By Levner E. (ed.)
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The arbitrator schedules the most urgent between the two submitted. It is interesting to note as this rule may produce . arbitrarily bad sequences in scenario Consider the instance of reported in Table 6, where n >> m >> 1. Figure 4 illustrates a global optimum. Note that there are m + 1 tardy jobs for Agent 1 and 0 for Agent 2. Table 6. An instance of A 1-optimal schedule is obtained if Agent 1 just skips A1 in the list of candidate jobs, and submits all the others, from 2 to m + 3. The resulting schedule is illustrated in Figure 5: in this case there are n + m + 1 tardy jobs and just one job of Agent 1 is tardy.
Hereafter this ratio is referred to as density . For a job i, with processing time pi and positive weight wi , . In classical single machine scheduling, a sequence of jobs in non-decreasing . By standard pairwise interchange order of density (WSPT-order) minimizes arguments, it is easy to prove the following: , if both agents propose WSPTProposition 4: In the scenario ordered candidate jobs, the resulting sequence is 1- optimal, 2-optimal, R -optimal and globally optimal. Figure 2. Schedule is a Nash Equilibrium 42 Multiprocessor Scheduling: Theory and Applications Note that if we view the scenario as a game in which each agent's strategy is the order in which the jobs are submitted, Proposition 4 can be equivalently stated saying that the pair of strategies consisting in ordering the jobs in WSPT is the only Nash equilibrium.
Generally speaking, each agent aims at maximizing its own utility (or minimizing its costs). To pursue this goal, the agents have to make their decisions in an environment which is strongly characterized by the presence of the other agents, and will therefore have to carry out a suitable negotiation process. As a consequence, a decision support model must suitably represent the way in which the agents will interact to reach a mutually acceptable allocation. The next two chapters present in some detail two major modeling and procedural paradigms to address bargaining issues in a scheduling environment.