Lattices are important as models for the node locations in wireless networks for two main reasons: (1) When network designers have control over the placement of the nodes, they often prefer a regular arrangement in a lattice for coverage and interference reasons. (2) If nodes are randomly distributed or mobile, good channel access schemes ensure that concurrent transmitters are regularly spaced, hence the locations of the transmitting nodes are well approximated by a lattice. In this paper, we introduce general interference bounding techniques that permit the derivation of tight closed-form upper and lower bounds for all lattice networks, and we present and analyze optimum or near-optimum channel access schemes for one-dimensional, square, and triangular lattices.
While a growing body of work studies interference in random networks (see, , @cite_2 @cite_4 and references therein), only few papers have addressed the issue of interference in lattice networks. In @cite_8 , bounds on the interference in triangular networks were derived using a relatively crude upper bound on the Riemann zeta function that is within 25 of @math to @math . We will derive a much tighter bound that is within 1.3 TDMA scheduling scheme for square lattices that is optimum for the case where the density of concurrent transmitters is @math is suggested in @cite_5 . Here, we provide near-optimum scheduling schemes for any density @math , @math . The interference distribution in one-dimensional networks with Rayleigh fading is analyzed in @cite_0 for the case where all nodes transmit, and @cite_1 derives outage results an throughput-optimum TDMA schedulers for the same type of network. Finally, the single-hop throughput for two-dimensional lattice networks with Rayleigh fading is approximated in @cite_6 . For non-fading channels, @cite_7 provides throughput results for general TDMA schemes in two-dimensional lattice networks. The interference is expressed using complicated infinite double sums (that are evaluated numerically), for which we will present tight bounds.
@cite_4 @cite_7 @cite_8 @cite_1 @cite_6 @cite_0 @cite_2 @cite_5
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Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communication-theoretic results accounting for the networks geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs -including point process theory, percolation theory, and probabilistic combinatorics-have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue. The throughput of large wireless networks with regular topologies is analyzed under two medium-access control schemes: synchronous array method (SAM) and slotted ALOHA. The regular topologies considered are square, hexagon, and triangle. Both nonfading channels and Rayleigh fading channels are examined. Furthermore, both omnidirectional antennas and directional antennas are considered. Our analysis shows that the SAM leads to a much higher network throughput than the slotted ALOHA. The network throughput in this paper is measured in either bits-hops per second per Hertz per node or bits-meters per second per Hertz per node. The exact connection between the two measures is shown for each topology. With these two fundamental units, the network throughput shown in this paper can serve as a reliable benchmark for future works on network throughput of large networks. Outage probabilities and single-hop throughput are two important performance metrics that have been evaluated for certain specific types of wireless networks. However, there is a lack of comprehensive results for larger classes of networks, and there is no systematic approach that permits the convenient comparison of the performance of networks with different geometries and levels of randomness. The uncertainty cube is introduced to categorize the uncertainty present in a network. The three axes of the cube represent the three main potential sources of uncertainty in interference-limited networks: the node distribution, the channel gains (fading), and the channel access scheme (set of transmitting nodes). For the performance analysis, a new parameter, the so- called spatial contention, is defined. It measures the slope of the outage probability in an ALOHA network as a function of the transmit probability p at p = 0. Outage is defined as the event that the signal-to-interference ratio (SIR) is below a certain threshold in a given time slot. It is shown that the spatial contention is sufficient to characterize outage and throughput in large classes of wireless networks, corresponding to different positions on the uncertainty cube. Existing results are placed in this framework, and new ones are derived. Further, interpreting the outage probability as the SIR distribution, the ergodic capacity of unit-distance links is determined and compared to the throughput achievable for fixed (yet optimized) transmission rates. We present closed-form expressions of the average link throughput for sensor networks with a slotted ALOHA MAC protocol in Rayleigh fading channels. We compare networks with three regular topologies in terms of throughput, transmit efficiency, and transport capacity. In particular, for square lattice networks, we present a sensitivity analysis of the maximum throughput and the optimum transmit probability with respect to the signal-to-interference ratio threshold. For random networks with nodes distributed according to a two-dimensional Poisson point process, the average throughput is analytically characterized and numerically evaluated. It turns out that although regular networks have an only slightly higher average link throughput than random networks for the same link distance, regular topologies have a significant benefit when the end-to-end throughput in multihop connections is considered. This paper deals with the distribution of cumulated instantaneous interference power in a Rayleigh fading channel for an infinite number of interfering stations, where each station transmits with a certain probability, independently of all others. If all distances are known, a necessary and sufficient condition is given for the corresponding distribution to be nondefective. Explicit formulae of density and distribution functions are obtained in the interesting special case that interfering stations are located on a linear grid. Moreover, the Laplace transform of cumulated power is investigated when the positions of stations follow a one- or two-dimensional Poisson process. It turns out that the corresponding distribution is defective for the two-dimensional models. For wireless ad hoc networks with stationary and deterministically placed nodes, finding the optimal placement of the nodes is an interesting and challenging problem, especially under energy and QoS constraints. We study and compare the performance of several networks with regular topologies utilizing a Rayleigh fading link model. For nearest neighbor and shortest path routing, analytical expressions of the path efficiency, delay, and energy consumption for a given end-to-end reception probability are derived. For the interference analysis, the maximum throughput and optimum transmit probability are determined, and a simple MAC scheme is compared with an optimum scheduler, yielding lower and upper performance bounds