A Unified Approach to Diffusion Analysis of Queues with General Patience-Time Distributions

• 成果类型: Article
• 署名作者: Huang, Junfei; Zhang, Hanqin; Zhang, Jiheng
• 署名单位: Chinese University of Hong Kong; National University of Singapore; Hong Kong University of Science & Technology
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2015.0772
• 发表日期: 2016
• 页码: 1135-1160
• 关键词: many-server queues; nondegenerate slowdown; impatient customers; queuing-networks; g/gi/n queue; heavy; ABANDONMENT; regime; approximation; asymptotics

摘要

We propose a unified approach to establishing diffusion approximations for queues with impatient customers within a general framework of scaling customer patience time. The approach consists of two steps. The first step is to show that the diffusion-scaled abandonment process is asymptotically close to a function of the diffusion-scaled queue length process under appropriate conditions. The second step is to construct a continuous mapping not only to characterize the system dynamics using the system primitives, but also to help verify the conditions needed in the first step. The diffusion approximations can then be obtained by applying the continuous mapping theorem. The approach has two advantages: (i) it provides a unified procedure to establish the diffusion approximations regardless of the structure of the queueing model or the type of patience-time scaling; and (ii) it makes the diffusion analysis of queues with customer abandonment essentially the same as the diffusion analysis of queues without customer abandonment. We demonstrate the application of this approach via the single-server system with Markov-modulated service speeds in the traditional heavy-traffic regime and the many-server system in the Halfin-Whitt regime and the nondegenerate slowdown regime.