximum-distance-separable (MDS) property of X-Code, and namely, the shortened X-Code is still MDS code with the optimal space efficiency. In the context of the shortening algorithm for X-Code, an in-depth performance analysis on X-Code at consecutive code lengths is conducted, and the impacts of the code shortening algorithm on the performance of X-Code in various performance metrics are revealed.

A cascading Latin scheme to tolerate double disk failures in raid architectures

In recent years, a lot of XOR-based coding schemes have been developed to tolerate double disk failures in Redundant Array of Independent Disks (RAID) architectures, such as EVENODD-code, X-code, B-code and BG-HEDP. Despite those researches, the decades-old strategy of Reed-Solomon (RS) code remains the only popular space-optimal Maximum Distance Separable (MDS) code for all but the smallest storage systems. The reason is that all those XOR-based schemes are too difficult to be implemented, it mainly because the coding-circle of those codes vary with the number of disks. By contrast, the coding-circle of RS code is a constant. In order to solve this problem, we develop a new MDS code named Latin code and a cascading scheme based on Latin code. The cascading Latin scheme is a nearly MDS code (with only one or two more parity disks compared with the MDS ones). Nevertheless, it keeps the coding-circle of the basic Latin code (i.e. a constant) and the low encoding/ decoding complexity similar to other parity array codes.

A Highly Accurate Method for Assessing Reliability of Redundant Arrays of Inexpensive Disks (RAID)

The statistical bases for current models of RAID reliability are reviewed and a highly accurate alternative is provided and justified. This new model corrects statistical errors associated with the pervasive assumption that system (RAID group) times to failure follow a homogeneous Poisson process, and corrects errors associated with assuming the time-to-failure and time-to-restore distributions are exponentially distributed. Statistical justification for the new model uses theory for reliability of repairable systems. Four critical component distributions are developed from field data. These distributions are for times to catastrophic failure, reconstruction and restoration, read errors, and disk data scrubs. Model results have been verified and predict between 2 to 1,500 times as many double disk failures as estimates made using the mean time to data loss method. Model results are compared to system level field data for RAID group of 14 drives and show excellent correlation and greater accuracy than either MTTDL.

Efficient erasure recovery algorithm for double disk failure correction

This paper proposes an efficient parity placement scheme, Horizontal‐Oblique Parity (HOP), for protecting against double disk failures in RAID‐5 disk array systems. It keeps all data unencoded, and uses only exclusive‐or (XOR) operations to compute parity. HOP is provably near optimal in computational complexity, both during construction and reconstruction. It is optimal in the amount of redundant information stored and is sub‐optimal in its accessing. HOP works within a single stripe of blocks of sizes normally used by file systems, databases, and disk arrays.

Modified Low-Density MDS Array Codes for Tolerating Double Disk Failures in Disk Arrays

In this paper, we present a new class of low-density MDS array codes for tolerating double disk failures in disk arrays. The proposed MDS array code has lower encoding and decoding complexity than the EVENODD code of Blaum et al

Independent Row-Oblique Parity for Double Disk Failure Correction

This paper proposes a parity placement scheme, Row-Oblique Parity (ROP), for protecting against double disk failure in disk array systems. It stores all data unencoded, and uses only exclusive-or (XOR) operations to compute parity. ROP is provably optimal in computational complexity, both during construction and reconstruction. It is optimal in the capacity of redundant information stored and accessed. The simplicity of ROP allowed us to implement it within the current available RAID framework.

EVENODD

We present a novel method, that we call EVENODD, for tolerating up to two disk failures in RAID architectures. EVENODD is the first known scheme for tolerating double disk failures that is optimal with regard to both storage and performance. EVENODD employs the addition of only two redundant disks and consists of simple exclusive-OR computations. A major advantage of EVENODD is that it only requires parity hardware, which is typically present in standard RAID-5 controllers. Hence, EVENODD can be implemented on standard RAID-5 controllers without any hardware changes. The only previously known scheme that employes optimal redundant storage (i.e. two extra disks) is based on Reed-Solomon (RS) error-correcting codes, requires computation over finite fields and results in a more complex implementation. For example, we show that the number of exclusive-OR operations involved in implementing EVENODD in a disk array with 15 disks is about 50% of the number required when using the RS scheme.