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on P Edge eij can be scheduled on P within [A, B] if B A B os (eij , Psrc ) + is (eij , L1 ) if P = Psrc , or (eij , Pdst ) + ir (eij , Ll ) if P = Pdst
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tf (ni , Psrc ) + os (eij , Psrc ) + is (eij , L1 ) if P = Psrc if P = Pdst tf (eij , Ll ) + or (eij , Pdst )
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This condition corresponds to Condition 73 for edge scheduling on the links It ensures that the time interval [A, B] adheres to the inequalities (83) and (85) of the causality Condition 81 For a given time interval, the start time of eij on Psrc and Pdst is determined as follows De nition 84 (Edge Scheduling on a Processor) Let [A, B] be an idle time interval on P adhering to Condition 82 The start time of eij on P is ts (eij , P) = if P = Psrc max{A, tf (ni )} max{A, tf (eij , Ll ) ir (eij , Ll )} if P = Pdst
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So, as with the scheduling on the links (De nition 79), the edge is scheduled as early as possible within the limits of the interval Of course, the choice of the interval should follow the same policy on the links and on the processors; that is, either the end or insertion technique should be used The Overhead and Involvement Functions In all de nitions and conditions related to scheduling under the involvement contention model, overhead and involvement are simply treated as functions for the sake of generality Nevertheless, these functions have a certain nature for most parallel systems, which is studied next Generally, the overhead and the involvement functions are at most linear in the communication volume for probably all parallel systems The overhead is of constant time for many parallel systems and their communication environment As mentioned before, the overhead is normally the path through the communication layer Thus, os (e, P) = os (P), or (e, P) = or (P), (89)
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that is, the overhead merely depends on the processor speed, so it is constant for homogeneous systems Sometimes, however, the data to be transferred is copied between buffers during the overhead by which the overhead becomes a linear function of the communication volume The involvement is by de nition not longer than the network activity On the sending side, the involvement stops at the latest when all data has been sent across
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the rst link, and on the receiving side, the involvement does not start before the communication begins on the last link Hence, the involvement must be smaller than the communication time on the respective link: is (e, L1 ) (e, L1 ), ir (e, Ll ) (e, Ll ) (810) (811)
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Since the involvement represents the actual data transfer performed by the processor, this time is typically proportional to the communication time of the rst/last link: is (e, L1 ) = Cs (e, L1 ), ir (e, Ll ) = Cr (e, Ll ), (812) (813)
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where Cs , Cr [0, 1] For example, Cs = Cr = 05 for the two-sided memory copy as shown in Figure 83(c) Consequently, in many common parallel systems, the overhead and the involvement are simply characterized by constants Furthermore, these constants are independent of the task graph and they only need to be determined once to describe the target system for the scheduling of any task graph On the other hand, the functions might also be used to describe in detail the behavior of interprocessor communication For instance, consider a parallel system, where the overhead of third-party communication is quite high For few data words, other means of communication (eg, memory copy by the processor) might be cheaper than setting up the communication device In this case, there is a threshold data size for which it is worthwhile to employ the communication device The overhead and involvement functions can be formulated accordingly Similar things can happen on the protocol level of the communication environment, where an optimized procedure is employed for short messages (White and Bova [199]) Note that the size of the involvement does not depend on the contention in the network The assumption is that if the processor has to wait to send or receive a communication due to contention, this wait is passive or nonblocking, which means it can perform other operations in the meantime
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822 Node and Edge Scheduling Few alterations are imposed by the new model on the edge scheduling on the links and on the scheduling of the nodes Edge Scheduling on Links While inequalities (83) and (85) of Condition 81 are newly introduced for the scheduling of the edges on processors, Eq (84) substitutes a constraint of edge scheduling The edge scheduling Condition 73 assures that an edge starts on the rst link of a route after the origin node has nished With Eq
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