Birth Process with Linear Birth Rate

Example

Consider a population whose members can give birth to new members but cannot die. Each member acts independently and takes an exponentially distributed amount of time with mean $1/\lambda$ to give birth.

Let $X(t)$ be the population size at time $t$. Then $\{X(t),t\ge 0\}$ is a pure birth process with:

${\lambda }_n=n\lambda ,n\ge 0$

This follows because if the population has $n$ persons and each gives birth at exponential rate $\lambda$, the total birth rate is $n\lambda$ (sum of $n$ independent exponentials).

This is known as a Yule Process.

Example Calculation

Given a pure birth process with $X(0)=0$ and birth parameters: ${\lambda }_0=1$, ${\lambda }_1=3$, ${\lambda }_2=2$, ${\lambda }_3=5$. Let $W_3$ be the time to reach state 3 and $T_i$ the sojourn time of the $i$-th birth.

(a) $\mathbb{E}[W_3]=\mathbb{E}[T_1+T_2+T_3]=\frac{1}{{\lambda }_1}+\frac{1}{{\lambda }_2}+\frac{1}{{\lambda }_3}=1+\frac{1}{3}+\frac{1}{2}=\frac{11}{6}$

(b) $\mathbb{E}[W_1+W_2+W_3]=2\mathbb{E}[T_1]+\mathbb{E}[T_2]+\mathbb{E}[W_3]=\frac{2}{1}+\frac{1}{3}+\frac{11}{6}=\frac{25}{6}$

(c) $\mathrm{Var}(W_3)=\mathrm{Var}(T_1+T_2+T_3)=\frac{1}{{\lambda }_1^2}+\frac{1}{{\lambda }_2^2}+\frac{1}{{\lambda }_3^2}=1+\frac{1}{9}+\frac{1}{4}=\frac{49}{36}$

Related

• Pure Birth Process
• Yule Process