Alexander Thomasian

A GRASP algorithm for the multi-objective knapsack problem

While the performance of individual disks is well understood, the performance of disk arrays configured as redundant arrays of independent disks - RAID requires further investigation. We report on a tool that can be used for the performance evaluation of k disk failure tolerant (kDFT) disk arrays. RAID0 is a 0DFT. RAID5 is a 1DFT utilizing a single check disk. RAID6, EVENODD, and RDP are 2DFTs with two check disks and a minimum level of redundancy, while RM2 is a 2DFT with a slightly higher redundancy level. The capacity overhead is less important for high capacity modern disks than the disk access overhead to update parities, especially the small write penalty. Degraded mode operation and rebuild processing incur extra overhead on surviving disks. The RAID level, configuration, operating mode (normal, degraded with 1 or 2 failures, and rebuild mode with various options), and workload characteristics have a significant effect on RAID performance, which can be evaluated using our tool. We consider here only discrete requests and derive cost functions to determine: (a) the volume of data to be transmitted; (b) the maximum attainable throughput for given disk characteristics; (c) the mean response time given the execution plans of RAID operations specified as dags (directed acyclic graphs). Analytic solutions are possible when arrivals are Poisson with the FCFS policy. Some interesting results obtained by the tool are given to illustrate its capabilities.In this article, we propose a greedy randomized adaptive search procedure (GRASP) to generate a good approximation of the efficient or Pareto optimal set of a multi-objective combinatorial optimization problem. The algorithm is based on the optimization of all weighted linear utility functions. In each iteration, a preference vector is defined and a solution is built considering the preferences of each objective. The found solution is submitted to a local search trying to improve the value of the utility function. In order to find a variety of efficient solutions, we use different preference vectors, which are distributed uniformly on the Pareto frontier. The proposed algorithm is applied for the 0/1 knapsack problem with r = 2, 3, 4 objectives and n = 250, 500, 750 items. The quality of the approximated solutions is evaluated comparing with the solutions given by three genetic algorithms from the literature.

Higher reliability redundant disk arrays: Organization, operation, and coding

Parity is a popular form of data protection in redundant arrays of inexpensive/independent disks (RAID). RAID5 dedicates one out of N disks to parity to mask single disk failures, that is, the contents of a block on a failed disk can be reconstructed by exclusive-ORing the corresponding blocks on surviving disks. RAID5 can mask a single disk failure, and it is vulnerable to data loss if a second disk failure occurs. The RAID5 rebuild process systematically reconstructs the contents of a failed disk on a spare disk, returning the system to its original state, but the rebuild process may be unsuccessful due to unreadable sectors. This has led to two disk failure tolerant arrays (2DFTs), such as RAID6 based on Reed-Solomon (RS) codes. EVENODD, RDP (Row-Diagonal-Parity), the X-code, and RM2 (Row-Matrix) are 2DFTs with parity coding. RM2 incurs a higher level of redundancy than two disks, while the X-code is limited to a prime number of disks. RDP is optimal with respect to the number of XOR operations at the encoding, but not for short write operations. For small symbol sizes EVENODD and RDP have the same disk access pattern as RAID6, while RM2 and the X-code incur a high recovery cost with two failed disks. We describe variations to RAID5 and RAID6 organizations, including clustered RAID, different methods to update parities, rebuild processing, disk scrubbing to eliminate sector errors, and the intra-disk redundancy (IDR) method to deal with sector errors. We summarize the results of recent studies of failures in hard disk drives. We describe Markov chain reliability models to estimate RAID mean time to data loss (MTTDL) taking into account sector errors and the effect of disk scrubbing. Numerical results show that RAID5 plus IDR attains the same MTTDL level as RAID6, while incurring a lower performance penalty. We conclude with a survey of analytic and simulation studies of RAID performance and tools and benchmarks for RAID performance evaluation.

