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(84) the start of the edge on the rst link must be further delayed by at least the time of the communication overhead Hence, the rst case (k = 1) of inequality (77) of Condition 73, B tf (ni ) + (eij , L1 ), becomes B ts (eij , Psrc ) + os (eij , Psrc ) + (eij , L1 ) (814)
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The calculation of the edge s start time on the rst link (De nition 79) must be modi ed accordingly Thus, the rst case (k = 1) of Eq (78), ts (eij , L1 ) = max{A, tf (ni )}, becomes ts (eij , L1 ) = max{A, ts (eij , Psrc ) + os (eij , Psrc )} (815)
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The rest of the edge scheduling procedure is completely unaffected by the scheduling of the edges on the processors and remains unmodi ed Node Scheduling To adapt the scheduling of the nodes to the new model, it is only necessary to rede ne the total nish time of the edge, which was originally de ned for the classic model in De nition 46 and rede ned for the contention model in De nition 711 De nition 85 (Edge Finish Time) Let G = (V, E, w, c) be a task graph and M = ((N, P, D, H, b), , o, i) a parallel system The nish time of eij E, ni , nj V, communicated from processor Psrc to Pdst , Psrc , Pdst P, is tf (eij , Psrc , Pdst ) = tf (ni , Psrc ) if Psrc = Pdst tf (eij , Pdst ) otherwise
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(816)
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As with the edge scheduling discussed earlier, the rest of the node scheduling procedure is completely unaffected by the scheduling of the edges on the processors and remains unmodi ed
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823 Task Graph For scheduling under the contention model, the communication cost (ie, the edge weight) was de ned to be the average time a link is occupied with the communication associated with the edge (De nition 77) As elaborated in Section 741, this de nition is compatible with the edge weight under the classic model, which represents the communication delay Scheduling under the involvement contention model is based on edge scheduling and adopts the communication cost de nition of edge scheduling Thus, it remains unmodi ed Yet, in practice, some caution is indicated Under the contention model, the time an edge spends on a link corresponds to the communication delay In consequence, preand postprocessing parts are either ignored or accredited to the communication cost
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represented by the edge weight Under the involvement contention model, the communication delay is also composed of the overheads in icted on the communicating processors; hence, these overheads are not included in the edge weights (see below) Thus, task graph weights determined for different models are not necessarily identical Path Length Most metrics used for analyzing a task graph and its nodes are based on the path length, for example, node levels or the critical pathAs stated in Section 44, the length of a path can be interpreted as its execution time, when all communications are interprocessor communications Under the involvement contention model, the total communication time, or communication delay, does not merely consist of the edge weight, but also of the overheads incurred by the sending and receiving processors Thus, the path length must be adapted in order to still represent the execution time of the path De nition 86 (Path Length with Overhead) Let G = (V, E, w, c) be a task graph and M = ((N, P, D, H, b), , o, i) a parallel system Let os,r (e) = P P os,r (e, P)/|P| be the average overhead The length of a path p in G is len( p) =
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