I'm modeling a stochastic process in continuous time where objects go through $n$ different states.
At any point in time $t$, there is one object in the system. If the object is in state $i \in \{1,\ldots,n\}$ it transitions to state $i+1$ via a Poisson process with intensity $\lambda_i$. If the object is in state $n$, then the object exits the system and a new object arrives to state $1$ via a Poisson process with intensity $\lambda_0$.
I want to compute the expected number of objects that have exited the system (or equivalently, reached the final state $n$) at time $T$.
Because the object moves from state $i$ to state $i+1$ with Poisson intensity $\lambda_i$, I know that the expected time it takes for an object to move from state $i$ to state $i+1$ is $\tau_i = \frac{1}{\lambda_i}$. Since expectations are additive, the expected time that it would take for an object to arrive to state 1, and then transition through all the states up to state $n$ would be $$\tau = \tau_0 + \cdots + \tau_{n-1} = \frac{1}{\lambda_0} + \cdots + \frac{1}{\lambda_{n-1}}.$$
I would like to say that the expected number of objects that exit the system in one unit of time is $\frac{1}{\tau}$. This would be true, for example, if objects exited the system as soon as they arrived (the distribution of the number of events would be Poisson), but I'm not sure this holds when one has Poisson processes linked in series as I described above.