Stability of the stochastic matching model

We introduce and study a new model that we call the {\em matching model}. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be {\em matched}. There is a finite set of classes $\maV$ for the items, and the allowed matchings depend on the classes, according to a {\em matching graph} on $\maV$. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a {\em matching policy}. When the sequence of classes of the arriving items is i.i.d., the sequence of buffer-contents is a Markov chain, whose stability is investigated. In particular, we prove that the model may be stable if and only if the matching graph is non-bipartite.