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To illustrate and compare performances of the rebuild algorithms we consider a system of N S = 5 active servers and one spare server. Each server has 200GB storage, so the system has a total of 1TB storage, including the redundant units. We assume a server transfer capacity of 600Mbps, e.g., using Gigabit Ethernet links. Figure 14.7 plots the data rebuild rate versus server utilization for all the rebuild algorithms. We include the data rebuild rate for reloading data from back-up for the sake of comparison. Note that this data rebuild rate is also the upper bound. We observe that for baseline rebuild, the data rebuild rate is constant at SS /(N S 1) for (1 1/(N S 1)), even if the remaining active servers are lightly loaded and have idle capacities available. As the system utilization approaches one, the rebuild rate drops quickly. Distributed rebuild performs better than baseline rebuild when the server utilization is low (e.g., 0.55), but it deteriorates earlier when the system utilization increases. This is because in distributed rebuild the active servers need to receive data transmissions from other servers in addition to sending data to other servers, and thus consume considerably more transfer capacity than baseline rebuild. Finally, as expected, the mixed distributed baseline rebuild gives the best performance in all cases.

Department of Mathematics and Computing Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands; and CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

600 Data Rebuild Rate (Mbps)

Since only fundamental constants are used for the chromatographic estimation of q, and no adjustable parameters are present, as exist in previous models, this is the best quantitative theory in existence that describes the chromatographic retention of polyelectrolytes on ionexchange columns Its applicability to other biopolymers, such as highly charged nucleic acids, has yet to be determined..

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Recently, there has been much interest in the behavior of queues with heavy-tailed service time distributions. This interest has been triggered by a large number of traf c measurements on modern communication network traf c (e.g., see Willinger et al. [38] for Ethernet LAN traf c, Paxson and Floyd [30] for WAN traf c, and Beran et al. [3] for VBR video traf c; see also various chapters in this volume). These measurements and their statistical analysis (e.g., see Leland et al. [26]) suggest that modern communication traf c often possesses the properties of selfsimilarity and long-range dependence. A natural possibility to introduce long-range dependence in an input traf c process is to take a uid queue and to assume that at least one of the input quantities I (on or off periods in the uid queue fed by on=off sources) has the following ``heavy-tail'' behavior: P I > t $ hn t n ;

Figure 14.8 shows the time required to completely rebuild a server s worth of data (200GB). Interestingly, for system utilization 0.75 performances of the baseline rebuild algorithm and the mixed distributed baseline algorithm converge. This is because at high system utilization the remaining active servers become the bottleneck, and so it is better to switch to the baseline rebuild algorithm which consumes less transfer capacities of the active servers than distributed rebuild. In this case the mixed distributed baseline algorithm simply allocates all transfer capacity to baseline rebuild. Finally, we plot in Figure 14.9 the rebuild time versus the number of servers in the system under a system utilization of = 0.5. The key observation is that the baseline rebuild algorithm is not scalable the rebuild time increases with the number of servers in the system. This is because in baseline rebuild the maximum rebuild rate is limited by the transfer capacity of the spare server. Thus, as the number of servers increases, so will the number of data/redundant units that need to be sent to the spare server to rebuild an unavailable data unit, thereby resulting in longer rebuild time. By contrast, the rebuild times of distributed rebuild and mixed distributed baseline rebuild do not increase signi cantly with increases in the number of servers and so are much more

6:1

W. S. Hancock, Ed., High Performance Liquid Chromatography in Biotechnology, John Wiley & Sons, Inc., New York, 1990. A. Fallon, R. F. G. Booth, and L. D. Bell, Applications of HPLC in Biochemistry, Elsevier, New York, 1987. J.-C. Janson and L. Ryden, Eds., Protein Puri cation: Principles, High-Resolution Methods and Applications, VCH Publishers, New York, 1989. P. R. Haddad and P. E. Jackson, Ion Chromatography: Principles and Applications, Elsevier, New York, 1987. S. Yamamoto, Ion-Exchange Chromatography of Proteins, Marcel Dekker, New York, 1988. I. M. Chaiken, Analytical Af nity Chromatography, CRC Press, Boca Raton, FL, 1987. P. D. G. Dean, W. S. Johnson, and F. A. Middle, Af nity Chromatography: A Practical Approach, Oxford University Press, Washington, DC, 1985. D. J. Winzor, From gel ltration to biosensor technology: the development of chromatography for the characterization of protein interactions. Review article in J. Mol. Recognit. 13, 2000, 279 298.

Self-Similar Network Traf c and Performance Evaluation, Edited by Kihong Park and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc.

Rebuild Time (hours)

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