distribution of times that a traveller passes by vertex

Question

a traveller is travelling on a map. arriving every vertex of the map, the traveller could choose to go to next vertex according to a constant probability. The probabilities are represented in a matrix \begin{equation} A=\left[\begin{matrix}a_{1,1}&\cdots& a_{n,1}\\ \vdots&\ddots&\vdots\\ a_{1,n}&\cdots&a_{n,n} \end{matrix} \right] \end{equation} , where $a_{i,j}$ is the probability of travelling from vertex $j$ to $i$. The length between every vertex pairs are represented in a matrix \begin{equation} L=\left[\begin{matrix}l_{1,1}&\cdots& l_{n,1}\\ \vdots&\ddots&\vdots\\ l_{1,n}&\cdots&l_{n,n} \end{matrix} \right] \end{equation} , where $l_{i,j}$ is the length of the path from vertex $j$ to $i$. The traveller moves at constant speed $v$. I want to calculate the the distributions of times the traveller passes by vertex $i$ within certain time period $t$. The distribution is represented by $f_i(n,t)$. Any hints to the solve the distribution? Thx in advance.