  Motivated by a wide range of assemble-to-order systems and systems of the
collaborative economy applications, we introduce a stochastic matching model on
hypergraphs and multigraphs, extending the model introduced by Mairesse and
Moyal 2016. In this thesis, the stochastic matching model
$(\mathbb{S},\Phi,\mu)$ on general graph structures are defined as follows:
given a compatibility general graph structure
$\mathbb{S}=(\mathcal{V},\mathcal{S})$ which of a set of nodes denoted by
$\mathcal{V}$ that represent the classes of items and by a set of edges denoted
by $\mathcal{S}$ that allows matching between different classes of items. Items
arrive at the system at a random time, by a sequence (assumed to be $i.i.d.$)
that consists of different classes of $\mathcal{V}$, and request to be matched
due to their compatibility according to $\mathbb{S}$. The compatibility by
groups of two or more (hypergraphical cases) and by groups of two with
possibilities of matching between the items of the same classes (multigraphical
cases). The unmatched items are stored in the system and wait for a future
compatible item and as soon as they are matched they leave it together. Upon
arrival, an item may find several possible matches, the items that leave the
system depend on a matching policy $\Phi$ to be specified. We study the
stability of the stochastic matching model on hypergraphs, for different
hypergraphical topologies. Then, the stability of the stochastic matching model
on multigraphs using the maximal subgraph and minimal blow-up to distinguish
the zone of stability.
