Average number of organisms at any point in time given birth and death rates.

Question

There is a population of organisms that get birthed at rate $\lambda$ and once they are born, they die at rate $\mu$. Once steady state is achieved, what will the average number of organisms at some random point in time ($a$) be? My intuition is $a=\frac{\lambda} {\mu}$. Will the answer depend on the actual point process defining the births and the distribution of time until death?

In general, steady state might never be achieved, so we have to think of $a$ as a function of time, $a(t)$. Is it possible to derive an expression for this?

EDIT: why is this relevant to me? I'm building a simulation of organisms that live and die by the rule stated. Need to figure out how many there will be so I can buy the appropriate amount of food for them.

Why is it relevant to the community? It's a math question that maps clearly to a well known problem in queueing theory as the accepted answer shows and this is a Math community.

Answer 1

The meaning of "population of organisms that get birthed at rate $\lambda$" is somewhat unclear. I assume you mean that, independent of the current population, a new organism is born at rate $\lambda$. If so, the state of this system is described using a Continuous Time Markov Chain (CTMC).

The state $N(t)$ represents the number of organisms alive at time $t$. If $n$ organisms are alive, then we transition next to

• $n+1$ organisms at rate $\lambda$ (the "birth" rate),
• $n-1$ organisms at rate $n \mu$ (the "death" rate is additive since each organism dies independently).

This is a simple M/M/$\infty$ queue, which has a steady state distribution, and the exact distribution for the time-varying probabilities are shown on the referenced Wiki page.