This article studies a continuous-time generalization
of the so-called secretary problem.
A man seeks an apartment. Opportunities to inspect apartments
arise according to
a homogeneous Poisson process of unknown intensity $\lambda$
having a Gamma prior density, $G(r,1/a)$,
where $r$ is natural number.
At any epoch he is able to rank a given apartment
amongst all those inspected to date, where all permutations
of ranks are equally likely and independent of the Poisson
process.The objective to maximize the probability of selecting
best apartment from those (if any) available in the interval
$[0,T]$, where $T$ is given. This problem is reformulated as
the optimal stopping problem and it is shown to be the monotone
case. The optimal strategy for the problem is
solved to be a threshold rule.