State the product theorem for Poisson random measures.

Consider a system of $n$ queues, each with infinitely many servers, in which, for $i=1, \ldots, n-1$, customers leaving the $i$ th queue immediately arrive at the $(i+1)$ th queue. Arrivals to the first queue form a Poisson process of rate $\lambda$. Service times at the $i$ th queue are all independent with distribution $F$, and independent of service times at other queues, for all $i$. Assume that initially the system is empty and write $V_{i}(t)$ for the number of customers at queue $i$ at time $t \geqslant 0$. Show that $V_{1}(t), \ldots, V_{n}(t)$ are independent Poisson random variables.

In the case $F(t)=1-e^{-\mu t}$ show that

$$\mathbb{E}\left(V_{i}(t)\right)=\frac{\lambda}{\mu} \mathbb{P}\left(N_{t} \geqslant i\right), \quad t \geqslant 0, \quad i=1, \ldots, n,$$

where $\left(N_{t}\right)_{t \geqslant 0}$ is a Poisson process of rate $\mu$.

Suppose now that arrivals to the first queue stop at time $T$. Determine the mean number of customers at the $i$ th queue at each time $t \geqslant T$.