In this paper, a tractable model of cellular wireless networks is considered, where both the basestation (BS) and mobile user (MU) locations are distributed as independent Poisson point processes, and each MU connects to its nearest BS. Each packet from the BS is transmitted using an automatic-repeat-request strategy until the signal-to-interference-plus-noise ratio (SINR) is larger than a threshold, and the packet delay is equal to the expected number of retransmissions required for successful reception. We define the network capacity as the product of the BS density and the reciprocal of the packet delay, maximized over all BS strategies. This definition of capacity, while being natural, is non-trivial to analyses because of the temporal correlations of SINRs and arbitrary BS strategies. An exact characterization (non-asymptotic) of this natural capacity metric is derived, which shows that the capacity increases polynomially with the BS density in the low BS density regime and then scales inverse exponentially with the increasing BS density.

In cellular networks, under ARQ and SINR model of transmission, the effective downlink rate of packet transmission is the reciprocal of the expected delay (number of retransmissions needed till success). We define the cellular network capacity as the ratio of the basestation (BS) density and the expected delay. Exact characterization of this natural and practical but non-trivial (because of SINR temporal correlations) capacity metric is derived. The capacity is shown to first increase polynomially with the BS density and then scale inverse exponentially with the increasing BS density. Two distinct upper bounds are derived that are relevant for the low and the high BS density regimes. A single power control strategy is shown to achieve the upper bounds in both the regimes up to constants. Our result is fundamentally different than the transport and transmission capacity for ad hoc networks that scale as the square root of the (high) BS density. Our results show that the strong temporal correlations of SINRs with PPP distributed BS locations model for cellular networks is limiting, and the realizable capacity is much smaller than previously thought.

In wireless networks, where each node transmits independently of other nodes in the network (the ALOHA protocol), the expected delay experienced by a packet until it is successfully received at any other node is known to be infinite for the signal-to-interference-plus-noise-ratio (SINR) model with node locations distributed according to a Poisson point process. Consequently, the information velocity, defined as the limit of the ratio of the distance to the destination and the time taken for a packet to successfully reach the destination over multiple hops, is zero, as the distance tends to infinity. A nearest neighbor distance based power control policy is proposed to show that the expected delay required for a packet to be successfully received at the nearest neighbor can be made finite. Moreover, the information velocity is also shown to be non-zero with the proposed power control policy. The condition under which these results hold does not depend on the intensity of the underlying Poisson point process.

Connectivity in an information-theoretically secure graph is considered where both the legitimate and the eavesdropper nodes are distributed as Poisson point processes. To allow concurrent transmissions from multiple legitimate nodes, a signal-to-interference plus noise ratio secure graph is introduced, and its percolation (having an unbounded connected component) properties are studied. It is shown that for a fixed eavesdropper node density, percolation happens for large enough (but finite) legitimate node density and small enough interference suppression parameter of the legitimate nodes. Conversely, a concrete bound is obtained that shows that if the legitimate node density is below a fixed threshold, then the probability of percolation is zero.

We propose a method to compute a probably approximately correct (PAC) normalized histogram of observations with a refresh rate of Θ(1) time units per histogram sample on a random geometric graph with noise-free links. The delay in computation is Θ(√n) time units. We further extend our approach to a network with noisy links. While the refresh rate remains Θ(1) time units per sample, the delay increases to Θ(√n log n). The number of transmissions in both cases is Θ(n) per histogram sample. The achieved Θ(1) refresh rate for PAC histogram computation is a significant improvement over the refresh rate of Θ(1/log n) for histogram computation in noiseless networks. We achieve this by operating in the supercritical thermodynamic regime where large pathways for communication build up, but the network may have more than one component. The largest component however will have an arbitrarily large fraction of nodes in order to enable approximate computation of the histogram to the desired level of accuracy. Operation in the supercritical thermodynamic regime also reduces energy consumption. A key step in the proof of our achievability result is the construction of a connected component having bounded degree and any desired fraction of nodes. This construction may also prove useful in other communication settings on the random geometric graph.

In a sensor network, the points in the operational area that are suitably sensed are a two-dimensional spatial coverage process. For randomly deployed sensor networks, typically, the network coverage of two-dimensional areas is analyzed. However, in many sensor network applications, e.g., tracking of moving objects, the sensing process on paths, rather than in areas, is of interest. With such an application in mind, we analyze the coverage process induced on a one-dimensional path by a sensor network that is modeled as a two-dimensional Boolean model. In the analysis, the sensor locations form a spatial Poisson process of density \lambda and the sensing regions are circles of i.i.d. random radii. We first obtain a strong law for the fraction of a path that is k{\hbox{-}}{\rm sensed}, i.e., sensed by (\geq k) sensors. Asymptotic path-sensing results are obtained under the same limiting regimes as those required for asymptotic coverage by a two-dimensional Boolean model. Interestingly, the asymptotic fraction of the area that is 1-sensed is the same as the fraction of a path that is 1-sensed. For k = 1, we also obtain a central limit theorem that shows that the asymptotics converge at the rate of \Theta(\lambda^{1/2}) for k = 1. For finite networks, the expectation and variance of the fraction of the path that is k{\hbox{-}}{\rm sensed} is obtained. The asymptotics and the finite network results are then used to obtain the critical sensor density to k{\hbox{-}}{\rm sense} a fraction \alpha_{k} of an arbitrary path with very high probability is also obtained. Through simulations, we then analyze the robustness of the model when the sensor deployment is nonhomogeneous and when the paths are not rectilinear. Other path coverage measures like breach, support, "length to first sense,? and sensing continuity measures like holes and clumps are also characterized. Finally, we discuss some generalizations of the results like characterization of the coverage process of m{\hbox{-}}{\rm dimensional} "straight line paths? by n{\hbox{-}}{\rm dimensional}, n > m, sensor networks.

We consider evolving exponential RGGs in one dimension and characterize the time dependent behavior of some of their topological properties. We consider two evolution models and study one of them detail while providing a summary of the results for the other. In the first model, the inter-nodal gaps evolve according to an exponential AR(1) process that makes the stationary distribution of the node locations exponential. For this model we obtain the one-step conditional connectivity probabilities and extend it to the k-step case. Finite and asymptotic analysis are given. We then obtain the k-step connectivity probability conditioned on the network being disconnected. We also derive the pmf of the first passage time for a connected network to become disconnected. We then describe a random birth-death model where at each instant, the node locations evolve according to an AR(1) process. In addition, a random node is allowed to die while giving birth to a node at another location. We derive properties similar to those above.

We consider the clustering properties of one-dimensional sensor networks where the nodes are randomly deployed. Unlike most other work on randomly deployed networks, ours assumes that the node locations are drawn from a non uniform distribution. Specifically, we consider an exponential distribution. We first obtain the probability that there exists a path between two labeled nodes in a randomly deployed network and obtain the limiting behavior of this probability. The probability mass function (pmf) for the number of components in the network is then obtained. We show that the number of components in the network converges in distribution. We also derive the probabilities for different locations of the components. We then obtain the probability for the existence of a k-sized component and components of size /spl ges/k. Asymptotics in the number of nodes in the network are computed for these probabilities. An interesting result is that, as the number of nodes, n, in the network tends to infinity, a giant component, in which a specific fraction, /spl alpha/, of the nodes form a component, almost surely does not exist for any 0n/sub 0/, the network almost surely does not have a giant component.