  We study numerically the maximum $z$-matching problems on ensembles of
bipartite random graphs. The $z$-matching problems describes the matching
between two types of nodes, users and servers, where each server may serve up
to $z$ users at the same time. By using a mapping to standard
maximum-cardinality matching, and because for the latter there exists a
polynomial-time exact algorithm, we can study large system sizes of up to
$10^6$ nodes. We measure the capacity and the energy of the resulting optimum
matchings. First, we confirm previous analytical results for bipartite regular
graphs. Next, we study the finite-size behaviour of the matching capacity and
find the same scaling behaviour as before for standard matching, which
indicates the universality of the problem. Finally, we investigate for
bipartite Erd\H{o}s-R\'enyi random graphs the saturability as a function of the
average degree, i.e., whether the network allows as many customers as possible
to be served, i.e. exploiting the servers in an optimal way. We find phase
transitions between unsaturable and saturable phases. These coincide with a
strong change of the running time of the exact matching algorithm, as well with
the point where a minimum-degree heuristic algorithm starts to fail.
