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. In this paper, a novel framework is proposed for optimizing the operation and performance of a large-scale multi-hop millimeter wave (mmW) backhaul within a wireless small cell network having multiple mobile network operators (MNOs). The proposed framework enables the small base stations to jointly decide on forming the multi-hop, mmW links over backhaul infrastructure that belongs to multiple, independent MNOs, while properly allocating resources across those links. In this regard, the problem is addressed using a novel framework based on matching theory composed of two, highly inter-related stages: a multi-hop network formation stage and a resource management stage. One unique feature of this framework is that it jointly accounts for both wireless channel characteristics and economic factors during both network formation and resource management. The multi-hop network formation stage is formulated as a one-to-many matching game, which is solved using a novel algorithm, that builds on the so-called deferred acceptance algorithm and is shown to yield a stable and Pareto optimal multi-hop mmW backhaul network. Then, a one-to-many matching game is formulated to enable proper resource allocation across the formed multi-hop network. This game is then shown to exhibit peer effects and, as such, a novel algorithm is developed to find a stable and optimal resource management solution that can properly cope with these peer effects. Simulation results show that, with manageable complexity, the proposed framework yields substantial gains, in terms of the average sum rate, reaching up to 27 and 54 , respectively, compared with a non-cooperative scheme in which inter-operator sharing is not allowed and a random allocation approach. The results also show that our framework improves the statistics of the backhaul sum rate and provides insights on how to manage pricing and the cost of the cooperative mmW backhaul network for the MNOs. The objective of the paper is to provide qualitative insight into the global effects of distributed mechanisms, such as carrier sense multiple access (CSMA) and rate control, on the performance and stability of multi-hop wireless networks. Toward this end, we introduce a linear queueing network model where the service capacity of each node is modulated by the transmission state of its neighbor. We derive lower bounds on the steady-state utilization at each queue of such networks and demonstrate the existence of a phase transition phenomenon, whereby infinitesimal traffic increase at a single node in the network can suddenly render the entire network instable. We also present NS simulation results that show how this phenomenon can actually take place in IEEE 802.11 multi-hop wireless networks. Our results have direct bearing on rate control schemes, in that they indicate a minimum admissible threshold rate required to prevent network instability. This paper presents a framework for the calculation of stochastic connectivity properties of wireless multihop networks. Assuming that n nodes, each node with transmission range r0, are distributed according to some spatial probability density function, we study the level of connectivity of the resulting network topology from three viewpoints. First, we analyze the number of neighbors of a given node. Second, we study the probability that there is a communication path between two given nodes. Third, we investigate the probability that the entire network is connected, i.e. each node can communicate with every other node via a multihop path. For the last-mentioned issue, we compute a tight approximation for the critical (r0 ,n )pairs that are required to keep the network connected with a probability close to one. In fact, the problem is solved for the general case of a k-connected network, accounting for the robustness against node failures. These issues are studied for uniformly distributed nodes (with and without ‘border effects’), Gaussian distributed nodes, and nodes that move according to the commonly used random waypoint mobility model. The results are of practical value for the design and simulation of wireless sensor and mobile ad hoc networks. This paper investigates two fundamental characteristics of a wireless multi -hop network: its minimum node degree and its k--connectivity. Both topology attributes depend on the spatial distribution of the nodes and their transmission range. Using typical modeling assumptions :--- :a random uniform distribution of the nodes and a simple link model :--- :we derive an analytical expression that enables the determination of the required range r0 that creates, for a given node density ρ, an almost surely k--connected network. Equivalently, if the maximum r0 of the nodes is given, we can find out how many nodes are needed to cover a certain area with a k--connected network. We also investigate these questions by various simulations and thereby verify our analytical expressions. Finally, the impact of mobility is discussed.The results of this paper are of practical value for researchers in this area, e.g., if they set the parameters in a network--level simulation of a mobile ad hoc network or if they design a wireless sensor network. The unstructured multi-hop radio network model, with asynchronous wake-up, no collision detection and little knowledge on the network topology, is proposed for capturing the particularly harsh characteristics of initially deployed wireless adhoc and sensor networks. In this paper, assuming such a practical model, we study a fundamental problem of both theoretical and practical interests--the local broadcasting problem. Given a set of nodes V where each node wants to broadcast a message to all its neighbors that are within a certain local broadcasting range R, the problem is to schedule all these requests in the fewest timeslots. By adopting the physical interference mode land without any knowledge on neighborhood, we give a new randomized distributed approximation algorithm for the local broadcasting problem with approximation ratio O (log n) where nis the number of nodes. This distributed approximation algorithm improves the state-of-the-art result in [22] by a logarithmic factor. In this paper, we investigate the connectivity for large-scale clustered wireless sensor and ad hoc networks. We study the effect of mobility on the critical transmission range for asymptotic connectivity in k-hop clustered networks and compare to existing results on nonclustered stationary networks. By introducing k -hop clustering, any packet from a cluster member can reach a cluster head within k hops, and thus the transmission delay is bounded as Θ(1) for any finite k. We