Semi-Markov Models

Appendix D

The Solution of the System of Equation (3.34)

Let us prove the system (3.34) to have the solution given by formula (3.35). Introduce the notation: $\rho (3111)={\rho }_0$.

By substituting the third equation of the system (3.34) in the fifth one, we get:

$\begin{array}{l}\rho (3\widehat{0}\widehat{1}0)=\underset{0}{\overset{\infty }{{\displaystyle \int }}}\rho (1011z)e^{-{\lambda }_{2}z}dz={\rho }_0{\lambda }_1\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-{\lambda }_{2}z}dz\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-({\lambda }_1+{\lambda }_2)t}r(z+t)dt=\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}={\rho }_0{\lambda }_1\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-{\lambda }_{2}z}dz\underset{z}{\overset{\infty }{{\displaystyle \int }}}{e}^{-({\lambda }_1+{\lambda }_2)(t-z)}r(t)dt={\rho }_0{\lambda }_1\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-({\lambda }_1+{\lambda }_2)t}r(t)dt\underset{0}{\overset{t}{{\displaystyle \int }}}{e}^{{\lambda }_{1}z}dz=\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}={\rho }_0\left(\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-{\lambda }_{2}t}r(t)dt-\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-({\lambda }_1+{\lambda }_2)t}r(t)dt\right).\end{array}$

(D.1)

Similarly, substituting the sixth equation of the system in the fourth, we have:

$\rho (3\widehat{1}\widehat{0}0)={\rho }_0\left(\underset{0}{\overset{\infty }{\int }}{e}^{-{\lambda }_{1}t}r(\right)$