Stability in stochastic hypergraph matching II: weighted matching, routing, and batch arrivals

Many real-life systems can be found as examples of stochastic matching on hypergraphs, such as production lines or assemble-to-order systems. Two common features are the number of items required may vary between matchings, and there may intermediary items which exist as a combination of other items and not of external arrivals. Both of these phenomena can be modelled by considering the weighted variant of stochastic matching.
In this work, we formalise the notion of stochastic weighted matching on hypergraphs, as well as allow batch arrivals, meaning multiple items of a given class may arrive at the same time. We then extend the results of Nguyen and Bušić (2006) to overcome the intricacies brought up by this new setting. Despite many differences, we derive necessary and sufficient criteria as direct generalisations of those in the unweighted setting. The constructive proofs also give a maximally stable, size-based, arrival-rate agnostic policy.