Scheduling games on uniform machines with activation cost

In this paper, we investigate job scheduling games on uniform machines. Activating each machine incurs an activation cost in accordance with its speed. Jobs are self-decision-making players choosing machines with the objective of minimizing their own individual cost. Here, a job's individual cost is defined as the load of its chosen machine plus its activation cost, which is proportionally shared with respect to its size. We first propose an algorithm when there are m kinds of unlimited number of uniform machines, and it is proved to be a Nash Equilibrium (NE) algorithm by using the induction method. Because of equilibria always being far from optimal solutions, the inefficiency of pure NE is measured with the social objective of minimizing the maximum individual cost. Assuming that the longest job is α   times as large as the unit activation cost, together with α≥a2α≥a2, we obtain that both the Price of Anarchy (PoA) and Price of Stability (PoS) are equal to 1. Otherwise, they are bounded by (α+1)/2α. Finally, we provide an example to illustrate the tight bound.