Heat Diffusion Algorithm for Resource Allocation and Routing in Multihop Wireless Networks Reza Banirazi, Edmond Jonckheere, Bhaskar Krishnamachari Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089 E-mail: {banirazi, jonckhee, bkrishna}@usc.edu Abstract—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. I. I NTRODUCTION Backpressure is a well-known algorithm for resource allo- cation and data routing in wireless networks, which achieves maximum throughput in the presence of time-varying network conditions and without precisely knowing arrival rates. It as- signs a weight to each link, equal to the maximum differential backlog between transmitting and receiving nodes, and then chooses link rates to maximize the sum of the products of link rates and link weights. The algorithm, first called max- weight, was originally proposed in [1] for routing traffic over a multi-hop packet radio network with random packet arrivals and fixed set of link selection options. Then the idea was extended to ad-hoc mobile networks and was also combined with optimization techniques [2]–[6]. In this paper, inspired by the heat diffusion process on smooth manifolds, we introduce a new paradigm for the problem of resource allocation and packet routing in multi- hop wireless networks, referred to as Heat-Diffusion (HD) protocol. We develop the HD protocol along the same line as backpressure, as the differential queue length plays a main role in the routing decision, and as we use a multiple stage strategy to implement the algorithm. Our approach differs from backpressure, as each stage of the algorithm is formulated according to a metaphorical referral to the heat equation. The ubiquitousness of heat equation in mathematics and physics not only gives us a deeper insight into the resource allocation and routing problem, but also provides us with a high flexi- bility to derive an optimal solution for a very general class of routing networks. However, for the heat equation to be more than a metaphor, we need to bring it from smooth geometry in which the heat lives to a purely combinatorial domain in which the data network runs. To this end, we utilize the theory of combinatorial calculus, which works with a cell complex as a countable discrete domain, and preserves such fundamental differential and integral operators as the Laplacian that convey most of the physics of the process. Before proceeding, it is valuable to have a heuristic com- parison between HD protocol and the original backpressure. Recall that the backpressure routing is anecdotally described by the flow of water through a network of pipes under pressure gradients. Certainly, the flow of packets in a data network is restricted by the capacity of each link, and hence the model is more relevant if pipes are replaced with buckets. Then we can imagine a barrel of packets at each node, where the packets may move from barrel i to barrel j if there is a communication link from node i to node j , and for transferring packets, each link uses a bucket whose capacity is equal to the link capacity. Having this analogy, the original backpressure is based on two assumptions: (i) all barrels are of the same storage capacity, and (ii) all buckets require the same amount of work to transfer the same number of packets. Technically, the first assumption implies that all nodes have equal capacity, which may reflect power, processing, and storage capability, while the second assumption means that all links are of equal cost, which may reflect power consumption, channel quality, and distance. In contrast to the backpressure, the HD protocol considers a variety of node capacities and link penalties on the network. In this sense, the backpressure policy becomes a special case of the HD protocol, while the latter also provides a better performance under the influence of its ancestor, the heat diffusion process. In particular, the HD protocol inherits from the heat equation the property of jointly optimizing queue backlogs versus capacity of each node, and routing cost versus communication expense of each link. Furthermore, it is proved that the HD protocol achieves maximum throughput of the network in the sense that it guarantees stability of the queue buffers under any arrival rate such that there exists a stabilizing routing policy for it. We also notice that the notions of node capacity and link cost may reflect different constraints from the network. An example is a wireless sensor network in which communications happen over lossy channels and nodes are subject to different memory restrictions. Obviously, the former can be modeled by link costs and the latter by node capacities. U.S. Government work not protected by U.S. copyright Globecom 2012 - Wireless Networking Symposium 5693