## Section: New Results

### Dynamic matching models

The model of First Come First Served infinite bipartite matching was introduced in Caldentey, Kaplan and Weiss, 2009. In this model, there is a sequence of items that are chosen i.i.d. from a finite set 𝒞 and an independent sequence of items that are chosen i.i.d. from a finite set 𝒮, and a bipartite compatibility graph G between 𝒞 and 𝒮. Items of the two sequences are matched according to the compatibility graph, and the matching is FCFS, meaning that each item in the one sequence is matched to the earliest compatible unmatched item in the other sequence. In Adan and Weiss, 2012, a Markov chain associated with the matching was analyzed, a condition for stability was derived, and a product form stationary distribution was obtained. In [2], we present several new results that unveil the fundamental structure of the model. First, we provide a pathwise Loynes’ type construction which enables to prove the existence of a unique matching for the model defined over all the integers. Second, we prove that the model is dynamically reversible: we define an exchange transformation in which we interchange the positions of each matched pair, and show that the items in the resulting permuted sequences are again independent and i.i.d., and the matching between them is FCFS in reversed time. Third, we obtain product form stationary distributions of several new Markov chains associated with the model. As a by-product, we compute useful performance measures, for instance the link lengths between matched items.

In [51], we propose an explicit construction of the stationary state of Extended Bipartite Matching (EBM) models, as defined in (Busic et. al., 2013). We use a Loynes-type backwards scheme similar in flavor to that in (Moyal et al., 2017), allowing to show the existence and uniqueness of a bi-infinite perfect matching under various conditions, for a large class of matching policies and of bipartite matching structures. The key algebraic element of our construction is the sub-additivity of a suitable stochastic recursive representation of the model, satisfied under most usual matching policies. By doing so, we also derive stability conditions for the system under general stationary ergodic assumptions, subsuming the classical markovian settings.

In [42], we consider holding costs for the items that are waiting to be matched. We model this problem as an MDP (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.