The usual infinitesimal analysis leading to a differential system applies, with a twist: for small $s>0$, $$ p_{i0}(k,t+s)=(1-\lambda s)p_{i0}(k,t)+\mu sp_{i1}(k-1,t), $$ and $$ p_{i1}(k,t+s)=(1-\mu s)p_{i1}(k,t)+\lambda sp_{i0}(k,t), $$ with the convention that $p_{ij}(-1,t)=0$ for every $t\geqslant0$. Hence, $$ \partial_tp_{i0}(k,t)=-\lambda p_{i0}(k,t)+\mu p_{i1}(k-1,t), $$ and $$ \partial_tp_{i1}(k,t)=-\mu p_{i1}(k,t)+\lambda p_{i0}(k,t). $$ With the initial condition $p_{ii}(0,0)=1$ for every $i$, $p_{ij}(0,0)=0$ for every $i\ne j$, and $p_{ij}(k,0)=0$ for every $(i,j)$ and every $k\geqslant1$, this determines uniquely $p_{ij}(k,t)$ for every $(i,j)$, $k\geqslant0$ and $t\geqslant0$.

The usual infinitesimal analysis leading to a differential system applies, with a twist: for small $s>0$, $$ p_{i0}(k,t+s)=(1-\lambda s)p_{i0}(k,t)+\mu sp_{i1}(k-1,t)+o(s), $$ and $$ p_{i1}(k,t+s)=(1-\mu s)p_{i1}(k,t)+\lambda sp_{i0}(k,t)+o(s), $$ with the convention that $p_{ij}(-1,t)=0$ for every $t\geqslant0$. Hence, $$ \partial_tp_{i0}(k,t)=-\lambda p_{i0}(k,t)+\mu p_{i1}(k-1,t), $$ and $$ \partial_tp_{i1}(k,t)=-\mu p_{i1}(k,t)+\lambda p_{i0}(k,t). $$ With the initial condition $p_{ii}(0,0)=1$ for every $i$, $p_{ij}(0,0)=0$ for every $i\ne j$, and $p_{ij}(k,0)=0$ for every $(i,j)$ and every $k\geqslant1$, this determines uniquely $p_{ij}(k,t)$ for every $(i,j)$, $k\geqslant0$ and $t\geqslant0$.