Let $X$ be the stochastic process corresponding the the number of cells.
$X \rightarrow X+1$ at rate $I$ due to immigration
$X \rightarrow X+1$ at rate $X B$ due to birth (X cells, each giving birth are rate B)
$X \rightarrow X-1$ at rate $XD$ due to death (X cells, each dying at rate D)
I need to derive the following PGF: $$F(z)=(\frac{B-D}{Bz-D})^{I/B}$$
Let $\pi_i$ be the long-time proportion of having $i$ cells. Then let $F(z)=\sum_{i=0} z^i \pi_i$
I build the following relationship using the detailed balance equations:
$$(I+iB)\pi_i=(i+1)D \pi_{i+1}$$
Then multiply both sides by $z^i$ and sum over $i$ to get:
\begin{align*} & I \sum \pi_i z^i+ \sum iB\pi_i z^i = \sum (i+1)D \pi_{i+1} z^i\\ & I F(z) + B z \dot F(z) = D F(z)\\ & \dot F(z) = (D-I)F(z)/Bz\\ & df/dz=(D-I)F(z)/Bz\\ & \int df/F(z)=\int (D-I)dz/Bz\\ & \log F(z) = (D-I)/B log z\\ & F =\exp{(D-I)/B} z \\ \end{align*}
Subject to inital conditions $F(0)=0$ and $F(1)=1$