ODE compartmental model: waiting time Here is a ODE compartmental model made of 3 equations :
$\frac{dX}{dt}=-\alpha X$
$\frac{dY}{dt}=\alpha X-\beta Y$
$\frac{dZ}{dt}=\beta Y$
$X$, $Y$, $Z$ represents, in my case, the total number of people being in the state/compartment/case $X$, $Y$ or $Z$ (it could be a SIR model) and we assume that each individual can move from $X$ to $Y$ and from $Y$ to $Z$ with the respective rates $\alpha$ and $\beta$.
Let's $w_{XY}$ be the average waiting time for an individual being in $X$ before moving to $Y$. This rate is often calculated/estimated as $w_{XY}=1/\alpha$.
First question: Is it an approximation or is the exact result? How is it obtained? Should we make more assumptions to find it?
Second question: Let's assume now that they are more exits that leave $X$. Will the waiting time $w_{XY}$ still be $w_{XY}=1/\alpha$? Again, how is it obtained?
 A: Here's the kicker: we're operating under the assumption that the population of compartment $X$ is exponentially distributed, and that its rate of change (probability of change per unit time) is $\alpha.$ Its probability density function is $$f(t)=\begin{cases}\alpha e^{-\alpha t}&t\ge0,\\0&t<0.\end{cases}$$ This means that the probability of a member of that compartment leaving prior to time $T$ will be $$\int_{-\infty}^Tf(t)\,dt$$ in general. For positive $T,$ this will be $$\int_0^T\alpha e^{-\alpha t}\,dt=\left[-e^{-\alpha t}\right]_{t=0}^T=1-e^{-\alpha T},$$ but it will otherwise be $0.$
To find the average waiting time, we need the expected value: $$\int_{-\infty}^\infty tf(t)\,dt=\int_0^\infty\alpha t e^{-\alpha t}\,dt=\left[-\frac1\alpha(\alpha t+1)e^{-\alpha t}\right]_{t=0}^\infty=\frac1\alpha.$$

Now, if we're talking about a "population" of a compartment, we're probably dealing with a discrete case, in actuality, so this should be considered an approximation. If there are more exits, I'm not sure $w_{XY}$ even makes sense.
