Outage probabilities in poisson and clumped poisson-distributed hybrid ad-hoc networks

We study coverage and outage in large networks containing two kinds of fixed wireless transceivers that we call nodes and base stations (BSs) respectively. The nodes have common wireless capabilities, with the BSs having in addition direct wideband connections to the wired infrastructure. Nodes can communicate with the "outside world" only through the BSs. Connections to nodes without a direct (i.e., a single-hop) wireless connection to any BS are established through other nodes serving as wireless repeaters. The locations of the nodes are assumed randomly and uniformly distributed over the entire service area, or clustered following a clumped Poisson process. The BSs are also assumed randomly and uniformly distributed over the given service area. We evaluate the probability of a potential node to have a "working" wireless connection to any of the BSs within a fixed but arbitrary number of hops, as a function of the densities of the BSs and nodes and the parameters of the wireless links, accounting for both fast and shadow fading. We provide exact expressions for the outage probability of an node, and lower bounds when evaluation of the exact expression is impractical. Our results allow simple comparisons with other means of extending the BSs' reach, thereby allowing network designers to choose the optimal solution. We study a large network of fixed wireless transceivers that we call nodes. These nodes have certain common wireless capabilities and some, called base stations (BSs), also have direct wideband connections to the wired infrastructure. Nodes can communicate with the "outside world" only through the BSs. To augment connectivity, connections to nodes without a direct (i.e., a single-hop) wireless connection to any BS are established through other nodes serving as wireless repeaters, as long as the number of hops does not exceed a prescribed limit. Regular nodes are installed at customer premises and their locations cannot be predicted ahead of time; we therefore assume that their locations are random. We further assume that due to practical constraints, availability of high speed wired connections and economic considerations, BSs are sparse, and often cannot be positioned based on coverage considerations only. This scenario applies in particular to service providers offering wideband wireless connectivity in an area where high speed optical cables are scarce and owned by different entities. To account for this reality, we assume that the BSs, like the regular nodes, are also placed randomly over the service area, recognizing that this is a "worst case" scenario. The focus of this paper is the t-hop outage probability, defined as the probability that a node (e.g., a sensor) cannot connect to any of the BSs in ≤ t hops, evaluated as a function of the densities of the nodes and the statistics of fast and shadow fades on the wireless links. Since t-hop outage is equivalent to the minimum number of hops from the node to any BS being greater than t, we see that the collection of t-hop outage probabilities for all t is equivalent to the complementary cumulative distribution function (cdf) of the minimum number of hops required for a packet to get from an arbitrary node to aB S. This paper is organized as follows: following a review of related work, we first state and prove for our reference a well- known result that will be extensively used later. We then de- scribe the location models to be used and the wireless channel model, and compute in closed form the exact probability that an arbitrary node is isolated from BSs and other nodes when the node locations are either homogeneous or clump Poisson points. We then derive bounds on the general t-hop outage probability for both node location models using two different approximations. These require the distribution of the distance between two nodes conditioned on their being connected, which is derived next. We compare our analytical results with simulations and end by summarizing our conclusions.