This paper deals with a scheduling problem with parallel-batching machines from a game theoretic perspective.There are m parallel-batching machines,each of which can handle up to b jobs simultaneously as a batch.The processing time of a batch is the time required for processing the longest job in the batch,and all the jobs in a batch start and complete at the same time.There are n jobs.Each job is owned by a rational and selfish agent.The agent of a job chooses a machine for processing its own job.The aim of each agent is to minimize the completion time of its job while the system tries tominimize the maximal completion time over all jobs,the makespan.We design a coordination mechanism for the scheduling game problem.We discuss the existence of Nash equilibria of the mechanism and showthat it has a price of anarchy 2.
Let G be a graph which can be embedded in a surface of nonnegative Euler characteristic.In this paper,it is proved that the total chromatic number of G is △(G)+1 if △(G)9,where △(G)is the maximum degree of G.
