This paper presents a tractable analytical framework for the exact calculation of probability of node isolation and minimum node degree distribution when @math sensor nodes are independently and uniformly distributed inside a finite square region. The proposed framework can accurately account for the boundary effects by partitioning the square into subregions, based on the transmission range and the node location. We show that for each subregion, the probability that a random node falls inside a disk centered at an arbitrary node located in that subregion can be expressed analytically in closed-form. Using the results for the different subregions, we obtain the exact probability of node isolation and minimum node degree distribution that serves as an upper bound for the probability of @math -connectivity. Our theoretical framework is validated by comparison with the simulation results and shows that the minimum node degree distribution serves as a tight upper bound for the probability of @math -connectivity. The proposed framework provides a very useful tool to accurately account for the boundary effects in the design of finite wireless networks.
Since many practical multi-hop networks are formed by distributing a finite number of nodes in a finite area, there has been an increasing interest to model and determine the connectivity properties in finite multi-hop networks @cite_14 @cite_24 @cite_11 @cite_9 @cite_18 @cite_16 @cite_23 @cite_20 @cite_3 . This is also due to the fact, established earlier in @cite_14 @cite_24 and recently in @cite_23 , that the asymptotic connectivity results for large-scale networks provide an extremely poor approximation for finite wireless networks. This poor approximation is due to the boundary effects experienced by the nodes near the borders of the finite region over which the nodes are deployed. Since the nodes located close to the physical boundaries of the network have a limited coverage area, they have a greater probability of isolation. Therefore, the boundary effects play an important role in determining the overall network connectivity.
@cite_18 @cite_14 @cite_9 @cite_3 @cite_24 @cite_23 @cite_16 @cite_20 @cite_11
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The class of Gupta-Kumar results give the asymptotic throughput in multi-hop wireless networks but cannot predict the throughput behavior in networks of typical size. This paper addresses the non-asymptotic analysis of the multihop wireless communication problem and provides, for the first time, closed-form results on multi-hop throughput and delay distributions. The results are non-asymptotic in that they hold for any number of nodes and also fully account for transient regimes, i.e., finite time scales, delays, as well as bursty arrivals. Their accuracy is supported by the recovery of classical single-hop results, and also by simulations from empirical data sets with realistic mobility settings. Moreover, for a specific network scenario and a fixed pair of nodes, the results confirm Gupta-Kumar's Ω(1√n log n) asymptotic scaling law. Motivated by low energy consumption in geographic routing in wireless networks, there has been recent interest in determining bounds on the length of edges in the Delaunay graph of randomly distributed points. Asymptotic results are known for random networks in planar domains. In this paper, we obtain upper and lower bounds that hold with parametric probability in any dimension, for points distributed uniformly at random in domains with and without boundary. The results obtained are asymptotically tight for all relevant values of such probability and constant number of dimensions, and show that the overhead produced by boundary nodes in the plane holds also for higher dimensions. To our knowledge, this is the first comprehensive study on the lengths of long edges in Delaunay graphs Connectivity and capacity are two fundamental properties of wireless multihop networks. The scalability of these properties has been a primary concern for which asymptotic analysis is a useful tool. Three related but logically distinct network models are often considered in asymptotic analyses, viz. the dense network model, the extended network model, and the infinite network model, which consider respectively a network deployed in a fixed finite area with a sufficiently large node density, a network deployed in a sufficiently large area with a fixed node density, and a network deployed in R2 with a sufficiently large node density. The infinite network model originated from continuum percolation theory and asymptotic results obtained from the infinite network model have often been applied to the dense and extended networks. In this paper, through two case studies related to network connectivity on the expected number of isolated nodes and on the vanishing of components of finite order k > 1 respectively, we demonstrate some subtle but important differences between the infinite network model and the dense and extended network models. Therefore, extra scrutiny has to be used in order for the results obtained from the infinite network model to be applicable to the dense and extended network models. Asymptotic results are also obtained on the expected number of isolated nodes, the vanishingly small impact of the boundary effect on the number of isolated nodes, and the vanishing of components of finite order k > 1 in the dense and extended network models using a generic random connection model. In multi-hop ad hoc networks, stations may pump more traffic into the networks than can be supported, resulting in high packet-loss rate, re-routing instability and unfairness problems. This paper shows that controlling the offered load at the sources can eliminate these problems. To verify the simulation results, we set up a real 6-node multi-hop network. The experimental measurements confirm the existence of the optimal offered load. In addition, we provide an analysis to estimate the optimal offered load that maximizes the throughput of a multi-hop traffic flow. We believe this is a first paper in the literature to provide a quantitative analysis (as opposed to simulation) for the impact of hidden nodes and signal capture on sustainable throughput. The analysis is based on the observation that a large-scale 802.11 network with hidden nodes is a network in which the carrier-sensing capability breaks down partially. Its performance is therefore somewhere between that of a carrier-sensing network and that of an Aloha network. Indeed, our analytical closed-form solution has the appearance of the throughput equation of the Aloha network. Our approach allows one to identify whether the performance of an 802.11 network is hidden-node limited or spatial-reuse limited.