Stability and performance analysis of multi-dimensional sequences of random variables, with applications

在许多应用中，感兴趣的过程的时间演化可以表示为一个随机向量，其中的组件是相关的，但其中一个具有易于预测和理解的行为，而另一个则很难分析。因此，利用其中一个组成部分的行为和组成部分之间的依赖关系的知识，开发整个序列行为的数学分析方法是非常重要的。我的主要重点将是稳定性，性能和这种随机变量序列的最优性，产生于各种应用感兴趣的系统。稳定性可以被粗略地定义为系统以良好控制和良好预测的方式工作的能力。稳定性保证了对客户的服务质量的保证和对服务器最大容量以下资源的使用。因此，稳定性问题是一个非常重要的问题，可以说是研究随机系统时需要问的第一个问题。一旦建立了稳定性，人们可能会提出性能和最优性的各种问题，并针对不同的性能指标比较不同的治理算法（策略、策略）。我的研究将导致创造新的先进的概率方法，因此将在随机建模领域向前迈进一步。它在许多应用领域也将是有益的：无线通信网络、经济、能源和潜在的其他领域。在无线通信网络中，我的研究结果将揭示一些已知传输协议所实现的稳定性和吞吐量，以及单个传输所经历的延迟。我还将致力于构建新的算法，在实际感兴趣的特定网络中实现更好的性能（在稳定性，吞吐量或延迟方面）。我计划构建去中心化（不需要中央控制实体的存在）算法，以实现最佳的稳定性特征。在经济学方面，从理论的角度来看，我的结果将会很有趣，并将有助于更好地理解众多经济模型的行为。在能源方面，我将研究不同的存储策略，并根据感兴趣的性能指标（例如最小化成本或最小化短缺概率）找到最优的存储策略。

In many applications the time-evolution of processes of interest may be represented as a random vector where the components are dependent but one of them has an easily predictable and understandable behaviour, while the other one is much more difficult to analyse. It is therefore of great importance to develop methods for the mathematical analysis of the behaviour of the entire sequence, using the knowledge on the behaviour of one of the components and on the dependence between the components.My main focus will be on stability, performance and optimality of such sequences of random variables, arising from various systems of applied interest. Stability may be loosely defined as the ability of a system to work in a well-controlled and well-predictable manner. Stability ensures a guaranteed quality of service to the customer and the use of resources below maximum capacity to the server. The question of stability is therefore a very important one and arguably the first question one needs to ask when studying a stochastic system. Once stability is established, one may ask various questions of performance and optimality and compare different governing algorithms (policies, strategies) with respect to different performance metrics.My research will lead to the creation of new advanced probability methods and hence will be a step forward in the area of stochastic modelling. It will also be beneficial in a number of application areas: wireless communication networks, economics, energy and potentially others. In wireless communication networks my results will shed light on the stability and throughputs achieved by some known transmission protocols and also on delays experienced by individual transmissions. I will also work on constructing new algorithms, achieving better performance (in terms of stability, throughput or delays) in specific networks of practical interest. I plan to construct decentralised (not requiring a presence of a central controlling entity) algorithms achieving the best possible stability characteristics. In economics, my results will be interesting from a theoretical point of view and will lead to a better understanding of the behaviour of numerous economic models. In energy I will investigate different storage strategies and find optimal ones with respect to a performance metric of interest (for example minimising the cost or minimising probability of a shortage).