[Paper Review] Loynes construction for the extended bipartite matching

Pascal Moyal, Ana Bušić+1 more|arXiv (Cornell University)|Mar 7, 2018

Advanced Queuing Theory Analysis11 references4 citations

TL;DR

This paper introduces a Loynes-type backward coupling construction for Extended Bipartite Matching (EBM) models, establishing the existence and uniqueness of a bi-infinite perfect matching under general stationary ergodic assumptions. By leveraging sub-additivity of a stochastic recursive representation, the method proves stability and stationary distribution existence for a broad class of matching policies and bipartite structures.

ABSTRACT

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.

Motivation & Objective

• To extend the Loynes construction—previously used for FCFS stochastic bipartite matching—to the more general Extended Bipartite Matching (EBM) model with arbitrary arrival distributions not necessarily product-form.
• To establish the existence and uniqueness of a bi-infinite perfect matching under general stationary ergodic assumptions, subsuming classical Markovian settings.
• To derive stability conditions for EBM systems by exploiting the sub-additivity of a stochastic recursive representation under common matching policies.
• To generalize coupling-from-the-past techniques to EBM models, enabling pathwise construction of the stationary state for various matching policies.
• To unify and extend previous results on product-form stationary measures and coupling convergence in matching systems under broader conditions.

Proposed method

• Adapts the Loynes backwards scheme to EBM models, constructing the stationary state by evolving backwards in time from a bi-infinite sequence of arrivals.
• Employs a stochastic recursive representation of the system state, where the evolution depends on the arrival process and matching policy.
• Establishes sub-additivity of the recursive state update function under most standard matching policies, enabling the use of coupling arguments.
• Uses the sub-additivity property to prove convergence of coupled processes and existence of a unique bi-infinite matching path.
• Applies the coupling-from-the-past framework to construct the stationary distribution pathwise, ensuring perfect simulation of the steady state.
• Analyzes multiple matching policy cases (e.g., Match the Longest, FCFS, Random) and proves stability under general ergodic assumptions via recursive sub-additivity.

Experimental results

Research questions

• RQ1 Can the Loynes construction be generalized from FCFS bipartite matching to the broader class of Extended Bipartite Matching (EBM) models with non-product-form arrival distributions?
• RQ2 Under what conditions does a bi-infinite perfect matching exist and remain unique in EBM models under general stationary ergodic arrival processes?
• RQ3 How can sub-additivity of the state recursion be established for various matching policies in EBM models to enable coupling-based existence proofs?
• RQ4 What are the stability conditions for EBM systems under general matching policies, and how do they generalize classical Markovian stability criteria?
• RQ5 Can the stationary distribution of EBM models be constructed explicitly via coupling-from-the-past techniques, and what policies allow such construction?

Key findings

• The Loynes construction is successfully extended to EBM models, proving the existence and uniqueness of a bi-infinite perfect matching under general stationary ergodic assumptions.
• Sub-additivity of the stochastic recursive state update is established for most standard matching policies, enabling the use of coupling arguments to prove convergence.
• Stability conditions for EBM systems are derived under general stationary ergodic arrival processes, subsuming classical Markovian settings and generalizing known results.
• The stationary distribution of the EBM model can be constructed pathwise via coupling-from-the-past, allowing perfect simulation of the steady state for a wide class of policies.
• The Match the Longest policy is shown to be stable under the same condition that guarantees existence of a perfect matching in the fluid limit, confirming its maximal stability region.
• The construction applies uniformly across policies including FCFS, LCFS, Match the Longest, and Uniform, demonstrating robustness of the method.