of a SM, as (\[A2\]), can be constructed using empirical data collected from previous observations and $$\mathbb{U}^{\mathrm{T}}\left( 0\right) =\left( 
\begin{array}{cccc}
U_{1}\left( 0\right) & U_{2}\left( 0\right) & U_{3}\left( 0\right) & 
U_{4}\left( 0\right)\end{array}\right) ,  \label{A22}$$ represents an initial state for the distribution of vehicles; the superscript $\mathrm{T}$ stands for transposition. The $n$-step process evolves of state (\[A22\]) as $$\mathbb{U}^{\mathrm{T}}(n)= \mathbb{U}^{\mathrm{T}}(0)\mathbb{M}^{n} \ ,
\label{A24}$$ that is presumed to forecast the distribution of vehicles at the $n$-th moment.

Properties of stochastic matrices (SM)
--------------------------------------

**(1)** The sum of the components of the vector $\mathbb{U}(n)$ is a conserved quantity, $$\sum_{i}U_{i} {\left( n\right) }=\sum_{i}U_{i}{\left( 0\right) } \quad
n=1,2,3,...\ .  \label{A25}$$ because $\mathbb{M}^{n}$ is a SM [@harary1966].

**(2)** The SM $\mathbb{M}\geq 0$, of dimension $N \times N$, has eigenvalues $\lambda _{1}=1$ and $\left\vert \lambda
_{2}\right\vert ,\left\vert \lambda _{3}\right\vert ,...,\left\vert \lambda
_{N}\right\vert <1$, therefore $\lim_{n\rightarrow \infty }\left\vert
\lambda _{k}\right\vert ^{n}=0$, for $k\neq 1$. Eigenvalue $\lambda _{1}$ is known in the literature as the Perron-Frobenius (PF) eigenvalue [@ching2006].

**(3)** If all the eigenvalues of $\mathbb{M}$ have multiplicity 1 with linearly independent eigenvectors $\mathbb{X}^{\left( 1\right) }$, $\mathbb{X}^{\left( 2\right) }$, ..., $\mathbb{X}^{\left( N\right) }$, we construct the matrix $$\mathbb{Q}=\left( 
\begin{array}{cccc}
\mathbb{X}^{\left( 1\right) } & \mathbb{X}^{\left( 2\right) } & \cdots & 
\mathbb{X}^{\left( 