Queueing Models and Performance Analysis -- Telecommunication Systems and Other Applications

这项研究计划的目标是揭示模式、结构和性质，寻找新的方法，发展和扩展先进的方法，并提出有效的近似和计算程序。我们期望新的发现可以用于各种多维随机系统的建模和分析，包括多个等待队列、串联队列和其他与批量队列、耦合服务器、自适应服务和优先级等机制相结合的队列模型。如上所述，这些随机系统提出了新的挑战，通过我们提出的先进方法来应对这些挑战，我们的目标是为工程和计算机问题以及其他排队应用提供解决方案。当期望显式性质时，对于本研究中提出的复杂系统，概率分布的显式解通常是不可用的。因此，近似、渐近性质、性能界成为研究的关键组成部分。本研究将使用、扩展和进一步发展以下方法和方法：渐近的核方法；边值问题（BVP）方法和显式解的解析（均匀化/代数）方法；矩阵分析法（MAM）用于求解具有无限多个相态的过程，等等。从核方法扩展到具有杀伤或吸收状态的随机漫步，到更一般的结构化漫步和高维模型，提出了新的发现和新的结果；推广了处理无穷多相态过程的矩阵解析方法从探索BVPs的幂极限，以及显式解的均匀化和代数方法；通过谱性质研究准平稳分布中的衰减率，有望在排队理论、随机建模与分析等方面取得新颖而重要的进展，并对系统设计与性能评价、电信系统等应用产生深远的影响。

The objective of this proposed research program is to reveal patterns, structures and properties, search for new approaches, develop and extend advanced methods, and propose efficient approximations and computational procedures. We expect that new discoveries can be employed in both modelling and analysis, for various multi-dimensional stochastic systems, including multiple waiting lines queues, tandem queues, and other queueing models which are incorporated with mechanisms such as bulk queues, coupled servers, adaptive services, and priorities. These stochastic systems offer new challenges, as outlined above, and through our development of proposed advanced methods in dealing with these challenges, we aim at providing solutions to engineering and computer problems and other queueing applications.When explicit properties are expected, explicit solutions for probability distributions are not, in general, available for the complex systems proposed in this research. Therefore, approximations, asymptotic properties, performance bounds, become key components in the study. The proposed research is to use, extend, and further develop the following methodology and approaches: the kernel method for asymptotics; boundary-value problems (BVP) approach and analytic (uniformization / algebraic) method for explicit solutions; the matrix analytic method (MAM) for processes with infinitely many phase states, among others.New discoveries made and new results obtained from the proposed program, from extending the kernel method to random walks with killing or absorbing states to more general structured walks and to high-dimensional models; from generalizing the matrix-analytic method for dealing with processes with infinitely many phase states; from exploring the limit of the power of the BVPs, and uniformization and algebraic methods for explicit solutions; from studying the decay rate in quasi-stationary distributions by spectral properties, are expected to be original and significant results in advances in queueing theory, stochastic modelling and analysis, and produce profound impact on system design and performance evaluation, telecommunication systems and other applications.