Performance of Two-Disk Failure-Tolerant Disk Arrays

RAID5 disk arrays use the rebuild process to reconstruct the contents of a failed disk on a spare disk, but this process is unsuccessful if latent sector failures are encountered or a second disk failure occurs. The high cost of data loss has led to two-disk failure-tolerant (2DFT) arrays: RAID6, EVENODD, row-diagonal parity (RDP), and RM2. RAID6 uses Reed-Solomon (RS) codes, whereas the latter three use parity encoding. This paper is concerned with the performance from the viewpoint of disk accesses, which, with an appropriate choice of symbol sizes, is the same for RAID6, EVENODD, and RDP, rather than the computational cost (number of XOR operations). We compare the performance of 2DFTs with each other, as well as RAIDO and RAID5 in normal and degraded operating modes. We derive cost functions for processing discrete disk accesses. The mean response time can be obtained analytically with Poisson arrivals and first-come, first-served (FCFS) scheduling. A simulation is used for validation, calibrating the approximate fork-join response analysis, and shortest-access-time-first (SATF) scheduling. The response time for read requests in RAID6 and RM2 is higher than RAID5 and RAIDO and increases with the fraction of write requests. When there is a single disk failure, RM2 outperforms RAID6 since it has a smaller parity group size than RAID6, but RAID6 outperforms RM2 with two disk failures because of its costlier recovery process. Disk loads in RM2 and RAID6 in degraded mode are unbalanced, and a solution based on pseudorandom permutations is proposed for this purpose

Mirrored Disk Organization Reliability Analysis

Disk mirroring or RAID level 1 (RAID1) is a popular paradigm to achieve fault tolerance and a higher disk access bandwidth for read requests. We consider four RAID1 organizations: basic mirroring, group rotate declustering, interleaved declustering, and chained declustering, where the last three organizations attain a more balanced load than basic mirroring when disk failures occur. We first obtain the number of configurations, A(n, i), which do not result in data loss when i out of n disks have failed. The probability of no data loss in this case is A(n, i)/matrix of(n, i). The reliability of each RAID1 organization is the summation over 1 les i les n/2 of A(n, i)rn-i (1 - r) i, where r denotes the reliability of each disk. A closed-form expression for A(n, i) is obtained easily for the first three organizations. We present a relatively simple derivation of the expression for A(n, i) for the chained declustering method, which includes a correctness proof. We also discuss the routing of read requests to balance disk loads, especially when there are disk failures, to maximize the attainable throughput

Multi-level RAID for very large disk arrays

Very Large Disk Arrays - VLDAs have been developed to cope with the rapid increase in the volume of data generated requiring ultrareliable storage. Bricks or Storage Nodes - SNs holding a dozen or more disks are cost effective VLDA building blocks, since they cost less than traditional disk arrays. We utilize the Multilevel RAID - MRAID paradigm for protecting both SNs and their disks. Each SN is a k-disk-failure-tolerant kDFT array, while replication or l-node failure tolerance - lNFTs paradigm is applied at the SN level. For example, RAID1(M)/5(N) denotes a RAID1 at the higher level with a degree of replication M and each virtual disk is an SN configured as a RAID5 with N physical disks. We provide the data layout for RAID5/5 and RAID6/5 MRAIDs and give examples of updating data and recovering lost data. The former requires storage transactions to ensure the atomicity of storage updates. We discuss some weaknesses in reliability modeling in RAID5 and give examples of an asymptotic expansion method to compare the reliability of several MRAID organizations. We outline the reliability analysis of Markov chain models of VLDAs and briefly report on conclusions from simulation results. In Conclusions we outline areas for further research.

Disk scheduling policies with lookahead

Advances in magnetic recording technology have resulted in a rapid increase in disk capacities, but improvements in the mechanical characteristics of disks have been quite modest. For example the access time to random disk blocks has decreased by a mere factor of two, while disk capacities have increased by several orders of magnitude. High performance OLTP applications subject disks to a very demanding workload, since they require high access rates to randomly distributed disk blocks and gain limited benef£t from caching and prefetching. We address this problem by re-evaluating the performance of some well known disk scheduling methods, before proposing and evaluating extensions to them. A variation to CSCAN takes into account rotational latency, so that the service time of further requests is reduced. A variation to SATF considers the sum of service times of several successive requests in scheduling the next request, so that the arm is moved to a (temporal) neighborhood with many requests. The service time of further requests is discounted, since their immediate processing is not guaranteed. A variation to the SATF policy prioritizes reads with respect to writes and processes winner write requests conditionally, i.e., when the ratio of their service time to that of the winner read request is smaller than a certain threshold. We review previous work to put our work into the proper perspective and discuss plans for future work.

