Math Problems Vault

Problem 6.11.1

Describe the jump chain for a birth-death process with rates $\lambda_{n}$ and $\mu_{n}$ .

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Problem 6.11.2

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

\pi _{n} = \frac{1}{2(n!)}\left(1+\frac{n}{\rho }\right) \rho ^{n} e^{-\rho }

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

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Problem 6.11.3

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

\eta ^{\prime }(t)+(\lambda +\mu ) \eta (t) = \mu +\lambda \eta (t)^{2}

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

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Problem 6.11.4

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.

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Problem 6.11.5

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

\mathbb {E}\left[T \mid T < \infty \right] = \begin{cases} \frac{1}{\lambda } \log \left(\frac{\mu }{\mu -\lambda }\right) & \text{ if } \lambda < \mu \\ \frac{1}{\mu } \log \left(\frac{\lambda }{\lambda -\mu }\right) & \text{ if } \lambda > \mu \end{cases}

What happens when $\lambda = \mu$ ?

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Problem 6.11.6

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 $\mathbb {E}\left[V_{r}(\infty )\right] = \lambda^{r-1} r^{-1}[\max \left\{ \lambda , \mu \right\} ]^{-r}$ .

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Problem 6.11.7

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

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Problem 6.11.8

(a)(b)

Consider a birth-death process $X$ with birth rates $\lambda_{n} > 0$ for $n \geq 0$ , death rates $\mu_{n} > 0$ for $n > 0$ , and $\mu_{0} = 0$ . Let $X(0) = n > 0$ , and let $D_{n}$ be the time until the process first takes the value $n-1$ .

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(a)

Show that $d_{n} = \mathbb {E}\left[D_{n}\right]$ satisfies

\lambda _{n} d_{n+1} = \mu _{n} d_{n}-1, \quad n \geq 1

(b)

Show that the moment generating function $M_{n}(\theta ) = \mathbb {E}\left[e^{\theta D_{n}}\right]$ satisfies

\left(\lambda _{n}+\mu _{n}-\theta \right) M_{n}(\theta ) = \mu _{n}+\lambda _{n} M_{n}(\theta ) M_{n+1}(\theta ), \quad n \geq 1

Problem 6.11.9

In a model for a biofilm population, we assume there are $n$ colonizable 'niches' (or 'food sources'). Let $X(t)$ be the number of occupied niches at time $t$ , and assume $X$ is a Markov chain that evolves as follows. The lifetime of any colony is exponentially distributed with parameter $\mu$ ; if $X(t) = i$ , the rate of establishment of a new colony in an empty niche is $\lambda i(n-i)$ . The usual independence may be assumed.

For $X(0) = 1,2, \ldots$ , find the mean time until the population is extinct, which is to say that no niches are occupied. Discuss the implications when $n$ is large.

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