0\right) \mathbb{C}_{1}+\sum_{l=2}^{N}\left[ \mathrm{sgn}\left( \lambda
_{l}\right) \right] ^{n}\left( \mathbb{U^{\mathrm{T}}}\left( 0\right) 
\mathbb{C}_{l}\right) \exp \left( -\frac{n}{T_{l}}\right) ,  \label{B30}$$ where $\mathbb{U}^{\mathrm{T}}\left( 0\right) \mathbb{C}_{1}=\lim_{n\rightarrow \infty }\mathbb{U}^{\mathrm{T}}\left( n\right) $ is the asymptotic distribution of vehicles and $\left[ \mathrm{sgn}\left( \lambda _{l}\right) \right] ^{n}\times \left( \mathbb{U^{\mathrm{T}}}\left( 0\right) \mathbb{C}_{l}\right) \exp \left( -n/T_{l}\right) $ are ($N-1$) partial distributions in the transient regimes and the matrices can have negative entries, although the entries of the vectors $\mathbb{U}^{\mathrm{T}}\left( 0\right) 
\mathbb{C}_{1} $ and $\mathbb{U}^{\mathrm{T}}\left( n\right) $ are positive. We illustrate the theory through an example. Let us consider the SM $$\mathbb{M}=\left( 
\begin{array}{ccc}
 1/2 & {0} &  1/2 \\ 
 1/4 &  1/2&  1/4\\ 
 4/10&  3/10&  3/10\end{array}\right) \ ,  \label{C1}$$ whose eigenvectors and eigenvalues are

\[C2\] $$\begin{aligned}
\mathbb{X}^{\left( 1\right) } &=&\left( 
\begin{array}{r}
{0.577350} \\ 
{0.577350} \\ 
{0.577350}\end{array}\right) ,\quad {\ \lambda }_{1}{\ =1},\quad \text{{\ stationary mode,}}
\label{C1a} \\
\mathbb{X}^{\left( 2\right) } &=&\left( 
\begin{array}{r}
{0.576331} \\ 
-{0.802915} \\ 
-{0.152215}\end{array}\right) ,\quad {\ \lambda }_{2}{\ =0.367945},{\ \quad }\text{{\ decaying
mode,}}  \label{C2c} \\
\mathbb{X}^{\left( 3\right) } &=&\left( 
\begin{array}{r}
{0.660268} \\ 
{0.039495} \\ 
-{0.749991}\end{array}\right) ,\quad {\ \lambda }_{3}{\ =-0.067945,\quad }\text{{\ decaying mode.}}
\label{C2e}\end{aligned}$$

The matrix $\mathbb{M}$ is regular because $\mathbb{M}^{2}$ is already positive (all entries are positive). We construct the matrix $\left( \begin{array}{ccc}
\mathbb{X}^{\left( 1\right) } & \mathbb{X}^{\left( 2\right) } & 