Inefficiency of equilibria for scheduling game with machine activation costs

In this paper, we study the scheduling game with machine activation costs. A set of jobs is to be processed on parallel identical machines. The number of machines available is unlimited, and an activation cost is needed whenever a machine is activated in order to process jobs. Each job chooses a machine on which it wants to be processed. The cost of a job is the sum of the load of the machine it chooses and its shared activated cost. The social cost is the total cost of all jobs. Representing the Price of Anarchy (PoA) and Price of Stability (PoS) as functions of the number of jobs, we get the tight bounds of PoA and PoS. Representing PoA and PoS as functions of the smallest processing time of jobs, asymptotically tight bound of PoA and improved lower and upper bounds of PoS are also given.