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 Gauss-Markov mobility model 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.

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

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