Stochastic matching model on the general graphical structures

TL;DR

This paper introduces a stochastic matching model on hypergraphs and multigraphs, extending previous models to more complex structures, and analyzes the stability conditions for these systems in various topologies.

Contribution

It extends the stochastic matching model to general hypergraph and multigraph structures, providing stability analysis for these complex topologies.

Findings

Stability conditions are derived for hypergraph-based matching models.

Stability zones are characterized for multigraph models using maximal subgraph and minimal blow-up techniques.

The model generalizes previous work by incorporating hypergraph and multigraph compatibility structures.

Abstract

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.