Max-min fairness is widely used in various areas of networking. In every case where it is used, there is a proof of existence and one or several algorithms for computing the max-min fair allocation; in most, but not all cases, they are based on the notion of bottlenecks. In spite of this wide applicability, there are still examples, arising in the context of wireless or peer-to-peer networks, where the existing theories do not seem to apply directly. In this paper, we give a unifying treatment of max-min fairness, which encompasses all existing results in a simplifying framework, and extend its applicability to new examples. First, we observe that the existence of max-min fairness is actually a geometric property of the set of feasible allocations (uniqueness always holds). There exist sets on which max-min fairness does not exist, and we describe a large class of sets on which a max-min fair allocation does exist. This class contains, but is not limited to the compact, convex sets of R N. Second, we give a general purpose centralized algorithm, called Max-min Programming, for computing the max-min fair allocation in all cases where it exists (whether the set of feasible allocations is in our class or not). Its complexity is of the order of N linear programming steps in R N , in the case where the feasible set is defined by linear constraints. We show that, if the set of feasible allocations has the free-disposal property, then Max-min Programming reduces to a simpler algorithm, called Water Filling, whose complexity is much lower. Free disposal corresponds to the cases where a bottleneck argument can be made, and Water Filling is the general form of all previously known centralized algorithms for such cases. Our derivations are based on the relation between max-min fairness and leximin ordering. All our results apply mutatis mutandis to min-max fairness. Our results apply to weighted, unweighted and util-max-min and min-max fairness. Distributed algorithms for the computation of max-min fair allocations are outside the scope of this paper.
Abstract-Ultra-Wide Band (UWB) is an emerging wireless physical layer technology that uses a very large bandwidth. We are interested in finding the design objectives of the medium access (MAC, namely, power control and scheduling) and routing protocols of a multi-hop, best-effort, UWB network. Our objective is to maximize flow rates (more precisely, log-utility of flow rates) given node power constraints. The specificity of UWB is expressed by the linear dependence between rate and signal-to-noise ratio at the receiver. It is known that, in wireless networks, different routing strategies can imply differences in MAC protocol design. Hence we search for the jointly optimal routing, scheduling and power control.We find that the optimal solution is characterized by the following. (1) When data is being sent over a link, it is optimal to have an exclusion region around the destination, in which all nodes remain silent during transmission, whereas nodes outside of this region can transmit in parallel, regardless of the interference they produce at the destination. Additionally, the source adapts its transmission rate according to the level of interference at the destination due to sources outside of the exclusion region.(2) The optimal size of this exclusion region depends only on the transmission power of the source of the link, and not on the length of the link nor on positions of nodes in its vicinity. (3) Each node in a given time slot either sends data at the maximum power, or does not send at all. As for the routing, we restrict ourselves to a subset of routes where on each successive hop we decrease the distance toward the destination, and we show that (4) relaying along a minimum energy and loss route is always better than using longer hops or sending directly, which is not obvious since we optimize rate and not power consumption. Finally (5), the design of the optimal MAC protocol is independent of the choice of the routing protocol. For narrow-band networks, (2), (4) and (5) do not hold, which shows that the design of an UWB network should be addressed in a different way than for narrow-band.Our technical approach is based on expressing the design requirements as a mathematical optimization problem. We solve it exactly for simple networks on a line and approximately on random topologies in a plane with up to 50 nodes with various power constraints, traffic matrices, and mobility parameters.
Due to environmental concerns and strict constraints on interference imposed on other networks, the radiated power of emerging pervasive wireless networks needs to be strictly limited, yet without sacrificing acceptable data rates. Pulsed Time-Hopping Ultra-Wideband (TH-UWB) is a radio technology that has the potential to satisfy this requirement. Although TH-UWB is a multi-user radio technology, non-zero cross-correlation between time-hopping sequences, time-asynchronicity between sources and a multipath channel environment make it sensitive to strong interferers and near-far scenarios. While most protocols manage interference and multiple-access through power control or mutual exclusion, we base our design on rate control, a relatively unexplored dimension for multiple-access and interference management. We further take advantage of the nature of pulsed TH-UWB to propose an interference mitigation scheme that alleviates the need for an exclusion scheme. A source is always allowed to send and continuously adapts its channel code (hence its rate) to the interference experienced at the destination. In contrast to power control or exclusion, our MAC layer is local to sender and receiver and does not need coordination among neighbors not involved in the transmission. We show by simulation that we achieve a significant increase in network throughput compared to alternative designs.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.