Load rebalancing games in dynamic systems with migration costs

We consider the following dynamic load balancing game: Given an initial assignment of jobs to identical parallel machines, the system is modified; specifically, some machines are added or removed. Each job's cost is the load on the machine it is assigned to; thus, when machines are added, jobs have an incentive to migrate to the new unloaded machines. When machines are removed, the jobs assigned to them must be reassigned. Consequently, other jobs might also benefit from migrations. In our job-extension penalty model, for a given extension parameterδ≥0, if the machine on which a job is assigned to in the modified schedule is different from its initial machine, then the job's processing time is extended by δ. We provide answers to the basic questions arising in this model. Namely, the existence and calculation of a Nash Equilibrium and a Strong Nash Equilibrium, and their inefficiency compared to an optimal schedule. Our results show that the existence of job-migration penalties might lead to poor stable schedules; however, if the modification is a result of a sequence of improvement steps or, better, if the sequence of improvement steps can be supervised in some way (by forcing the jobs to play in a specific order) then any stable modified schedule approximates well an optimal one. Our work adds two realistic considerations to the study of job scheduling games: the analysis of the common situation in which systems are upgraded or suffer from failures, and the practical fact according to which job migrations are associated with a cost.