Abstract: As an effective tool to improve system performance and regulate queue congestion, the working vacation policy is widely used in queueing systems due to its practicality and conventionality. For example, regular system upgrades can ensure the server functions well. In complex queueing systems, the arrivals and the service processes typically evolve at a much faster rate than changes in the server status. Thus, discrete customers can be considered as the continuous fluid that flows into and out of a storage space. In addition to physical queues such as reservoirs, flexible manufacturing systems, and transportation flow, fluid queues are also widely applied to communication fields such as big data, cloud computing, and electronic information transfers. Considering that the fluid is rational and tends to maximize its interest, the strategic behavior of the fluid has received extensive attention from the perspective of economics, sociology, and cognitive psychology. The main purpose of economic analysis is to investigate the decentralized behavior and socially optimal strategies, and rational suggestions can be provided for individuals and managers to accurately make more advisable decisions in complex queues.
Based on the extensive application background of the vacation strategy and the important theoretical research value of the fluid queue, this paper makes a game-theoretic analysis of a continuous fluid queue with setup times and working vacations. The individual balking behavior and social utility maximization are derived based on the noncooperative game theory and optimization theory under different information precision levels, namely fully observable and almost observable cases. The performance indicators are obtained based on renewal processes, total-probability decomposition techniques, and the standard theory of ordinary differential equations. The admission fee strategy for the social designers could be imposed on the fluid and regulated system parameters dynamically, which can maximize social benefits without compromising individual interests. The numerical results show that the higher balking threshold may reduce the maximal admission fees, and the fees are also related to the effective arrival rate and the unit price.
The constructed fluid model can be applied to explore the issue of how to dynamically control the rapid transmission of continuous data and information to maximize social benefits. P2P networks, are also known as peer-to-peer networks, in which each node can either provide resources services or send service requests. Due to the large number of nodes involved in P2P networks, how to reduce the total energy consumption of the system has become a hot research topic for scholars. In a network service system, the resources requests are equivalent to fluid arrivals, the normal transmission process of resource requests is abstracted as the regular busy period, the semi-dormant process of ordinary nodes as the working vacation period, and the activation process of ordinary nodes as the set-up period. Besides, the nodes process data is at a lower rate under the semi-sleep strategy. Therefore, the information transmissions of nodes can be modeled as a fluid queueing system, which operates between three modes: the setup period, working vacation period, and normal working period.
This paper extends the existing theoretical research results of fluid vacation queuing systems and provides appropriate management insights for individuals and service providers. It should be noted that the equilibrium analysis of the fluid queuing model in the invisible case seems intractable. Its solvability and subsequent performance analysis are interesting subjects for future research.

Key words: fluid queue, setup times, working vacations, balking strategy, equilibrium analysis

摘要： 针对在大数据技术背景下,如何动态控制连续数据和信息的快速传输从而实现社会利益最大化这一问题,本文从经济学的角度研究带有启动期和工作休假期的流体排队模型中顾客的止步策略及社会最优收益等问题。基于完全可见和几乎可见两种信息水平,依据流体在缓冲期内的平均逗留时间构建个体平均净收益,进而揭示流体的纳什均衡行为规律。运用更新过程理论,全概率分解技术和常微分方程组的标准型理论,结合流体水平在外部随机环境下的动态变化规律和边界条件导出排队系统稳态下的主要性能指标,研究服务系统的优化设计问题。为保证社会福利最大化,探究最优服务定价策略,利用数值算例对P2P网络中资源传输过程进行仿真分析,进一步优化网络节点运行中的激活时长和排队拥塞等问题,研究结果可为复杂网络服务系统的管理调控提供理论指导和实践参考。

关键词: 流体排队, 启动时间, 工作休假, 止步策略, 均衡分析