Join the Shortest Queue with Many Servers. The Heavy-Traffic Asymptotics

• 成果类型: Article
• 署名作者: Eschenfeldt, Patrick; Gamarnik, David
• 署名单位: Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT)
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2017.0887
• 发表日期: 2018
• 页码: 867-886
• 关键词: limit-theorems; INEQUALITY; customers; systems

摘要

We consider queueing systems with n parallel queues under a Join the Shortest Queue (JSQ) policy in the Halfin-Whitt heavy-traffic regime. We use the martingale method to prove that a scaled process counting the number of idle servers and queues of length exactly two weakly converges to a two-dimensional reflected Ornstein-Uhlenbeck process, while processes counting longer queues converge to a deterministic system decaying to zero in constant time. This limiting system is comparable to that of the traditional Halfin-Whitt model, but there are key differences in the queueing behavior of the JSQ model. In particular, only a vanishing fraction of customers will have to wait, but those who do incur a constant order waiting time.