Expected Time to Extinction

Definition

The expected time to extinction ${\tau }_m=\mathbb{E}[T_{0,m}]$ is the expected time for a birth and death process to reach state $0$ starting from state $m$.

Theorem 6.3 (Allen). Suppose ${\lambda }_0={\mu }_0=0$ and ${\lambda }_i,{\mu }_i>0$ for $i\ge 1$, with ${\lim }_{t\to \infty }{p}_0(t)=1$ (certain extinction). Then:

${\tau }_m=\{\begin{array}{ll}\frac{1}{{\mu }_1}+\sum _{i=2}^{\infty }\frac{{\lambda }_1\cdots {\lambda }_{i-1}}{{\mu }_1\cdots {\mu }_i},& m=1\\ {\tau }_1+\sum _{s=1}^{m-1}\left[\frac{{\mu }_1\cdots {\mu }_s}{{\lambda }_1\cdots {\lambda }_s}\sum _{i=s+1}^{\infty }\frac{{\lambda }_1\cdots {\lambda }_{i-1}}{{\mu }_1\cdots {\mu }_i}\right],& m=2,3,\dots \end{array}$

Interpretation

The expected extinction time decomposes as: time to go from 1 to 0, plus the time to regress from each higher state. The double sum accounts for the possibility of moving up before moving down.

For Finite State Space $\{0,1,\dots ,N\}$

Replace $\infty$ with $N$ in the formulas above, or solve the linear system:

$D\mathbf{\tau }=\mathbf{d}$

where $D$ is the $N\times N$ tridiagonal matrix of birth-death rates and $\mathbf{d}=(-1,-1,\dots ,-1{)}^T$.

Recursive Formula for First Passage Times

For moving upward ($i\to i+1$):

$\mathbb{E}[T_{i+1,i}]=\frac{1}{{\lambda }_i}+\frac{{\mu }_i}{{\lambda }_i}\mathbb{E}[T_{i,i-1}]$

For moving downward ($i\to i-1$):

$\mathbb{E}[T_{i-1,i}]=\frac{1}{{\mu }_i}+\frac{{\lambda }_i}{{\mu }_i}\mathbb{E}[T_{i,i+1}]$

Example: Simple Birth and Death Process

For ${\lambda }_i=\lambda i$, ${\mu }_i=\mu i$ with $\lambda <\mu$ (certain extinction):

${\tau }_1=-\frac{1}{\lambda }\ln \left(1-\frac{\lambda }{\mu }\right)$

Related

• Probability of Population Extinction
• Birth and Death Process
• Stochastic Logistic Growth Process