Birth-death processes: transient laws, extinction and generating functions

Definition 17. A homogeneous Markov process $X = \left\{ X(t), t \geqslant 0\right\}$ with state space $S = \left\{ 0,1, \ldots \right\}$ is called a birth-and-death process if its infinitesimal matrix $Q$ has the following form

where $\mu_{0} = 0, \mu_{i} > 0, i \geqslant 1$ and $\lambda_{i} > 0, i \geqslant 0$. If $\mu_{i} = 0$ for all $i \geqslant 0$, one speaks of a birth process, and if $\lambda_{i} = 0, i \geqslant 0$, of a death process.

The backward and forward systems of Kolmogorov equations for the matrices $P(t) = \left(p_{i j}(t)\right)_{i, j \in \mathrm{S}}$, $t \geqslant 0$, have, respectively, the form

and

Solutions are sought in the class of semistochastic matrices $(P(t))_{t \geqslant 0}$, for which $\sum_{j} p_{i j}(t) \leqslant 1, t \geqslant 0, i \in \mathrm{S}$. If it turns out that $\sum_{j} p_{i j}(t) = 1$ for all $t \geqslant 0, i \in \mathrm{S}$, then one speaks of a proper solution.

Suppose a birth process is such that $\sum_{i} \lambda_{i}^{-1} < \infty$. Prove that then the system of forward Kolmogorov equations has a unique solution, while the system of backward equations has infinitely many. Prove that if $\sum_{i} \mu_{i}^{-1} < \infty$ for a death process, then the system of backward equations has a unique solution, while the system of forward equations has infinitely many, exactly one of which is proper.

Consider an immigration-death process $X$, being a birth-death process with birth rates $\lambda_{n} = \lambda$ and death rates $\mu_{n} = n \mu$. Find the transition matrix of the jump chain $Y$, and show that it has as stationary distribution

where $\rho = \lambda / \mu$. Explain why this differs from the stationary distribution of $X$.

Consider the birth-death process $X$ with $\lambda_{n} = n \lambda$ and $\mu_{n} = n \mu$ for all $n \geq 0$. Suppose $X(0) = 1$ and let $\eta (t) = \mathbb {P}_{1}\left(X(t) = 0\right)$. Show that $\eta$ satisfies the differential equation

Hence find $\eta (t)$, and calculate $\mathbb {P}_{1}\left( X(t) = 0 \mid X(u) = 0 \right)$ for $0 < t < u$.

For the birth-death process of the previous exercise with $\lambda < \mu$, show that the distribution of $X(t)$, conditional on the event $\left\{ X(t) > 0\right\}$, converges as $t \rightarrow \infty$ to a geometric distribution.

Let $X$ be a birth-death process with $\lambda_{n} = n \lambda$ and $\mu_{n} = n \mu$, and suppose $X(0) = 1$. Show that the time $T$ at which $X(t)$ first takes the value 0 satisfies

What happens when $\lambda = \mu$?

Let $X$ be the birth-death process of Exercise (6.11.5) with $\lambda \neq \mu$, and let $V_{r}(t)$ be the total amount of time the process has spent in state $r \geq 0$, up to time $t$. Find the distribution of $V_{1}(\infty )$ and the generating function $\sum_{r} s^{r} \mathbb {E}\left[V_{r}(t)\right]$. Hence show in two ways that $\mathbb {E}\left[V_{1}(\infty )\right] = [\max \left\{ \lambda , \mu \right\} ]^{-1}$. Show further that $\mathbb {E}\left[V_{r}(\infty )\right] = \lambda^{r-1} r^{-1}[\max \left\{ \lambda , \mu \right\} ]^{-r}$.

Repeat the calculations of Exercise (6.11.6) in the case $\lambda = \mu$.

Immigrants arrive at the instants of a Poisson process of rate $\nu$, and each independently founds a simple birth process of rate $\lambda$. At the instants of an independent Poisson process of rate $\delta$, the population is annihilated. Find the probability generating function of the population $X(t)$, given that $X(0) = 0$.

In the framework of Exercise (6.12.2), suppose that each immigrant gives rise to a simple birth-death process of rates $\lambda$ and $\mu$. Show that the mean population size stays bounded if and only if $\delta > \lambda -\mu$.

Consider an immigration-death process $X(t)$, being a birth-death process with rates $\lambda_{n} = \lambda$, $\mu_{n} = n \mu$. Show that its generating function $G(s, t) = \mathbb {E}\left[s^{X(t)}\right]$ is given by

where $\rho = \lambda / \mu$ and $X(0) = I$. Deduce the limiting distribution of $X(t)$ as $t \rightarrow \infty$.

Let $X$ be a simple symmetric birth-death process with $\lambda_{n} = \mu_{n} = n \lambda$, and let $T$ be the time until extinction. Show that

and deduce that extinction is certain if $\mathbb {P}\left(X(0) < \infty \right) = 1$. Show that $\mathbb {P}\left(\lambda T / I \leq x \mid X(0) = I\right) \rightarrow e^{-1 / x}$ as $I \rightarrow \infty$.

Let $X$ be an immigration-death-disaster process, that is, a birth-death process with parameters $\lambda_{i} = \lambda , \mu_{i} = i \mu$, and with the additional possibility of 'disasters' which reduce the population to 0. Disasters occur at the times of a Poisson process with intensity $\delta$, independently of all previous births and deaths.