Motivated by applications in job scheduling, queuing networks, and load balancing in cyber-physical systems, we develop and analyze a game-theoretic framework to balance the load among servers in both static and dynamic settings. In these applications, jobs/tasks are often held by selfish entities that do not want to coordinate with each other, yet the goal is to balance the load among servers in a distributed manner. First, we provide a static game formulation in which each player holds a job with a certain processing requirement and wants to schedule it fractionally among a set of heterogeneous servers to minimize its average processing time. We show that this static game is a potential game and admits a pure Nash equilibrium (NE). In particular, the best-response dynamics converge to such an NE after $n$ iterations, where $n$ is the number of players. We then extend our results to a dynamic game setting, where jobs arrive and get processed in the system, and players observe the load (state) on the servers to decide how to schedule their jobs among the servers in order to minimize their averaged cumulative processing time. In this setting, we show that if the players update their strategies using dynamic best-response strategies, the system eventually becomes fully load-balanced and the players' strategies converge to the pure NE of the static game. In particular, we show that the convergence time scales only polynomially with respect to the game parameters. Finally, we provide numerical results to evaluate the performance of our proposed algorithms under both static and dynamic settings.

翻译：受作业调度、排队网络及信息物理系统负载均衡等应用的启发，我们开发并分析了一个博弈论框架，用于在静态与动态场景下实现服务器间的负载均衡。在这些应用中，作业/任务通常由不愿相互协调的自利实体持有，而目标是以分布式方式平衡服务器间的负载。首先，我们提出一种静态博弈模型，其中每个参与者持有一个具有特定处理需求的作业，并希望将其按比例分配到一组异构服务器上，以最小化其平均处理时间。我们证明该静态博弈是一个势博弈，且存在纯纳什均衡。特别地，最优响应动态经过 $n$ 次迭代后收敛至该均衡，其中 $n$ 为参与者数量。随后，我们将结果扩展至动态博弈场景，其中作业持续到达系统并被处理，参与者通过观测服务器负载状态来决定如何在服务器间调度其作业，以最小化其平均累计处理时间。在此场景下，我们证明若参与者采用动态最优响应策略更新其策略，系统最终将达到完全负载均衡状态，且参与者的策略将收敛至静态博弈的纯纳什均衡。特别地，我们证明收敛时间仅随博弈参数呈多项式规模增长。最后，我们通过数值实验结果评估了所提算法在静态与动态场景下的性能。