随机排队网络的强逼近及其相关渐近分析

Motivated by the need for tools to model and analyze communication networks and large call centers, this project will develop strong approximations and heavy-traffic limit theories for stochastic queueing networks, in both the conventional single-server heavy-traffic regime and the modern multi-server Halfin-Whitt regime. The results on strong approximations will serve as important building blocks for the applications of the heavy-traffic fluid and diffusion approximations in some related areas, such as communication networks, customer contact centers and healthcare systems, which have been proven to yield useful engineering values and further asymptotically solve the corresponding problems. There are two main steps in this research. First, the strong approximations will be developed in both the single-server and multi-server heavy-traffic regimes; these strong approximation results are legitimate results in their own rights because they have remained open problems. Next, the strong approximations will be applied to characterize the convergence rate of the heavy-traffic fluid and diffusion limits and the functional law of iterated logarithm for desired queueing network models. In this project, analytic results of all key performance measures will be developed, including the size of the waiting lines, number of busy servers, workload, busy-time and idle-time processes for each queue in the networks. Two innovations are as follows, we firstly focus on open problems of the strong approximation of many-server in Halfin-Whitt regime, and we secondly advance a strong approximation approach for the related asymptotic analysis.

本项目以通信网络和大型电话中心网络为背景，研究随机排队网络的强逼近及相关渐近分析，在经典逼近模式和Halfin-Whitt逼近模式下建立强逼近,并基于此建立且完善适合相关渐近分析的强逼近方法，为流逼近和扩散逼近的应用提供理论支持，渐近刻画并近似解决实际网络中一些相应问题。具体为(1)以强逼近为研究内容，利用随机过程极限，在经典逼近模式下研究一些单服务员排队网络的强逼近，在Halfin-Whitt逼近模式下研究一些多服务员排队网络的强逼近；(2)以所得强逼近结果为工具，结合布朗运动的性质，分别研究相应逼近模式下排队网络中各指标过程的流逼近的收敛速度，扩散逼近的收敛速度和泛函重对数律，涉及队长、负荷、忙期、忙服务员数等指标过程。创新性:（1）研究Halfin-Whitt逼近模式下多服务员排队的强逼近；（2）将前期所得强逼近结果转化为研究工具，提出建立并完善适合相关渐近分析的强逼近方法。

本项目以通信网络和大型电话中心网络为背景，研究随机排队网络的强逼近及相关渐近分析，在经典逼近模式和Halfin-Whitt逼近模式下建立强逼近，并基于此建立且完善适合相关渐近分析的强逼近方法，为流逼近和扩散逼近的应用提供理论支持，渐近刻画并近似解决实际网络中一些相应问题。针对具体的随机排队网络模型，我们本着拓扑结构由简单到复杂，由单类顾客到多类顾客的研究思路，研究了标准的GI/G/1排队模型，带有反馈机制的GI/GI/n多服务排队,每个服务台上具有多个服务员的推广了的Jackson排队网络，两阶段的串联排队模型以及先到先服务排队服务规则下的单服务台排队模型等，针对这些排队模型，我们首先建立队长、负荷、忙期、忙服务员数等指标过程的强逼近，然后利用所得到的强逼近结果，借助于布朗运动的渐近性质，刻画了排队系统的震荡行为(重对数律和泛函重对数律)和指数收敛速度等问题。通过上述相应排队模型的研究方法和所得结果可以得出，我们建立的以强逼近为基础的渐近震荡分析方法，简称强逼近分析方法，在研究排队网络的渐近行为上是可行的，我们希望此强逼近分析方法能够应用到更多的排队网络中去，为排队网络的研究作出更大贡献。