A dynamic routing policy, referred to as Heat-Diffusion (HD), is developed for multihop uniclass wireless networks subject to random traffic, time-varying topology and inter-channel interference. The policy uses only current condition of queue occupancies and channel states, with requiring no knowledge of traffic and topology. Besides throughput optimality, HD minimizes an average quadratic routing cost defined by endowing each channel with a time-varying cost factor. Further, HD minimizes average network delay in the class of routing policies that base decisions only on current condition of traffic congestion and channel states. Further, in this class of routing policies, HD provides a Pareto optimal tradeoff between average routing cost and average network delay, meaning that no policy can improve either one without detriment to the other. Finally, HD fluid limit follows graph combinatorial heat equation, which can open a new way to study wireless networks using heat calculus, a very active area of pure mathematics.

The next generation of cellular networks (5G) will provide dense millimeter-wave backhaul architectures to wirelessly forward heterogeneous data traffic in a multihop fashion. In this paper, we present a general optimization framework for the design of delay-aware (DA) policies in multihop wireless networks, providing proportional prioritization of traffic. We develop three throughput-optimal DA algorithms (BP-DA, BPE-DA, and HD-DA) for joint dynamic routing and dynamic link-scheduling problems with good to optimal average network delay performance. Our DA framework considers both the classical back-pressure (BP) and the recent heat-diffusion (HD) algorithms, since queue back-pressure algorithms are being considered for mmWave backhauling management. We provide analytical results for the throughput-optimality of the proposed policies and average delay minimization of HD-DA within the class of DA policies. These are policies that make decisions at each timeslot based only on current channel state, current network queue sizes, and flow priorities. We discuss the applications of our proposals to backhaul management of mmWave cellular networks in light of recent works in the literature. Finally, we present extensive simulations of our proposed algorithms, which confirm the theoretical results and show how the algorithms effectively differentiate data traffic in terms of delay while satisfying flow rate requirements.

A new routing policy, named Heat-Diffusion (HD), is developed for multihop wireless networks subject to stochastic arrivals, time-varying topology, and inter-channel interference, using only current queue congestion and current channel states, without requiring the knowledge of topology and arrivals. Besides throughput optimality, HD minimizes a quadratic routing cost defined by endowing each channel with a cost-factor. It also minimizes average total queue congestion, and so average network delay, within the class of routing policies that base decision only on current queue lengths and current channel states. Further, within this class, HD provides a Pareto optimal tradeoff between average delay and average routing cost, meaning that no policy can improve either one without detriment to the other. Finally, HD fluid limit follows graph combinatorial heat equation that opens a new way to study wireless networks using heat calculus, a very active area of pure mathematics.

Minimum cost routing is considered on multiclass multihop wireless networks influenced by stochastic arrivals, inter-channel interference, and time-varying topology. Endowing each air link with a cost factor, possibly time-varying and different for different classes, we define the Dirichlet routing cost as the square of the link packet transmissions weighted by the link cost-factors. Our recently-proposed Heat-Diffusion (HD) routing protocol [3] is extended to minimize this cost, while ensuring queue stability for all stabilizable traffic demands, and without requiring any information about network topology or packet arrivals. This is the first time in literature that such a multiclass routing penalty can be minimized at network layer subject to queue stability. Further, when all links are of unit cost factor, our protocol here reduces to the one in our recent paper [4], leading to minimum average network delay among all routing protocols that act based only on current queue congestion and current channel states. Our approach is based on mapping a communication network into an electrical network by showing that the fluid limit of wireless network under our routing protocol follows Ohm's law on a nonlinear resistive network.

We propose a new scheduling and routing approach, the Heat Diffusion (HD) protocol, using combinatorial analogue of the heat equation in mathematical physics. The algorithm holds for systems subject to time-varying network conditions with general packet arrivals and random topology states, including ad-hoc networks with mobility. Compared to the well-known backpressure policy, the HD protocol is generalized in form and optimized in performance, which considers link penalties and node capacities in the routing. It mitigates the packet looping behavior of backpressure and attempts to communicate less over links of higher costs and with the nodes of lower capacities. While HD policy shows benefits over backpressure, it is developed using the same underlying control laws. Therefore, it can easily leverage all the theoretical works that have been done in improving the original backpressure. For the same reason, it provides a relatively easy path-way to modify existing applications of backpressure to the optimized versions using HD protocol.