NeTS: Small: Pareto-Optimized Heat Diffusion Protocol on Ollivier-Ricci Curvature Controlled Wireless Networks

虽然有各种吞吐量最优的无线网络协议，但更具挑战性的问题是如何在保持吞吐量最优的同时，相对于队列占用(与延迟相关)和路由成本(与功率管理相关)等相互冲突的目标使协议达到最优。在这个方向上的第一步是设计一个模仿热扩散的协议，因为著名的迪里克莱特热演算原理已经赋予了路由具有最小路由代价的性质。下一步需要显著偏离热扩散，以便使协议相对于路由成本和队列占用而言是帕累托最优的，同时实施干扰限制、链路方向性和容量。从这一点开始，研究工作将沿着两条不同的调查路线进行。首先，由于经典的热演算必须以一种不平凡的方式进行修改，以使其成为可实现的无线网络协议，因此有必要理解热方程？在有向图的上下文中，受干扰限制和容量限制。经典热演算涉及到经典的拉普拉斯算子，它是一个线性算子，而这里这种新的“热演算”关注的中心数学对象是流体极限中的非线性拉普拉斯算子，它描述了随机无线网络的速率级行为，而不是分组级行为。第二条研究路线致力于寻找能够预测较大队列占用率和/或较大路由成本的网络不变量。我们将把Ollivier-Ricci曲率发展为一个可计算实现的网络参数，即使在方向性和其他网络约束下，它也与队列占用率和路由代价成反比。此外，还将研究奥利维尔-里奇曲率作为容量区域大小的预测因子。最后，研究将使用一些Ricci流技术来优化网络以获得最大的Ollivier-Ricci曲率，从而获得最大的通行能力区域。该方案的框架适用于资源相互依赖的一类随机问题，其中资源是相互依赖的服务器的集合，只有在一定的约束条件下才能被访问，而消费者的服务时间是随机的，并且异步完成。这个通用模型描述了各种各样的问题，包括排队网络、产品组装系统、内存或处理器管理、呼叫中心、代理分配、数据交换机、医疗保健系统以及电力系统中的传输规划或存储分配，仅举几例。具体地说，这项研究将努力实现与经典热力学和电路理论中的欧姆·S定律之间的交叉滋养，为分析和优化这些复杂的问题开辟一条新的途径。

While there are a variety of throughput-optimal wireless network protocols, more challenging is the problem of making the protocol optimal relative to such conflicting objectives as queue occupancy (related to latency) and routing cost (related to power management), while holding throughput-optimality. The first step in that direction is the design of a protocol mimicking heat diffusion, as the celebrated Dirichlet principle of heat calculus already endows the routing with minimum routing cost property. The next step requires a significant departure from heat diffusion in order to make the protocol Pareto-optimal relative to routing cost and queue occupancy, while enforcing interference restrictions, link directionality and capacity. From this point onwards, the research effort will be conducted along two different lines of investigation. First, since the classical heat calculus had to be modified in a nontrivial way to make it an implementable wireless network protocol, there is a need to understand the ?heat equation? in the context of directed graphs, subject to interference restrictions and capacity constraints. Classical heat calculus involves the classical Laplacian, which is a linear operator, while here the central mathematical object of concern of this new "heat calculus" is a nonlinear Laplacian in the fluid limit, which describes the rate-level, rather than packet-level, behavior of the stochastic wireless network. The second line of investigation is dedicated to finding the network invariant that could anticipate the potential for large queue occupancy and/or large routing cost. We will develop the Ollivier-Ricci curvature as a computationally implementable network parameter inversely proportional to queue occupancy and routing cost, even under directionality and other network constraints. Also, the Ollivier-Ricci curvature will be investigated as a predictor of the size of the capacity region. Finally, the research will culminate with some Ricci flow technique to optimize the network for maximum Ollivier-Ricci curvature, hence attaining the largest capacity region. The framework of this proposal is applicable to a wide family of stochastic problems with interdependent resources, where the resources are a collection of interdependent servers that can only be accessed under certain constraints, and the consumers are of random service time with asynchronous completion. This general model describes a wide variety of problems including queuing networks, product assembly systems, memory or processor managements, call centers, agent allocations, data switches, healthcare systems, and transmission planning or storage allocation in power systems just to name a few. Specifically, the research will strive to achieve cross-fertilization among this broad set of problems with classical thermodynamics and Ohm?s law in circuit theory that open a new way to analyze and optimize these complicated problems.