A stochastic birth-death process is given by the master equation

$$\frac{d p_{n}}{d t}=\lambda\left(p_{n-1}-p_{n}\right)+\mu\left[(n-1) p_{n-1}-n p_{n}\right]+\beta\left[(n+1) p_{n+1}-n p_{n}\right]$$

where $p_{n}(t)$ is the probability that there are $n$ individuals in the population at time $t$ for $n=0,1,2, \ldots$ and $p_{n}=0$ for $n<0$. Give a brief interpretation of $\lambda, \mu$ and $\beta$.

Derive an equation for $\frac{\partial \phi}{\partial t}$, where $\phi$ is the generating function

$$\phi(s, t)=\sum_{n=0}^{\infty} s^{n} p_{n}(t)$$

Now assume that $\beta>\mu$. Show that at steady state

$$\phi=\left(\frac{\beta-\mu}{\beta-\mu s}\right)^{\lambda / \mu}$$

and find the corresponding mean and variance.