# Multi-Wavelength Path Loss Model for Indoor VLC with Mobile Human Blockage
## 4 Blockage Probability Analysis and Modeling
### Human Moving Trajectory Generating  
In this subsection, we present a human moving trajectory generating algorithm to characterize realistic human walking behavior. In order not to lose authenticity, the semi-Markov renewal process and the Levy-walk are used to describe human mobility. In a semi-Markov renewal process, a human moving trajectory can be divided into two states: the moving state before reaching a position and the sojourn state at the position. The two states can be used alternately to generate the human moving trajectory in a continuous time period. In the moving state, three steps need to be performed. Firstly, select the next destination according to the step length and orientation. Secondly, calculate the duration of the moving state based on the step length and the moving speed. Thirdly, select the moving route from the current position to the next destination. The step length indicates the radial distance between the next destination and the current position. According to [13], the step length is subject to truncated heavy-tailed distribution such as the Pareto distribution, which can be expressed by  
\[p(r)=\frac{\xi\times(r_{min})^{\xi}\times r^{-(\xi+1)}}{1-(r_{min}/r_{max})^{ \xi}}, \tag{17}\]  
in which \(p(r)\) is the PDF of the step length \(r\). \(\xi\) is a positive parameter. \(r_{min}\) and \(r_{max}\) are the minimum and maximum step lengths related to the distance of elements in the destination set. Human movements share similar patterns on any scale since the key  
Figure 6: Path loss for the 3-LED scenario with \(\lambda=455\,\mathrm{nm}\). statistical characteristics of re-scaled individual trajectories are identical. Hence, this truncated heavy-tail distribution contributes prominently to the indoor trajectories. The orientation is the direction of the next destination in the local coordinate system (LCS) of the current position. The random orientation of a mobile terminal can be modeled by a polar angle and an azimuth angle. In this paper, we assume that the human body maintains upright while walking, thus the polar angle is stationary at 0 degrees. According to [30], the azimuth angle can be regarded as a random variable subject to uniform distribution between 0 and 360 degrees. After the selection of the next destination, the human body moves in a straight line from the current position to the next destination. Generally, for the human body moving indoors, the maximum and minimum walking speed is 2.1 m/s [31] and 1 m/s, respectively. The walking speed can be regarded as evenly distributed between the maximum speed and the minimum speed. When \(p(r)\) is a heavy-tailed function, the mobility specified by \(\phi(t_{m}|r)\) is a Levy-walk, which is a kind of continuous-time random walk that has the joint space-time probability density function (PDF) [13]:  
\[\Phi(r,t_{m})=\phi(t_{m}|r)p(r), \tag{18}\]  
where \(t_{m}\) is the duration of the moving state can be obtained by dividing the step length by the moving speed. Due to the uniform distribution of moving speed, the duration of the moving state also follows a uniform distribution under the condition that the step length has been obtained. \(\phi(t_{m}|r)\) is the conditional probability density that step length \(r\) takes \(t_{m}\) time in movement. In the general movement, the bounded Levy-walk plays the decisive role in each transition [32]. After reaching the next destination, the sojourn state starts. Within a certain sojourn time, the human body remains stationary in the current position. From [32], the sojourn time is also subjected to a truncated Pareto distribution as  
\[p(t_{\rm s})=\frac{\eta\times(t_{s,min})^{\eta}\times t_{\rm s}^{-(\eta+1)}}{1 -(t_{s,min}/t_{s,max})^{\eta}}, \tag{19}\]  
in which \(t_{\rm s}\) denotes the sojourn time. \(\eta\) corresponds to the positive parameter. \(t_{s,min}\) and \(t_{s,max}\) are the minimum and the maximum sojourn times at the current location, respectively.