 Solution

Treat each arriving caterpillar as one customer in an M-G-infinity queue, with arrival rate $\lambda$ and service time equal to the sum of its three independent exponential stage lifetimes. This is an Erlang distribution of shape three and rate $\mu$, of mean $3/\mu$. Poisson arrivals independently retained according to whether their service has ended give, in stationarity, a Poisson number in service with meanThis is exactly the marginal law of $N$ found above. Each individual remains in service across both metamorphoses; counting a fresh service arrival at each stage would be incorrect.