Non-monetary fair scheduling: a cooperative game theory approach

We consider a multi-organizational system in which each organization contributes processors to the global pool but also jobs to be processed on the common resources. The fairness of the scheduling algorithm is essential for the stability and even for the existence of such systems (as organizations may refuse to join an unfair system). We consider on-line, non-clairvoyant scheduling of sequential jobs. The started jobs cannot be stopped, canceled, preempted, or moved to other processors. We consider identical processors, but most of our results can be extended to related or unrelated processors. We model the fair scheduling problem as a cooperative game and we use the Shapley value to determine the ideal fair schedule. In contrast to the current literature, we do not use money to assess the relative utilities of jobs. Instead, to calculate the contribution of an organization, we determine how the presence of this organization influences the performance of other organizations. Our approach can be used with arbitrary utility function (e.g., flow time, tardiness, resource utilization), but we argue that the utility function should be strategy resilient. The organizations should be discouraged from splitting, merging or delaying their jobs. We present the unique (to within a multiplicative and additive constants) strategy resilient utility function. We show that the problem of fair scheduling is NP-hard and hard to approximate. However, for unit-size jobs, we present a fully polynomial-time randomized approximation scheme (FPRAS). We also show that the problem parametrized with the number of organizations is fixed parameter tractable (FPT). In cooperative game theory, the Shapley value is considered in many contexts as "the" fair solution. Our results show that, although the problem for the large number of organizations is computationally hard, this solution concept can be used in scheduling (for instance, as a benchmark for measuring fairness of heuristic algorithms).