Incidence Rates Calculations in Wireless Mobile Delay-Tolerant Networks

We present the calculations of the incidence rates used in [1]. I. GENERAL CASE With respect to the main text, here we assume that χ ∈ [θ, θ + 2π] instead of χ ∈ [−π, π]. This is adopted for reasons of mathematical convenience, as we show later on. As all incidence rates are periodic, with a period 2π, with respect to χ, this assumption is not in any way restrictive. We expand the description of the forwarding region F that appears in [1] as follows: We use two functions r(φ), p(φ). The function r(φ) is such that (r(φ), φ) traces the boundary of F , in polar coordinates, whereas p(φ) is the angle formed between the positive x-axis and the vector that is perpendicular to the boundary of F at the location (r(φ), φ). We assume that r(φ) is continuous and p(φ) is increasing in φ (but not necessarily continuous). Although it would suffice to define r(φ) and p(φ) for φ ∈ [−π, π], for mathematical convenience we define them for φ ∈ R, therefore r(φ) is periodic (with period 2π). We also require that p(φ+ 2π) = p(φ) + 2π for all φ ∈ R. Also, let x1 and x2, with x1 < x2, be the locations on the x-axis where F intersects it. We have also assumed that x0 ≤ x1. Finally, let y1(x) and y2(x) be the functions describing the boundary of F below and above the x-axis respectively, in Cartesian coordinates. All these quantities we use to describe F appear in Fig. 1. A. Calculation of γ(χ, φ; θ) Note that the relative speed with which a node B with direction χ moves, as viewed by F , is, using phasor notation, vne jχ − vne = vne + vne = 2vn exp ( j [ χ+ θ + π 2 ]) × exp ( j [ χ−θ−π 2 ]) + exp ( j [ θ+π−χ 2 ])