We consider the capacity problem for wireless networks. Networks are modeled as random unit-disk graphs, and the capacity problem is formulated as one of finding the maximum value of a multicommodity flow. In this paper, we develop a proof technique based on which we are able to obtain a tight characterization of the solution to the linear program associated with the multiflow problem, to within constants independent of network size. We also use this proof method to analyze network capacity for a variety of transmitter receiver architectures, for which we obtain some conclusive results. These results contain as a special case (and strengthen) those of Gupta and Kumar for random networks, for which a new derivation is provided using only elementary counting and discrete probability tools.
This work is primarily motivated by our struggle to understand the results of Gupta and Kumar on the capacity of wireless networks @cite_19 . And the main idea behind our approach is simple: the transport capacity problem posed in @cite_19 , in the context of random networks, is essentially a throughput stability problem---the goal is to determine how much data can be injected by each node into the network while keeping the system stable---, and this throughput stability problem admits a very simple formulation in term of flow networks. Note also that because of the mechanism for generating source destination pairs, all connections have the same average length (one half of one network diameter), and thus we do not need to deal with the bit-meters sec metric considered in @cite_19 .
@cite_19
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We develop a new metric for quantifying end-to-end throughput in multihop wireless networks, which we term random access transport capacity, since the interference model presumes uncoordinated transmissions. The metric quantifies the average maximum rate of successful end-to-end transmissions, multiplied by the communication distance, and normalized by the network area. We show that a simple upper bound on this quantity is computable in closed-form in terms of key network parameters when the number of retransmissions is not restricted and the hops are assumed to be equally spaced on a line between the source and destination. We also derive the optimum number of hops and optimal per hop success probability and show that our result follows the well-known square root scaling law while providing exact expressions for the preconstants, which contain most of the design-relevant network parameters. Numerical results demonstrate that the upper bound is accurate for the purpose of determining the optimal hop count and success (or outage) probability. The class of Gupta-Kumar results, which predict the throughput capacity in wireless networks, is restricted to asymptotic regimes. This tutorial presents a methodology to address a corresponding non-asymptotic analysis based on the framework of the stochastic network calculus, in a rigorous mathematical manner. In particular, we derive explicit closed-form results on the distribution of the end-to-end capacity and delay, for a fixed source-destination pair, in a network with broad assumptions on its topology and degree of spatial correlations. The results are non-asymptotic in that they hold for finite time scales and network sizes, as well as bursty arrivals. The generality of the results enables the research of several interesting problems, concerning for instance the effects of time scales or randomness in topology on the network capacity. We consider the problem of determining rates of growth for the maximum stable throughput achievable in dense wireless networks. We formulate this problem as one of finding maximum flows on random unit-disk graphs. Equipped with the max-flow min-cut theorem as our basic analysis tool, we obtain rates of growth under three models of communication: (a) omnidirectional transmissions; (b) "simple" directional transmissions, in which sending nodes generate a single beam aimed at a particular receiver; and (c) "complex" directional transmissions, in which sending nodes generate multiple beams aimed at multiple receivers. Our main finding is that an increase of Θlog2n in maximum stable throughput is all that can be achieved by allowing arbitrarily complex signal processing (in the form of generation of directed beams) at the transmitters and receivers. We conclude therefore that neither directional antennas, nor the ability to communicate simultaneously with multiple nodes, can be expected in practice to effectively circumvent the constriction on capacity in dense networks that results from the geometric layout of nodes in space. In this paper, we present a methodology to analytically compute the throughput capacity, or the maximum end-to-end throughput of a given source and destination pair in a multi-hop wireless network. The end-to-end throughput capacity is computed by considering the interference due to neighboring nodes, as well as various modes of hidden node interference. Knowing the throughput capacity is important because it facilitates the design of routing policy, admission control for realtime traffic, as well as load control for wireless networks. We model location-dependent neighboring interference and we use a contention graph to represent these interference relationships. Based on the contention graph, we formulate the individual link capacity as a set of fixed point equations. The end-to-end throughput capacity can then be determined once these link capacities are obtained. To illustrate the utility of our proposed methodology, we present two important applications: (a) route optimization to determine the path with the maximum end-to-end throughput capacity and, (b) optimal offered load control for a given path so that the maximum end-to-end capacity can be achieved. Extensive simulations are carried out to verify and validate the proposed analytical methodology.