Semi-Markov Models

Appendix A

The Solution of the System of Integral Equations (2.24)

Let us prove the system of equation (2.24) to have the solution determined by the formula (2.25).

Introduce the operators:

$\left(A_r\varphi \right)(x)=\underset{x}{\overset{\infty }{\int }}r(y-x)\varphi (y)dy,\left({\overline{A}}_r\varphi \right)(x)=\underset{0}{\overset{\infty }{\int }}r(+y)\varphi (y)dy,$

$\left(A_v\varphi \right)(x)=\underset{x}{\overset{\infty }{\int }}v(y-x)\varphi (y)dy,\left({\overline{A}}_v\varphi \right)(x)=\underset{0}{\overset{\infty }{\int }}v(+y)\varphi (y)dy.$

Then the system of equation (2.24) takes the following form:

$\left\{\begin{array}{l}{\rho }_0=\rho (111)=\rho (222),\\ \rho (210x)={\rho }_0{A}_{r}f+A_r\rho (211x),\\ \rho (211x)=A_v\rho (210x),\\ \rho (100x)={\overline{A}}_v\rho (210x),\\ \rho (222)=\underset{0}{\overset{\infty }{\int }}\rho (100)d+\rho (200),\\ \rho (101x)={\rho }_0{\overline{A}}_{r}f+{\overline{A}}_r\rho (211x),\\ \rho (200)=\underset{0}{\overset{\infty }{\int }}\rho (101)d,\\ 2{\rho }_0+\rho (200)+\underset{0}{\overset{\infty }{\int }}\left(\rho (210x)+\rho (211x)+\rho (100x)+\rho (101x)\right)dx=1.\end{array}\right.$

(A.1) ...