  Balls-in-bins models describe a random sequential allocation of infinitely
many balls into a finite number of bins. In these models a ball is placed into
a bin with probability proportional to a given function (feedback function),
which depends on the number of existing balls in the bin. Typically, the
feedback function is the same for all bins (symmetric feedback), and there are
no constraints on the number of balls in the bins. In this paper we study
versions of BB models with two bins, in which the above assumptions are
violated. In the first model of interest the feedback functions can depend on a
bin (BB model with asymmetric feedback). In the case when both feedback
functions are power law and superlinear, a single bin receives all but finitely
many balls almost surely, and we study the probability that this happens for a
given bin. In particular, under certain initial conditions we derive the normal
approximation for this probability, which generalizes the result in [5]
obtained in the case of the symmetric feedback. The main part of the paper
concerns the BB model with asymmetric feedback evolving subject to certain
constraints on the numbers of allocated balls. The model can be interpreted as
a transient reflecting random walk in a curvilinear wedge, and we obtain a
complete classification of its long term behavior.
