We study the performance of general dynamic matching models. This model is defined by a connected graph, where nodes represent the class of items and the edges the compatibilities between items. Items of different classes arrive one by one to the system according to a given probability distribution. Upon arrival, an item is matched with a compatible item according to the First Come First Served discipline and leave the system immediately, whereas it is enqueued with other items of the same class, if any. We show that such a model may exhibit a non intuitive behavior: increasing the services ability by adding new edges in the matching graph may lead to a larger average population. This is similar to a Braess paradox. We first consider a quasicomplete graph with four nodes and we provide values of the probability distribution of the arrivals such that when we add an edge the mean number of items is larger. Then, we consider an arbitrary matching graph and we show sufficient conditions for the existence or non-existence of this paradox. We conclude that the analog to the Braess paradox in matching models is given when specific independent sets are in saturation, i.e., the system is close to the stability condition.

A dynamic bipartite matching model is given by a bipartite matching graph which determines the possible matchings between the various types of supply and demand items. Both supply and demand items arrive to the system according to a stochastic process. Matched pairs leave the system and the others wait in the queues, which induces a holding cost. We model this problem as a Markov Decision Process and study the discounted cost and the average cost case. We first consider a model with two types of supply and two types of demand items with an N matching graph. For linear cost function, we prove that an optimal matching policy gives priority to the end edges of the matching graph and is of threshold type for the diagonal edge. In addition, for the average cost problem, we compute the optimal threshold value. According to our preliminary numerical experiments, threshold-type policies performs also very well for more general bipartite graphs.

We study the performance of stochastic matching models with general compatibility graphs. Items of different classes arrive to the system according to independent Poisson processes. Upon arrival, an item is matched with a compatible item according to the First Come First Matched discipline and both items leave the system immediately. If there are no compatible items, the new arrival joins the queue of unmatched items of the same class. Compatibilities between item classes are defined by a connected graph, where nodes represent the classes of items and the edges the compatibilities between item classes. We show that such a model may exhibit a non intuitive behavior: increasing the matching flexibility by adding new edges in the matching graph may lead to a larger average population at the steady state. This performance paradox can be viewed as an analog of the Braess paradox. We show sufficient conditions for the existence or non-existence of this paradox. This performance paradox in matching models appears when specific independent sets are in saturation, i.e., the system is close to the stability condition.

We extend the general matching graph model to deal with matching graph where every node has a self loop. Thus the states on the Markov chain are associated with the independent sets of the matching graph. We prove that under i.i.d. arrivals assumptions the steady-state distribution of the Markov chain has a product form solution.