Semi-Markov Models

$\begin{array}{l}\psi ()=\underset{0}{\overset{\infty }{\int }}f(x+y)h(y)dy+\underset{0}{\overset{\infty }{\int }}\pi (x,y)dy\underset{0}{\overset{\infty }{\int }}f(y+t)h(t)dt,\hfill \\ {\varphi }_1(x_1,x_2)={\rho }_0\left(\underset{0}{\overset{\infty }{\int }}f(x_1+y)f(x_2+y)h_r(y)dy+\right.\hfill \\ \left.+\underset{0}{\overset{\infty }{\int }}\psi (x_1+t)f(x_2+t)h_r(t)dt+\underset{0}{\overset{\infty }{\int }}\psi (x_2+t)f(x_1+t)h_r(t)dt\right),\hfill \\ {\varphi }_2(x,z)={\varphi }_3(x,z)={\rho }_0\left(\underset{0}{\overset{\infty }{\int }}f(y)f(x+y){\nu }_r(y,z)dy+\right.\hfill \\ \left.+\underset{0}{\overset{\infty }{\int }}\psi (t)f(x+t){\nu }_r(t,z)dt+\underset{0}{\overset{\infty }{\int }}\psi (x+t)f(t){\nu }_r(t,z)dt\right),\hfill \\ {\varphi }_6(z)={\varphi }_7(z)={\rho }_0\left(\underset{0}{\overset{\infty }{\int }}dt\underset{0}{\overset{\infty }{\int }}f(y)f(t+y){\nu }_r(y,t+z)dy\right.+\hfill \\ \left.+\underset{0}{\overset{\infty }{\int }}dt\underset{0}{\overset{\infty }{\int }}\psi (y)f(t+y){\nu }_r(y,t+z)dy+\underset{0}{\overset{\infty }{\int }}dt\underset{0}{\overset{\infty }{\int }}\psi (t+y)f(y){\nu }_r(y,t+z)dy\right),\hfill \end{array}$

where $h_r(t)=\sum _{n=1}^{\infty }{r}^{*(n)}(t)$ is the density of renewal function H_r(t) of the renewal process generated by RV δ; ${\nu }_r(z,x)=r(z+x)+\underset{0}{\overset{z}{\int }}r(z+x-s)h_r(s)ds$ is the distribution density of the direct residual time for the renewal process ...