We investigate the long-run behavior of single-server queues with Hawkes arrivals and general service distributions and related optimization problems. In detail, utilizing novel coupling techniques, we establish finite moment bounds for the stationary distribution of the workload and busy period processes. In addition, we are able to show that, those queueing processes converge exponentially fast to their stationary distribution. Based on these theoretic results, we develop an efficient numerical algorithm to solve the optimal staffing problem for the Hawkes queues in a data-driven manner. Numerical results indicate a sharp difference in staffing for Hawkes queues, compared to the classic GI/GI/1 model, especially in the heavy-traffic regime.

翻译：本文研究了具有霍克斯到达过程和一般服务分布的单服务器队列的长期行为及相关优化问题。具体而言，利用新颖的耦合技术，我们建立了工作负荷和忙期过程平稳分布的有限矩界。此外，我们证明了这些排队过程以指数速度收敛到其平稳分布。基于这些理论结果，我们开发了一种高效数值算法，以数据驱动的方式解决霍克斯队列的最优人员配置问题。数值结果表明，与经典的GI/GI/1模型相比，霍克斯队列在人员配置上存在显著差异，尤其是在重流量区域。