Distance Between Random Points in Two Rectangular Cities

Abstract

The travel distance between two cities of rectangular shape is considered. Two uniformly distributed random points, one from each city, are taken. Their straight path travel distance is then measured. The explicit forms for the probability density function of this distance and its expected value are obtained. Numerical results of calculating the exact expected distance and the estimated distance as well as computer simulation are given for various cases. The integer moments of the distance are also discussed.

Distance Between Bivariate Beta Random Points in Two Rectangular Cities

“…In Table 2, the entries under the column "E z " are computed from the Maple computer programming mentioned above using (9), (7), and (2) with a = b = c = a * = b * = c * = 1. The entries under Table 2 Comparisons of exact values E z for independence case with computed and simulated values for the case of correlation = 0 478 when Table 3 Exact, computed and simulated values of E z for the case of independent and uniformly distributed random points when a = b = c = d = a * = b * = c * = d * = 1 A, B, A * , B * , , Exact E z Simulation the column "Exact" are taken from Chu (2006) for the purpose of comparisons.…”

“…Table 3 shows the values of E z when the random points are all independently uniformly distributed, i.e., x, y, x * , and y * are independent and uniformly distributed over 0 1 . The entries under the column "Exact" are taken from Chu (2006) for verification, while the entries under the column "E z " are computed using 14 terms in the infinite series (9) for the special case of uniform distribution when a = b = c = d = a * = b * = c * = d * = 1. Therefore, the joint moment of x and y shown in (12) is used, instead of (2), in the computer programming.…”

“…also considered the distance between random points in a cube. Chu (2006) examined the situations when two independent random points are uniformly distributed on two rectangular cities. He obtained the explicit forms for the probability density function of the distance between these random points as well as for the expected distance.…”

A Geometric Probability Model for Capacity Analysis and Interference Estimation in Wireless Mobile Cellular Systems

“…1(a), the circle x 2 +y 2 = d 2 is entirely inside the hexagon. Since area S = πd 2 6 of the circle intersects with OAB, we have…”

“…The distance between users is thus highly important, which calls for an accurate model that can lead to all statistical characteristics of a cellular system. However, although the geometric probability model based on distance distributions has been developed and applied to the area of city planning and transportation [2], forestry and chemistry [3], relatively little research has appeared in the field of wireless communications.…”