Fluid Approximations for Quantitative Analysis

定量模型对于设计信息和通信技术、通信网络、传感器网络和其他物流领域的有效系统至关重要，例如业务流程、生化系统和医疗保健资源分配或调度。然而，在高级别描述中指定的模型在其数学解方面往往效率低下。相反，可以有效解决的模型通常很小，人为的和/或非常具体的问题。在具有非常多组件的许多系统中，通常最好将许多相同类型的实体聚集成单个数量，从而在数学上产生更容易处理的、连续的(实数)状态模型，参看。以体积表示的大量气体分子。由此产生的所谓的流体方法已经被研究了很多年，例如在物理、生物、化学和金融建模中。最近，类似的流体方法已经被用于随机业绩模型。例如，对于传统的排队网络，存在流体近似，其中(整数)队列群被连续的流体水平所取代，并且离散的客户移动被连续的流体所近似。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(SPN)和随机进程代数(SPA)。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。在缺乏解析解或有效的数值解的情况下，人们不得不求助于模拟，这对于大型问题来说是众所周知的低效的，即使当使用流体来逼近大量原本离散的客户时也是如此。当前的流体SPA模型导致了时间变量的耦合微分方程组的确定性系统，这些系统可以用传统的数值解方法求解。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变特性，平衡解(如果存在)也是确定性的。它们对概率变化的输入和服务时间的概括需要复杂得多的分析，涉及一阶或二阶偏微分方程组，这些方程很少能被精确求解。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排队进行了较为深入的研究，其结果存在于关于输入过程的相当一般的假设下。然而，即使是非常简单的排队网络也有难以处理的复杂解，而非平凡的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，因为它们构成了找到数值容易处理的解的唯一途径。这通常涉及到简化假设或使用分层排队网络(LQN)范例等分层方法。支持这一建议的中心思想是在流体系统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。其目的是为比目前能够处理的更广泛的流体模型开发有效的精确和近似求解方法。所提出的工作本身具有重要的理论价值，但也具有巨大的实践潜力，因为它打开了求解大规模流体模型的可能性，而不必求助于基于模拟的效率低得多的求解方法。潜在的应用范围从互联网通信系统和存储区域网络(SAN)的分析和优化，到涉及诸如泵送系统和河流网络等“真实”流体的系统的研究。

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.