Assume we have n by n square area and a movable object initially located at a random position in the specified area. If the object mobility modeled by a Gauss-Markov mobility model with a random speed(S) and a random direction (R), what is the probability the object gets out of the area in time t?

The problem Detail.

The current speed and direction is related to the previous speed and direction as the following equation.

$S_{t} = \alpha S_{t−1} + (1−\alpha)\check{S} + (1−\alpha^2) \sqrt{S_{x_{t−1}}}$

$d_{t} = \alpha d_{t−1} + (1−\alpha)\check{d} + (1−\alpha^2) \sqrt{d_{x_{t−1}}}$

As $S_{t}$ and $d_{t}$ are values of speed and direction for movement in the period time t. $S_{t−1}$ and $d_{t−1}$ are values of speed and direction for movement in the period time t−1. α is a constant value in the range [0,1]. $\check{S}$ and $\check{d}$ are constants representing the mean speed and direction. $S_{x_{t−1}}$ and $d_{x_{t−l}}$ are random variables from a Gaussian distribution. α is a single tuning parameter that represents the different levels of randomness or degree of random.

The destination position of the motion at time t is calculated by the following equations.

$x_{t}=x_{t−1} + x_{t-1}\cos{d_{t−1}}$

$y_{t} = y_{t−1} + S_{t−1}\sin{d_{t−1}}$