Clustered RAID Arrays and Their Access Costs

RAID5 (resp. RAID6) are two popular RAID designs, which can tolerate one (resp. two) disk failures, but the load of surviving disks doubles (resp. triples) when failures occur. Clustered RAID5 (resp. RAID6) disk arrays utilize a parity group size G, which is smaller than the number of disks N, so that the redundancy level is 1/G (resp. 2/G). This enables the array to sustain a peak throughput closer to normal mode operation; e.g. the load increase for RAID5 in processing read requests is given by α = (G - 1)/(N - 1). Three methods to realize clustered RAID are balanced incomplete blocks designs and nearly random permutations, which are applicable to RAID5 and RAID6, and RM2 where each data block is protected by two parity disks. We derive cost functions for the processing requirements of clustered RAID in normal and degraded modes of operation. For given disk characteristics, the cost functions can be translated into disk service times, which can be used for the performance analysis of disk arrays. Numerical results are used to quantify the level of load increase in order to determine the value of G which maintains an acceptable level of performance in degraded mode operation.

Reconstruct versus read-modify writes in RAID

RAID5 (Redundant Arrays of Independent Disk level 5) is a popular paradigm, which uses parity to protect against single disk failures. A major shortcoming of RAID5 is the small write penalty, i.e., the cost of updating parity when a data block is modified. Read-modify writes and reconstruct writes are alternative methods for updating small data and parity blocks. We use a queuing formulation to determine conditions under which one method outperforms the other. Our analysis shows that in the case of RAID6 and more generally disk arrays with k check disks tolerating k disk failures, RCW outperforms RMW for higher values of N and G. We note that clustered RAID and variable scope of parity protection methods favor reconstruct writes. A dynamic scheme to determine the more desirable policy based on the availability of appropriate cached blocks is proposed.

Reliability and Performance of Mirrored Disk Organizations

Disk mirroring or redundant array of independent disk (RAID) level 1 is a popular paradigm to achieve fault-tolerance and a higher disk access bandwidth for read requests. We consider four RAID1 organizations: basic mirroring (BM), group rotate declustering (GRD), interleaved declustering (ID) and chained declustering (CD). The last three organizations provide a more balanced disk load than BM when a single disk fails, but are more susceptible to data loss than BM when additional disks fail. We compare the four organizations from the viewpoint of: (i) reliability [we quote results from [Thomasian, A. and Blaum, M. (2006) Reliability analysis of mirrored disks. IEEE Trans. Comput., 55, 1640–1644.]] (ii) performability, (iii) performance. In (ii) and (iii), we postulate discrete requests to small randomly placed blocks. For (ii), we compute the mean number of disk requests processed to the point where data loss occurs. For the sake of tractability in (iii), the response time is obtained assuming Poisson arrivals and a first come first serve policy. The ranking from the viewpoint of reliability and performability is: BM, CD, GRD, ID (with two clusters). BM and CD provide the worst performance, ID has a better performance than BM and CD, but is outperformed by GRD. These results are also shown using an asymptotic expansion method. Areas of further research are also discussed, which include the applicability of the mirroring techniques to storage bricks.

Some new disk scheduling policies and their performance

Advances in magnetic recording technology have resulted in a rapid increase in disk capacities, but improvements in the mechanical characteristics of disks have been quite modest. For example, the access time to random disk blocks has decreased by a mere factor of two, while disk capacities have increased by several orders of magnitude. OLTP applications subject disks to a very demanding workload consisting of accesses to randomly distributed disk blocks and gain limited benefit from caching and prefetching (at the onboard disk cache). We propose some new disk scheduling methods to address the limited disk access bandwidth problem.Some well-known disk scheduling methods are: (i) FCFS. (ii) Shortest Seek Time First (SSTF). (iii) SCAN and Cyclical SCAN (CSCAN). The latter moves the disk arm to its beginning point after each SCAN so that requests at all disk cylinders are treated symmetrically. (iv) CSCAN with a lookahead of next i requests (CSCAN-LAi) takes into account latency to reorder their processing to minimize the sum of their service times. (v) Shortest Access Time First (SATF), which provides the best performance [2]. (vi) SATF with lookahead for i requests (SATF-LAi).In the case of SATF-LAi with i = 2 after the completion of request X the scheduler chooses requests A and B such that the sum of their service times processed consecutively, i.e., tX,A + atA,B, is minimized. In SATF with flexible lookahead only request A is definitely processed and request B is processed provided that it is selected in the next round. We refer to a as the discount factor (0 ≤ a ≤ 1), because less weight is attached to the service time of request B, since it may not be processed after request A. The case a = 0 corresponds to pure SATF. When a = 1 we consider a variant called SATF with fixed lookahead where B is processed unconditionally after A before any other other (perhaps more favorable recent) requests. Thus requests are processed two at a time, unless only one request is available. More generally requests in the temporal neighborhood of request A are given higher priority.