Self-Repairing Disk Arrays

As the prices of magnetic storage continue to decrease, the cost of replacing failed disks becomes increasingly dominated by the cost of the service call itself. We propose to eliminate these calls by building disk arrays that contain enough spare disks to operate without any human intervention during their whole lifetime. To evaluate the feasibility of this approach, we have simulated the behavior of two-dimensional disk arrays with n parity disks and n(n-1)/2 data disks under realistic failure and repair assumptions. Our conclusion is that having n(n+1)/2 spare disks is more than enough to achieve a 99.999 percent probability of not losing data over four years. We observe that the same objectives cannot be reached with RAID level 6 organizations and would require RAID stripes that could tolerate triple disk failures.Comment: Part of ADAPT Workshop proceedings, 2015 (arXiv:1412.2347

Decentralized Erasure Codes for Distributed Networked Storage

We consider the problem of constructing an erasure code for storage over a network when the data sources are distributed. Specifically, we assume that there are n storage nodes with limited memory and k<n sources generating the data. We want a data collector, who can appear anywhere in the network, to query any k storage nodes and be able to retrieve the data. We introduce Decentralized Erasure Codes, which are linear codes with a specific randomized structure inspired by network coding on random bipartite graphs. We show that decentralized erasure codes are optimally sparse, and lead to reduced communication, storage and computation cost over random linear coding.Comment: to appear in IEEE Transactions on Information Theory, Special Issue: Networking and Information Theor

EVENODD: An Efficient Scheme for Tolerating Double Disk Failures in RAID Architectures

We present a novel method, that we call EVENODD, for tolerating up to two disk failures in RAID architectures. EVENODD employs the addition of only two redundant disks and consists of simple exclusive-OR computations. This redundant storage is optimal, in the sense that two failed disks cannot be retrieved with less than two redundant disks. 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 most commonly used scheme that employes optimal redundant storage (i.e., two extra disks) is based on Reed-Solomon (RS) error-correcting codes. This scheme requires computation over finite fields and results in a more complex implementation. For example, we show that the complexity of implementing EVENODD in a disk array with 15 disks is about 50% of the one required when using the RS scheme. The new scheme is not limited to RAID architectures: it can be used in any system requiring large symbols and relatively short codes, for instance, in multitrack magnetic recording. To this end, we also present a decoding algorithm for one column (track) in error

Alpha Entanglement Codes: Practical Erasure Codes to Archive Data in Unreliable Environments

Data centres that use consumer-grade disks drives and distributed peer-to-peer systems are unreliable environments to archive data without enough redundancy. Most redundancy schemes are not completely effective for providing high availability, durability and integrity in the long-term. We propose alpha entanglement codes, a mechanism that creates a virtual layer of highly interconnected storage devices to propagate redundant information across a large scale storage system. Our motivation is to design flexible and practical erasure codes with high fault-tolerance to improve data durability and availability even in catastrophic scenarios. By flexible and practical, we mean code settings that can be adapted to future requirements and practical implementations with reasonable trade-offs between security, resource usage and performance. The codes have three parameters. Alpha increases storage overhead linearly but increases the possible paths to recover data exponentially. Two other parameters increase fault-tolerance even further without the need of additional storage. As a result, an entangled storage system can provide high availability, durability and offer additional integrity: it is more difficult to modify data undetectably. We evaluate how several redundancy schemes perform in unreliable environments and show that alpha entanglement codes are flexible and practical codes. Remarkably, they excel at code locality, hence, they reduce repair costs and become less dependent on storage locations with poor availability. Our solution outperforms Reed-Solomon codes in many disaster recovery scenarios.Comment: The publication has 12 pages and 13 figures. This work was partially supported by Swiss National Science Foundation SNSF Doc.Mobility 162014, 2018 48th Annual IEEE/IFIP International Conference on Dependable Systems and Networks (DSN

