Semi-Markov Models

Appendix B

The Solution of the System of Integral Equations (2.74)

To solve the system (2.74), let us substitute its third equation into the second one:

$\rho (2\widehat{1}0)={\rho }_0\underset{0}{\overset{\infty }{\int }}f(x+t)r(t)dt+{\overline{p}}_1\underset{x}{\overset{\infty }{\int }}\rho (2\widehat{1}0y)r(y-x)dy.$

(B.1)

Denoting $\rho (2\widehat{1}0)={\varphi }_1(x)$, we get the equation (B.1) in the following form:

${\varphi }_1(x)={\rho }_0\underset{x}{\overset{\infty }{\int }}f(t)r(t-)dt+{\overline{p}}_1\underset{x}{\overset{\infty }{\int }}{\varphi }_1(y)r(y-x)dy.$

(B.2)

Introduce the function $\tilde{r}(x)={\overline{p}}_{1}r(x)$ and the operator

$\left(A_{\tilde{r}}\varphi \right)(x)=\underset{x}{\overset{\infty }{\int }}\tilde{r}(y-x)\varphi (y)dy,$

then the equation (B.2) is as follows: ...