  An occupancy problem with an infinite number of bins and a random probability
vector for the locations of the balls is considered. The respective sizes of
bins are related to the split times of a Yule process. The asymptotic behavior
of the landscape of first empty bins, i.e., the set of corresponding indices
represented by point processes, is analyzed and convergences in distribution to
mixed Poisson processes are established. Additionally, the influence of the
random environment, the random probability vector, is analyzed. It is
represented by two main components: an i.i.d. sequence and a fixed random
variable. Each of these components has a specific impact on the qualitative
behavior of the stochastic model. It is shown in particular that for some
values of the parameters, some rare events, which are identified, play an
important role on average values of the number of empty bins in some regions.
