Theoretical Issues in Modeling of Large-Scale Wireless Sensor Networks

In large-scale WSNs (wireless sensor networks), packets of a source node are often transmitted via multi-hop relays to reach their sink nodes. The hop-count, h, of a sink node is related to a particular hop-count originator (namely the source node) and it is defined as the least number of multi-hop relays required to send one packet from the source to the sink in this chapter. A source node can use a simple controlled flooding to set up the hop-counts relative to itself for all other nodes [1]. Let d denote the Euclidean distance between a source and its sink (the source-to-sink distance is denoted as SS-distance in the sequel), various statistical models that characterize the relationships between h and d are some of the fundamental problems in modeling large-scale WSNs. These results can be applied to address many other WSN research issues such as range-free localization, communication protocol design and evaluation, throughput optimization, transmission power control and etc. Given a WSN whose nodes are distributed randomly according to a two-dimensional homogeneous Poisson point process of density λ, we investigate two statistical relationships between hop-count h and SS-distance d in this chapter. First of all, we propose a method termed CSP (Convolution of Successive Progress) to compute the K-hop connection probability for a two-dimensional network. The K-hop connection probability is defined as the conditional probability that a sink has a hop-count h = K with respect to a source given that the SS-distance is d. Mathematically, the K-hop connection probability is defined as the conditional probability1 P(h = K|d). The CSP method is also extended into threedimensional networks. Secondly, based on the results of K-hop connection probability, we also present a method to compute the PDF (probability density function) of the SS-distance d conditioned on hopcount h = K, namely the PDF of SS-distance d of all nodes with a hop-count h = K. Mathematically, this conditional PDF2 is denoted as f(d|h = K). Simulation studies show that the proposed methods are able to achieve significant error reduction in computing these hop-distance statistics (i.e. the conditional probability/PDF) compared with existing methods.