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The start time ts (e, P) of an edge e E on a processor P P is the function ts : E P Q+ Let R = L1 , L2 , , Ll be the route for the communication of edge e E from Psrc P to Pdst P, Psrc = Pdst The nish time of e on Psrc is tf (e, Psrc ) = ts (e, Psrc ) + os (e, Psrc ) + is (e, L1 ) and on Pdst is tf (e, Pdst ) = ts (e, Pdst ) + or (e, Pdst ) + ir (e, Ll ) (82) (81)
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Figure 87 illustrates scheduling under the involvement contention model The edge eAB of the simple task graph is not only scheduled on the two links between the communicating processors, but also on both processors in order to re ect the incurred overhead and their involvement Consequently, the execution time of an edge on a processor is the sum of the overhead and the involvement (see Eqs (81) and (82)) Clearly, for a sensible scheduling of the edges on the processors, some conditions must be formulated Like a node, an edge scheduled on a processor represents computation, precisely the computation necessary for the communication of the edge Thus, the exclusive processor allocation Condition 41 also applies to edges scheduled on the processors Remember, this is a requirement imposed by Property 2 of the system model (De nition 81) An edge is only scheduled once on a processor, from which it follows that a separation between the overhead and the involvement is not possible This is already ensured by the de nition of the nish time of an edge on a processor (Eqs (81) and (82)) Even for systems where the separation of overhead and involvement might
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Figure 87 Scheduling under the involvement contention model: edges are also scheduled on the processors S is a switch or other processor
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PROCESSOR INVOLVEMENT IN COMMUNICATION
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occur, the improvement on accuracy by allowing the separation would presumably not outweigh the more complicated scheduling For a meaningful and feasible schedule, the scheduling of the edges on the processors must obey Condition 81 Condition 81 (Causality in Involvement Scheduling) Let G = (V, E, w, c) be a task graph and M = ((N, P, D, H, b), , o, i) a parallel system Let R = L1 , L2 , , Ll be the route for the communication of edge eij E, ni , nj V, from Psrc P to Pdst P, Psrc = Pdst To assure the node strictness (De nition 38) of ni ts (eij , Psrc ) tf (ni , Psrc ) (83)
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Edge eij can be transferred on the rst link L1 only after the overhead is completed on the source processor Psrc : ts (eij , L1 ) ts (eij , Psrc ) + os (eij , Psrc ) (84)
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To assure the causality of the direct involvement on the destination processor Pdst , ts (eij , Pdst ) tf (eij , Ll ) ir (eij , Ll ) (85)
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The three inequalities can be observed in effect in Figure 87 Edge eAB starts on P1 after the origin node A nishes (inequality (83)) On the rst link L1 , eAB starts after the overhead nishes on P1 (inequality (84)), at which time the involvement of P1 begins And last, eAB starts on P2 so that the involvement nishes at the same time as eij on L2 (inequality (85)) Discussion of Condition 81 Order of Overhead Involvement For simplicity, Condition 81 xes the temporal order between overhead and involvement in the way it has been assumed so far: rst the overhead then the involvement on the source processor (established by Eq (84)) and the other way around on the destination processor (established by Eq (85)) Freedom of Node Edge Order It is not required by Condition 81 that an edge eij is scheduled on the source processor immediately after the termination of the origin node ni If ni has multiple leaving edges, this is not possible anyway only one edge can start immediately after ni but other edges, not leaving ni , or even nodes might be scheduled between ni and eij The same holds for eij and the receiving node nj , whereby eij must of course nish before nj starts This freedom in the scheduling order of the nodes and edges on the processors grants a high degree of exibility to scheduling algorithms Communications can be delayed, while executing other nodes
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