  In this paper, we introduce an over-time variant of the well-known
prophet-inequality with i.i.d. random variables. Instead of stopping with one
realized value at some point in the process, we decide for each step how long
we select the value. Then we cannot select another value until this period is
over. The goal is to maximize the expectation of the sum of selected values. We
describe the structure of the optimal stopping rule and give upper and lower
bounds on the prophet-inequality. - Which, in online algorithms terminology,
corresponds to bounds on the competitive ratio of an online algorithm.
  We give a surprisingly simple algorithm with a single threshold that results
in a prophet-inequality of $\approx 0.396$ for all input lengths $n$.
Additionally, as our main result, we present a more advanced algorithm
resulting in a prophet-inequality of $\approx 0.598$ when the number of steps
tends to infinity. We complement our results by an upper bound that shows that
the best possible prophet-inequality is at most $1/\varphi \approx 0.618$,
where $\varphi$ denotes the golden ratio. As part of the proof, we give an
advanced bound on the weighted mediant.