first characterize the critical transmission range for connectivity in mobile k-hop clustered networks where all nodes move under either the random walk mobility model with nontrivial velocity or the i.i.d. mobility model. By the term nontrivial velocity, we mean that the velocity of a node v is ω(r(n)), where r(n) is the transmission range of the node. We then compare with the critical transmission range for stationary k-hop clustered networks. In addition, the critical number of neighbors is studied in a parallel manner for both stationary and mobile networks. We also study the transmission power versus delay tradeoff and the average energy consumption per flow among different types of networks. We show that random walk mobility with nontrivial velocities increases connectivity in k-hop clustered networks, and thus significantly decreases the energy consumption and improves the power-delay tradeoff. The decrease of energy consumption per flow is shown to be Θ([(logn) (nd)]) in clustered networks. These results provide insights on network design and fundamental guidelines on building a large-scale wireless network. This paper presents the evaluation of the multi-hop aggregate information efficiency of the slotted and unslotted ALOHA protocols. We consider a multi-hop wireless network where the nodes are spatially characterized by a Poisson point process and the traffic generation also follows a Poisson distribution. By applying the properties of stochastic geometry, we derive a closed-form lower bound on the outage probability as a function of the required communication rate, the single-hop distance, the number of hops and the maximum number of retransmissions. The results indicate that slotted ALOHA always outperforms its unslotted version, demonstrating the importance of synchronization in distributed networks. In addition, we show that it is always possible to optimize the network efficiency by properly setting the required rate for a given packet density. Finally, in the scenario considered, the use of retransmissions and multiple hops never achieves the best performance if compared to the option of single-hop links without retransmissions. Gupta and Kumar (see IEEE Transactions an Information Theory, vol.46, no.2, p.388-404, 2000) determined the capacity of wireless networks under certain assumptions, among them point-to-point coding, which excludes for example multi-access and broadcast codes. We consider essentially the same physical model of a wireless network under a different traffic pattern, namely the relay traffic pattern, but we allow for arbitrarily complex network coding. In our model, there is only one active source destination pair, while all other nodes assist this transmission. We show code constructions leading to achievable rates and derive upper bounds from the max-flow min-cut theorem. It is shown that lower and upper bounds meet asymptotically as the number of nodes in the network goes to infinity, thus proving that the capacity of the wireless network with n nodes under the relay traffic pattern behaves like log n bits per second. This demonstrates also that network coding is essential: under the point-to-point coding assumption considered by , the achievable rate is constant, independent of the number of nodes. Moreover, the result of this paper has implications' and extensions to fading channels and to sensor networks. Many analytic results for the connectivity, coverage, and capacity of wireless networks have been reported for the case where the number of nodes, n, tends to infinity (large-scale networks). The majority of these results have not been extended for small or moderate values of n; whereas in many practical networks, n is not very large. In this article, we consider finite (small-scale) wireless sensor networks. We first show that previous asymptotic results provide poor approximations for such networks. We provide a set of differences between small-scale and large-scale analysis and propose a methodology for analysis of finite sensor networks. Furthermore, we consider two models for such networks: unreliable sensor grids and sensor networks with random node deployment. We provide easily computable expressions for bounds on the coverage and connectivity of these networks. With validation from simulations, we show that the derived analytic expressions give very good estimates of such quantities for finite sensor networks. Our investigation confirms the fact that small-scale networks possess unique characteristics different from their large-scale counterparts, necessitating the development of a new framework for their analysis and design. There is a rich recent literature on information-theoretically secure communication at the physical layer of wireless networks, where secret communication between a single transmitter and receiver has been studied extensively. In this paper, we consider how single-hop physical layer security techniques can be extended to multi-hop wireless networks. We show that guaranteed security can be achieved in multi-hop networks by augmenting physical layer security techniques, such as cooperative jamming, with the higher layer network mechanisms, such as routing. Specifically, we consider the secure minimum energy routing problem, in which the objective is to compute a minimum energy path between two network nodes subject to constraints on the end-to-end communication secrecy and goodput over the path. This problem is formulated as a constrained optimization of transmission power and link selection, which is proved to be NP-hard. Nevertheless, we show that efficient algorithms exist to compute both exact and approximate solutions for the problem. In particular, we develop an exact solution of pseudo-polynomial complexity, as well as an e-optimal approximation of polynomial complexity. Simulation results are also provided to show the utility of our algorithms and quantify their energy savings compared to a combination of (standard) security-agnostic minimum energy routing and physical layer security. In the simulated scenarios, we observe that, by jointly optimizing link selection at the network layer and cooperative jamming at the physical layer, our algorithms reduce the network energy consumption by half. During and immediately after their deployment, ad hoc and sensor networks lack an efficient communication scheme rendering even the most basic network coordination problems difficult. Before any reasonable communication can take place, nodes must come up with an initial structure that can serve as a foundation for more sophisticated algorithms. In this paper, we consider the problem of obtaining a vertex coloring as such an initial structure. We propose an algorithm that