On a hot summer night, opening my window brings some relief. This attracts hordes of mosquitoes who manage to negotiate a dense window net. But, luckily, I have a mosquito trapping device in my room.

Assume the mosquitoes arrive in a Poisson process at rate $\lambda$; afterwards they wander around for independent and identically distributed random times with a finite mean $\mathbb{E} S$, where $S$ denotes the random wandering time of a mosquito, and finally are trapped by the device.

(a) Identify a mathematical model, which was introduced in the course, for the number of mosquitoes present in the room at times $t \geqslant 0$.

(b) Calculate the distribution of $Q(t)$ in terms of $\lambda$ and the tail probabilities $\mathbb{P}(S>x)$ of the wandering time $S$, where $Q(t)$ is the number of mosquitoes in the room at time $t>0$ (assuming that at the initial time, $Q(0)=0$ ).

(c) Write down the distribution for $Q^{\mathrm{E}}$, the number of mosquitoes in the room in equilibrium, in terms of $\lambda$ and $\mathbb{E} S$.

(d) Instead of waiting for the number of mosquitoes to reach equilibrium, I close the window at time $t>0$. For $v \geqslant 0$ let $X(t+v)$ be the number of mosquitoes left at time $t+v$, i.e. $v$ time units after closing the window. Calculate the distribution of $X(t+v)$.

(e) Let $V(t)$ be the time needed to trap all mosquitoes in the room after closing the window at time $t>0$. By considering the event $\{X(t+v) \geqslant 1\}$, or otherwise, compute $\mathbb{P}[V(t)>v]$.

(f) Now suppose that the time $t$ at which I shut the window is very large, so that I can assume that the number of mosquitoes in the room has the distribution of $Q^{E}$. Let $V^{E}$ be the further time needed to trap all mosquitoes in the room. Show that

$$\mathbb{P}\left[V^{E}>v\right]=1-\exp \left(-\lambda \mathbb{E}\left[(S-v)_{+}\right]\right),$$

where $x_{+} \equiv \max (x, 0)$.