Generalized max-weight policies in stochastic matching

Jonckheere, Matthieu; Moyal, Pascal; Ramírez, Claudia; Soprano-Loto, Nahuel

doi:10.48550/arxiv.2011.04535

Cited by 1 publication

(3 citation statements)

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“…Our work is part of a broader research effort on the stochastic matching model that will be described in details in Section 2.1 [5,6,11,14,20,23,24]. Among these works, the following are particularly relevant because directly related to our results on stability.…”

Section: Non-bipartite or General Stochastic Matchingmentioning

confidence: 99%

“…Other variants of the model were studied recently, and an interesting future work would consist of generalizing our results to these variants. In particular, the paper [20] consider item abandonments, the paper [6] considers graphs with loops, and the papers [19,25,26] allow matches between several items by replacing the graph with a hypergraph.…”

Section: Non-bipartite or General Stochastic Matchingmentioning

confidence: 99%

“…On the other hand, if (4) is satisfied, then the random walk is unbiased, but the system is still unstable because the corresponding Markov chain is null recurrent. (Existing studies of matching in bipartite graphs usually solve this issue by coupling arrivals in both components [2, 8,12] or by assuming that items have a finite patience time [20].) If ( 4) is satisfied, the solutions of the conservation equation (2) can be described with a parameter α as shown in Figure 5.…”

Section: Bipartite Graph (That Is Neither Injective Nor Surjective)mentioning

confidence: 99%

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Stochastic dynamic matching: A mixed graph-theory and linear-algebra approach

Comte¹,

Mathieu²,

Bušíć³

2021

Preprint

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The stochastic dynamic matching problem has recently drawn attention in the stochastic-modeling community due to its numerous applications, ranging from supply-chain management to kidney exchange programs. In this paper, we consider a matching problem in which items of different classes arrive according to independent Poisson processes. Unmatched items are stored in a queue, and compatibility constraints are described by a simple graph on the classes, so that two items can be matched if their classes are neighbors in the graph. We analyze the efficiency of matching policies, not only in terms of system stability, but also in terms of matching rates between different classes.Our results rely on the observation that, under any stable policy, the matching rates satisfy a conservation equation that equates the arrival and departure rates of each item class. Our main contributions are threefold. We first introduce a mapping between the dimension of the solution set of this conservation equation, the structure of the compatibility graph, and the existence of a stable policy. In particular, this allows us to derive a necessary and sufficient stability condition that is verifiable in polynomial time. Secondly, we describe the convex polytope of non-negative solutions of the conservation equation. When this polytope is reduced to a single point, we give a closed-form expression of the solution; in general, we characterize the vertices of this polytope using again the graph structure. Lastly, we show that greedy policies cannot, in general, achieve every point in the polytope. In contrast, non-greedy policies can reach any point of the interior of this polytope, and we give a condition for these policies to also reach the boundary of the polytope. ACM Subject ClassificationMathematics of computing → Queueing theory; Mathematics of computing → Markov processes; Mathematics of computing → Matchings and factors

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