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.