Construction of Extended Steiner Systems for Information Retrieval

A multiset batch code is a variation of information retrieval where a t-multiset of items can be retrieved by reading at most one bit from each server. We study a problem at the other end of the spectrum, namely that of retrieving a t-multiset of items by accessing exactly one server. Our solution to the problem is a combinatorial notion called an extended Steiner system, which was first studied by Johnson and Mendelsohn [11]. An extended Steiner system ES(t; k; v) is a collection of k-multisets (thus, allowing repetition of elements in a block) of a v-set such that every t-multiset belongs to exactly one block. An extended triple system, with t = 2 and k = 3, has been investigated and constructed previously [3, 11]. We study extended systems over v elements with k = t + 1, denoted as ES(t, t + 1, v). We show constructions of ES(t, t + 1, v) for all t &#8805; 3 and v &#8805; t + 1.A multiset batch code is a variation of information retrieval where a t-multiset of items can be retrieved by reading at most one bit from each server. We study a problem at the other end of the spectrum, namely that of retrieving a t-multiset of items by accessing exactly one server. Our solution to the problem is a combinatorial notion called an extended Steiner system, which was first studied by Johnson and Mendelsohn [11]. An extended Steiner system ES(t, k , v ) is a collection of k-multisets (thus, allowing repetition of elements in a block) of a v -set such that every t-multiset belongs to exactly one block. An extended triple system, with t = 2 and k = 3, has been investigated and constructed previously [3, 11]. We study extended systems over v elements with k = t + 1, denoted as ES(t, t + 1, v ). We show constructions of ES(t, t + 1, v ) for all t 3 and v t + 1

Stochastic Analysis on RAID Reliability for Solid-State Drives

Solid-state drives (SSDs) have been widely deployed in desktops and data centers. However, SSDs suffer from bit errors, and the bit error rate is time dependent since it increases as an SSD wears down. Traditional storage systems mainly use parity-based RAID to provide reliability guarantees by striping redundancy across multiple devices, but the effectiveness of RAID in SSDs remains debatable as parity updates aggravate the wearing and bit error rates of SSDs. In particular, an open problem is that how different parity distributions over multiple devices, such as the even distribution suggested by conventional wisdom, or uneven distributions proposed in recent RAID schemes for SSDs, may influence the reliability of an SSD RAID array. To address this fundamental problem, we propose the first analytical model to quantify the reliability dynamics of an SSD RAID array. Specifically, we develop a "non-homogeneous" continuous time Markov chain model, and derive the transient reliability solution. We validate our model via trace-driven simulations and conduct numerical analysis to provide insights into the reliability dynamics of SSD RAID arrays under different parity distributions and subject to different bit error rates and array configurations. Designers can use our model to decide the appropriate parity distribution based on their reliability requirements.Comment: 12 page

Durability and Availability of Erasure-Coded Storage Systems with Concurrent Maintenance

This initial version of this document was written back in 2014 for the sole purpose of providing fundamentals of reliability theory as well as to identify the theoretical types of machinery for the prediction of durability/availability of erasure-coded storage systems. Since the definition of a "system" is too broad, we specifically focus on warm/cold storage systems where the data is stored in a distributed fashion across different storage units with or without continuous operation. The contents of this document are dedicated to a review of fundamentals, a few major improved stochastic models, and several contributions of my work relevant to the field. One of the contributions of this document is the introduction of the most general form of Markov models for the estimation of mean time to failure. This work was partially later published in IEEE Transactions on Reliability. Very good approximations for the closed-form solutions for this general model are also investigated. Various storage configurations under different policies are compared using such advanced models. Later in a subsequent chapter, we have also considered multi-dimensional Markov models to address detached drive-medium combinations such as those found in optical disk and tape storage systems. It is not hard to anticipate such a system structure would most likely be part of future DNA storage libraries. This work is partially published in Elsevier Reliability and System Safety. Topics that include simulation modelings for more accurate estimations are included towards the end of the document by noting the deficiencies of the simplified canonical as well as more complex Markov models, due mainly to the stationary and static nature of Markovinity. Throughout the document, we shall focus on concurrently maintained systems although the discussions will only slightly change for the systems repaired one device at a time.Comment: 58 pages, 20 figures, 9 tables. arXiv admin note: substantial text overlap with arXiv:1911.0032