Modeling and connectivity analysis in obstructed wireless ad hoc networks

Abstract

Connectivity properties of wireless networks in open space are typically modeled using geometric random graphs and have been analyzed in depth in different studies. Such scenarios, however, do not often represent situations encountered in practice, like urban environments or indoor spaces, which are deeply affected by obstacles. In this work, we present a model for obstructed wireless ad hoc networks consisting of a set of n nodes, deployed at random in a lattice square of size g×g, with a common transmission range r. For positioning the nodes in the field, all segments are considered as one-dimensional, but for communication purposes, we add a parameter to model the segments' width. Our model can be used to study the structure of obstructed networks analytically, as well as to simulate and evaluate a variety of node deployment strategies and the resulting network topologies. We derive analytical forms for the probability of existing crossing links between parallel and perpendicular segments sharing an intersection, toward a first topological characterization of our model. Moreover, we compute a lower bound for the probability of connectivity at intersections between segments, and apply percolation theory to derivate the Critical Transmission Range for connectivity in the overall network, i.e., the minimum transmission range that generates communication graphs that are connected with high probability.

Connectivity in obstructed wireless networks

“…A preliminary analysis of the MaxNorm model was presented in [1]. However, as shown in Section 5, it resulted in much higher (worse) values for the critical street width (i.e., for the minimal street width that induces a valid CTR for connectivity) and low-quality approximation to the Euclidean model, which is more realistic but leads to a very complex analysis.…”

“…W.l.o.g., we consider that positions of nodes reflect the Euclidean distances from each node to the intersection point between segments. There exists at least one link between two nodes in different perpendicular segments if and only if there is a link between X (1) and Y (1) . Considering r ≥ √ 8 , we can compute p ⊥ (μ) by…”

“…where fX (1) Y (1) (x, y) is the joint distribution function of X (1) and Y (1) . It is well known that the k-th order statistic of a family of μ independent standard uniform random variables is a Beta random variable U (k) ∼ B(k, μ + 1 − k).…”

“…It is well known that the k-th order statistic of a family of μ independent standard uniform random variables is a Beta random variable U (k) ∼ B(k, μ + 1 − k). Since all random variables are independent, we have that the joint distribution function of X (1) and…”

“…W.l.o.g., we consider that positions of nodes reflect the Euclidean distances from each node to the intersection point between segments. Consider the order statistics X (1) , Y (1) , W (1) and Z (1) , from the families {Xi} 1≤i≤μ , {Yj} 1≤j≤μ , {W k } 1≤k≤μ and {Zm} 1≤m≤μ , respectively. There exists a path between at least one node in a segment connecting at least one node in each one of the three other segments if and only if there exists a path between X (1) , Y (1) , W (1) and Z (1) .…”