基于流逼近的排队网络的渐近震荡和优化分析

Motivated by the analysis for the problems appearing from the computer-communication, cloud-computing and large call center networks, we study the asymptotic variability and asymptotic optimization based on the fluid approximation in this project. We firstly model some queueing networks from our real life and then transform some problems of operations and management of practical networks into the performance analysis and asymptotic optimization of single-server and many-server queueing networks; Secondly, we analyze and solve them based on the stochastic process limit and asymptotic optimization, and try to provide some cognitions and intuitions for the corresponding real problems. As the first step of approximating the networks, the fluid approximation is both the fundamental for others and the research point, and focuses on the network with stochastic arrival and service rates, which are usually growing up with the operations of the networks, such as pricing. The asymptotic variability will quantify the magnitude of asymptotic stochastic fluctuations of the indexed stochastic processes (i.e. the queue length process) compensated by their fluid limits with the help of strong approximation, it is embodied by the concepts of functional law of the iterated logarithm (LIL), Levy’s LIL, Csörgő’s LIL and convergence rate in this project. The asymptotic optimization, based on the fluid approximation, aims some problems in the regime of the operations and management of networks, including optimal capacity or staffing, pricing and outsourcing etc., and will provide the network manager some direct suggestion and guidance with some simple and intuitive research results.

本项目以计算机通信、云计算和大型电话中心网络为实际背景，以流逼近为理论基础，研究随机排队网络的渐近震荡行为和渐近优化分析。我们将现实中的网络运行和管理问题模型化为单服务员排队和多服务员排队网络的行为分析和渐近优化问题，从随机过程极限和随机优化的角度进行分析求解，以此获得对实际相应问题的认知和启发。流逼近作为网络逼近的第一步，既是分析基础又是一个研究点，它将研究由运作管理模式(比如定价)催生的带有随机到达服务率的排队网络。渐近震荡刻画网络中队长等指标过程围绕其流逼近的震荡行为，我们结合强逼近和布朗运动，利用泛函重对数律，Levy型重对数律，Csörgő型重对数律和收敛速度等对其进行刻画。流逼近下的渐近优化处理服务网络运作管理中的资源或人员配置、定价和外包等问题，其结论简单直观，对网络管理者具有直接的指导作用。

本项目以计算机通信、云计算和大型电话中心网络为实际背景，以流逼近为理论基础，研究随机排队网络的渐近震荡行为和渐近优化分析。我们将现实中的网络运行和管理问题模型化为单服务员排队和多服务员排队网络的行为分析和渐近优化问题，从随机过程极限和随机优化的角度进行分析求解，以此获得对实际相应问题的认知和启发。流逼近作为网络逼近的第一步，既是分析基础又是一个研究点。渐近震荡刻画网络中队长等指标过程围绕其流逼近的震荡行为，我们结合强逼近和布朗运动，利用泛函重对数律，Levy型重对数律，Csörg型重对数律和收敛速度等对其进行刻画。流逼近下的渐近优化处理服务网络运作管理中的资源配置、定价和调度等问题，其结论简单直观，对网络管理者具有直接的指导作用。研究结果主要通过下列模型给出：强占优先权或先到先服务服务规则下的单（多）服务台排队，广义Jackson网络，W-排队网络，重试排队，速率不确定的排队系统，等等。