Fluid Approximations for Quantitative Analysis

定量模型对于设计信息通信技术、通信网络、传感器网络和其他物流领域（如业务流程、生化系统和医疗资源分配或调度）的高效系统至关重要。然而，在高层次描述中指定的模型在其数学解方面通常是低效的。相反，可以有效解决的模型通常是小的、人为的和/或非常具体的问题。在许多具有大量组件的系统中，通常最好将许多相同类型的实体聚合为单个数量，从而导致数学上更易于处理的连续（实数）状态模型，例如由体积表示的大量气体分子。由此产生的所谓“流体方法”已被研究多年，例如在物理学、生物学、化学和金融建模领域。最近，类似的流体方法被用于性能的随机模型。例如，对于传统排队网络存在流体近似，其中（整数）队列人口由连续的液位代替，离散的“客户”运动由连续的流体流动近似。类似的流体近似也被引入到其他随机建模形式中，例如随机Petri网（SPNs）和随机过程代数（SPAs）。一般来说，流体模型的困难在于为有用的模型规范找到精确的或良好的近似解析解。在没有解析解或有效的数值解的情况下，人们不得不求助于模拟，这对于大问题是出了名的低效，即使当流体被用来近似大量的离散客户时也是如此。当前的流体SPA模型导致在时间变量上的耦合微分方程的确定性系统，这些系统可以用传统的数值求解方法求解。然而，这些方程仅由确定性参数（例如恒定到达和处理速率）定义，因此它们的相关性在于瞬态特性，平衡解（当它们存在时）也是确定性的。将它们推广到概率变化的输入和服务时间需要更复杂的分析，包括一阶或二阶偏微分方程，这些方程很少能得到精确解。在流体排队网络的背景下，对具有马尔可夫开/关过程的单流体队列进行了深入的研究，结果存在于对输入过程的相当一般的假设下。然而，即使是非常简单的队列网络也有难以解决的复杂解决方案，而非平凡网络通常通过某种形式的流体流动模拟来分析。在传统的离散状态模型中，大型系统的求解方法利用了组合方法，通常是近似方法，因为它们是找到数值上可处理的解的唯一方法。这通常涉及简化假设或使用分层方法，如分层排队网络（LQN）范例。支撑这一提议的中心思想是在流体系统的分析中寻求分层方法，类似于传统离散状态模型中发展良好的方法。其目的是开发有效的精确和近似求解方法，以处理比目前更广泛的流体模型。所提出的工作本身具有重要的理论价值，但也具有巨大的实际潜力，因为它开辟了求解大规模流体模型的可能性，而不必诉诸效率低得多的基于模拟的求解方法。潜在的应用范围从分析和优化互联网通信系统和存储区域网络（san）到研究涉及“真实”流体的系统，如泵系统和河流网络。

Quantitative models are vital for the design of efficient systems in ICT, communication networks, sensor networks and other logistical areas such as business processes, biochemical systems and healthcare resource allocation or scheduling. However, models specified at a high level of description are often inefficient in terms of their mathematical solutions. Conversely, models which can be solved efficiently are often small, contrived and/or very problem specific. In many systems with very large numbers of components, it is often preferable to aggregate many entities of the same type into a single quantity, leading to a mathematically more tractable, continuous (real number) state model, cf. large numbers of gas molecules represented by a volume. The resulting so-called 'fluid methods' have been studied for many years, for example in physics, biology, chemistry and financial modelling.More recently, similar fluid methods have been used in stochastic models of performance. For example, fluid approximations exist for conventional queueing networks where the (integer) queue populations are replaced by a continuous fluid level and where discrete 'customer' movements are approximated by a continuous fluid flow. Similar fluid approximations have also been introduced in other stochastic modelling formalisms, for example Stochastic Petri Nets (SPNs) and Stochastic Process Algebras (SPAs). The difficulty with fluid models in general is finding exact, or good approximate, analytical solutions for useful model specifications. In the absence of analytical solutions, or efficient numerical solutions, one has to resort to simulation, which is notoriously inefficient for large problems, even when a fluid is used to approximate large numbers of otherwise discrete customers.Current fluid SPA models lead to deterministic systems of coupled differential equations in the time variable that are solveable by conventional numerical solution methods. However, the equations are defined solely by deterministic parameters (e.g. constant arrival and processing rates) and so their relevance lies in transient properties, equilibrium solutions (when they exist) also being deterministic. Their generalisation to probabilistically varying inputs and service times requires much more complex analysis, involving first- or second-order partial differential equations, which only rarely can be solved exactly. In the context of fluid queueing networks, the single fluid queue with Markovian on/off processes has been studied in some depth and results exist under quite general assumptions about the input processes. However, even very simple networks of queues have intractably complex solutions, and non-trivial networks are usually analysed by some form of fluid flow simulation. In traditional discrete-state models, solution methods for large systems have exploited compositional, usually approximate, approaches as they constitute the only way to find numerically tractable solutions. This typically involves making simplifying assumptions or using a hierarchical methodology such as the layered queueing network (LQN) paradigm.The central idea that underpins this proposal is to seek hierarchical approaches in the analysis of fluid systems, analogous to those that are well developed in traditional discrete-state models. The aim is to develop efficient solution methods, both exact and approximate, for a much broader range of fluid models than can be handled at present. The proposed work has significant theoretical value in its own right, but also has enormous practical potential as it opens up the possibility of solving large-scale fluid models, without having to resort to much less efficient solution methods based on simulation. Potential applications range from the analysis and optimisation of internet communication systems and storage area networks (SANs) to the study of systems involving 'real' fluids such as pumping systems and river networks.