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= (c2,1 f 2,3 v10 f 2,4 v11 ) + f 2,3 v5 + f 2,4 v6 + f 2,5 v7 However, in this case the number of data block transmissions is ve, two blocks more than that of reusing c1,1 . Thus, in the SRDU algorithm, the system will rst compute the redundant data update overhead for all reusable redundant blocks and select the one with the lowest overhead for reuse.



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Note that defined-variable assignments relate current pj values on the lefthand side to past values of the state variables qi and to already computed current values of other defined variables pj This means that the defined-variable assignments (2-1a) must be properly sorted into a procedural order that produces successive pj values without algebraic loops, just as in Section 1-92 Following the defined-variable assignments (2-1a), we compute the current values Qi of our state variables qi with N difference-equation assignments





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In addition to reusing old redundant data, we can also reduce data block movements by caching and reusing data blocks already received by the redundant node. Reconsider the previous example of computing c1,1 (cf. equation (15.11)), the data blocks needed are v4 , v8 , and v9 . However, v4 has already been sent to the redundant node when computing c0,1 (cf. equation (15.7)) and thus can simply be reused if it is cached. As a redundant block is computed from a parity group of (N h) data blocks, we only need to cache the (N h) most recently received data blocks.

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In this section we evaluate performance of the SRDU algorithm using simulation Beginning with a small system, we add new nodes to the system and then apply the SRDU algorithm to update the redundant data blocks Performance is measured by the number of data blocks that need to be sent to the redundant nodes for redundant data update or simply called redundant data update overhead The total number of data blocks is 40,000 and is xed throughout the simulation For simplicity the redundant data update overhead for updating one redundant node is presented We will return to the issue of updating multiple redundant data nodes in Section 155 In the rst experiment, we investigate the redundant data update overhead in continuous system growth We begin with a system of ve data nodes and one redundant node.

Qi = Fi(t; q1, q2, ,qN; p1, p2, ) (i = 1, 2, , N)

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Then we add a new node to the system one by one, each time the redundant data blocks are completely updated using the SRDU algorithm This continues until the system grows to 400 data nodes Figure 1513 plots the redundant data update overhead versus data node size from 6 to 400 As expected, the simple algorithm (sending all data blocks to the redundant node) performs the worst On the other hand, regenerating redundant data using a centralized archive server incurs the least overhead These two curves serve respectively as the upper bound and lower bound for redundant data update overhead Surprisingly, direct reuse of the original redundant data does not result in signi cant savings This is because the algorithm maintains the same data order within the parity group in computing the redundant data, and thus signi cantly reduces the opportunity for redundant data reuse.

(2-1b)

Once this restriction is relaxed by reshuf ing the parity group, the redundant data update overhead is reduced by half to around 20,000 blocks Reuse of cached data blocks further reduces the overhead by half to around 10,000 blocks With all three techniques applied, the SRDU algorithm can reduce the redundant data update overhead by as much as 70% In the previous experiment, we completely update all redundant data blocks before adding another new node Clearly this is inef cient if new nodes are added frequently or added to the system in a batch To address this issue, we conduct a second experiment where multiple nodes, say W , are added to the system before the redundant data are updated Figure 1514 plots the redundant data update overhead versus the batch size W for initial system size of 80 nodes.

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