I work in mail delivery and often have to ring at multiple tenants until someone opens the front door. With modern intercoms, only the last person I have rang up can open the door, the others get blocked.
I asked myself, if for any tenant the probability distribution of the time which passes between me ringing the bell and them trying to open the door is identical (and independent between different tenants), what would be the bell ringing strategy minimizing my expected waiting time?
To put in mathematical terms, let $(X_i)_i$ be an infinite (for reasons of simplicity I assume there is an infinite number of tenants) series of i.i.d. random variables $X_i:\Omega\to(0,\infty)$. For any strategy $s:{\bf N}\to(0,\infty)$ (which means I wait $s_0$ seconds after ringing at $X_0$, $s_1$ seconds after ringing at $X_1$ if $X_0$ doesn't open, etc.), let the waiting time be denoted by $w_s(\omega)=\sum_{i<k}s_i+X_k(\omega)$ where $k$ is the smallest index with $X_k(\omega)\le s_k$.
The question is then, which $s$ minimizes the expected waiting time $\int_{\omega} w_s(\omega)$? What would change, if we allow $\infty$ as value for the $X_i$ (i.e. some tenants went out)? Especially, I'd like to know if an optimal $s$ has to be constant.
First, let us assume that the optimal $s$ has to be a constant (we will treat it as a real value, rather than vector for now). Let us define the probability $p(s)$ to be $$p(s) = \mathbb{P}(X_i \le s).$$ Therefore, the distribution of the smallest index $K(\omega)$ is given by a standard geometric distribution with success parameter $p$. Hence, the expected wait time is $$\begin{align} \mathbb{E}(w_s) = \mathbb{E}\left((K-1)s + X_K\right) &= s\left(\mathbb{E}(K-1)\right) + \mathbb{E}X_K \\ &= \frac{s(1-p(s))}{p(s)} + \mathbb{E}(X_i\mid X_i\leq s).\end{align}$$ Therefore, you want to minimize this, w.r.t. $s$ (depending on your distribution of $X_i$ you may be able to get a nice form for such an $s$).
Next, to note that $s$ must be a constant vector, we can formulate this problem as a two-state MDP (states are whether or not we're done waiting), and utilize standard results to state that there must be a stationary, non-randomized optimal policy.