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Question:

Consider a continuous time Markov birth-death process describing a population $n = 0,1,2,\dots$, where the birth rate in state $n$ is $\lambda$ and the death rate in state $n$ is $\sigma n$. The master equation for the probabilities $P(n,t)$ to be in the $n$th state at the time $t$ is $$\frac{d}{dt} P(n,t) = -\lambda P(n-1,t)-\sigma(n+1) P(n+1,t). $$ In steady state, the solution is the Poisson distribution: $$P(n) = \frac{(\lambda/\sigma)^n}{n!}e^{-\lambda/\sigma}.$$

I would like to know the mean time between death events.

Attempt:

One can write $$ F(n,t+\delta t) = F(n,t)(1-\sigma n \delta t)$$ for the probability that the population is $n$ and there hasn't been a death between $t$ and $t+\delta t$. Accordingly, as $\delta t \rightarrow 0$ one obtains a differential equation for an exponential distribution: $$ \frac{d}{dt}F(n,t) = -\sigma n F(n,t)$$ This has solution $$F(n,t) = \sigma n \exp(-\sigma n t).$$ The probability that there is no death in time $t$ in the $n$th state is an exponential distribution with mean time $t_n = (\sigma n)^{-1 }$ ,$n=1,2,3,\dots$.

I conclude that each state has its own mean time between deaths, and this time is inversely proportional to the number of individuals. The mean rate of deaths increases with more individuals. This makes sense to me.

But I'm interested in the overall mean time between deaths, averaged across all populations, so I thought to do this $$ \tau = \sum_{n=1}^\infty t_n P(n) = \frac{1}{\sigma}e^{-\lambda/\sigma} \sum_{n=1}^\infty \frac{(\lambda/\sigma)^n}{n n!}$$

I just don't know if this is right. I'm seeking the time between death transitions in this simple birth-death model

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