works in the unstructured radio network model. This model captures the characteristics of newly deployed ad hoc and sensor networks, i.e. asynchronous wake-up, no collision-detection, and scarce knowledge about the network topology. When modeling the network as a graph with bounded independence, our algorithm produces a correct coloring with O(Δ) colors in time O(Δ log n) with high probability, where n and Δ are the number of nodes in the network and the maximum degree, respectively. Also, the number of locally used colors depends only on the local node density. Graphs with bounded independence generalize unit disk graphs as well as many other well-known models for wireless multi-hop networks. They allow us to capture aspects such as obstacles, fading, or irregular signal-propagation. We consider transmission of packets in two-hop wireless ad hoc networks in which relay nodes are deployed between the source-destination pairs. Based on results from extreme value theory and product tails, we derive throughput scaling laws when opportunistic relay selection is performed. Assuming partial channel state information at each transmitter (CSIT) and decode- and-forward, half-duplex relays, we investigate how the per-hop throughput depends on the channel gain asymptotic distribution and the relay deployment. In dense networks with lambda t nodes per m 2 and fixed relay distances, we provide specific scaling laws for Rayleigh, lognormal, and Weibull fading, showing that the throughput is upper bounded by thetas(radic(lambda t )). Interestingly, with variable relay distances and location-aware relay selection, we analytically show that regularly varying channel distributions result in enhanced multi-relay diversity gain, achieving linear throughput scaling thetas(radic(lambda t )). In a wireless network, the set of transmitting nodes changes frequently because of the MAC scheduler and the traffic load. Analyzing the connectivity of such a network using static graphs would lead to pessimistic performance results. In this paper, we consider an ad hoc network with half-duplex radios that uses multihop routing and slotted ALOHA for the network MAC contention and introduce a random dynamic multi-digraph to model its connectivity. We first provide analytical results about the degree distribution of the graph. Next, defining the path formation time as the minimum time required for a causal path to form between the source and destination on the dynamic graph, we derive the distributional properties of the connection delay using techniques from first-passage percolation and epidemic processes.We show that the delay scales linearly with the distance and provide asymptotic results (with respect to time) for the positions of the nodes which are able to receive information from a transmitter located at the origin. We also provide simulation results to support the theoretical results. Long-distance multi-hop wireless networks have been used in recent years to provide connectivity to rural areas. The salient features of such networks include TDMA channel access, nodes with multiple radios, and point-to-point long-distance wireless links established using high-gain directional antennas mounted on high towers. It has been demonstrated previously that in such network architectures, nodes can transmit concurrently on multiple radios, as well as receive concurrently on multiple radios. However, concurrent transmission on one radio, and reception on another radio causes interference. Under this scheduling constraint, given a set of source-destination demand rates, we consider the problem of satisfying the maximum fraction of each demand (also called the maximum concurrent flow problem). We give a novel joint routing and scheduling scheme for this problem, based on linear programming and graph coloring. We analyze our algorithm theoretically and prove that at least 50 of a satisfiable set of demands is satisfied by our algorithm for most practical networks (with maximum node degree at most 5). Understanding the maximum end-to-end throughput available with any given wireless multi-hop network is an important, but very challenging task. It may be approximated up to certain degree for several simplified topologies such as chain, cross, and others. Typically, in case of chain-topology wireless multi-hop networks employing IEEE 802.11 DCF (distributed coordination function), the maximum throughput can be derived by calculating 1-hop average throughput considering the so-called bottleneck region. However, this approximation method that utilizes only the 1-hop average throughput is limited in terms of accuracy. Thus, in this paper, after verifying the limitation of 1-hop average approximation for the throughput calculation of chain-topology multi-hop wireless networks, a refined calculation scheme is proposed by additionally considering the deviation of throughput. Especially, we focus on the fact that the throughput deviation could lead to collisions, and the throughput deviation happens due to the summation of uniformly distributed backoff times of IEEE 802.11 DCF. With several ns-2-based network simulations, we verify the impact of selected temporal variations to the throughput calculation and thus validate the improved accuracy of proposed calculation scheme. In this paper we investigate the connectivity for large-scale clustered wireless sensor and ad hoc networks. We study the effect of mobility on the critical transmission range for asymptotic connectivity in k-hop clustered networks, and compare to existing results on non-clustered stationary networks. By introducing k-hop clustering, any packet from a cluster member can reach a cluster head within k hops, and thus the transmission delay is bounded as Θ(1) for any finite k. We first characterize the critical transmission range for connectivity in mobile k-hop clustered networks where all nodes move under either the random walk mobility model with non-trivial velocity or the i.i.d. mobility model. By the term non-trivial velocity, we mean that the velocity of nodes v is Θ(1). We then compare with the critical transmission range for stationary k-hop clustered networks. We also study the transmission power versus delay trade-off and the average energy consumption per flow among different types of networks. We show that random walk mobility with non-trivial velocity increases connectivity in k-hop clustered networks, and thus significantly decreases the energy consumption and improves the power-delay trade-off. The decrease of energy consumption per flow is shown to be Θ(logn nd ) in clustered networks. These results provide insights on network design and fundamental guidelines on building a large-scale wireless network.