On the instability of matching queues

Abstract

A matching queue is described via a graph, an arrival process, and a matching policy. Specifically, to each node in the graph there is a corresponding arrival process of items, which can either be queued or matched with queued items in neighboring nodes. The matching policy specifies how items are matched whenever more than one matching is possible. Given the matching graph and the matching policy, the stability region of the system is the set of intensities of the arrival processes rendering the underlying Markov process positive recurrent. In a recent paper, a condition on the arrival intensities, which was named Ncond, was shown to be necessary for the stability of a matching queue. That condition can be thought of as an analogue to the usual traffic condition for traditional queueing networks, and it is thus natural to study whether it is also sufficient. In this paper, we show that this is not the case in general. Specifically, we prove that, except for a particular class of graphs, there always exists a matching policy rendering the stability region strictly smaller than the set of arrival intensities satisfying Ncond. Our proof combines graph-and queueing-theoretic techniques: After showing explicitly, via fluid-limit arguments, that the stability regions of two basic models is strictly included in Ncond, we generalize this result to any graph inducing either one of those two basic graphs.

Asymptotically Optimal Control of a Centralized Dynamic Matching Market with General Utilities

“…They show that the optimal policy is of threshold form under vertical and unidirectionally horizontal differentiated types. Ding et al (2021) allow the matching utilities to depend on the class of buyer and seller and perform a fluid analysis of a greedy policy, and Bušić and Meyn (2015) minimize linear holding costs in a system without classdependent matching utilities or abandonment but also find that matches are not made until there is a sufficient number of agents in the market (see Moyal and Perry 2017, where these systems are referred to as matching queues, for other references to these types of models). Gurvich and Ward (2014) and Nazari and Stolyar (2019) study a control problem in a more general setting than the aforementioned studies, where arriving customers wait to be matched to agents of other classes.…”

A stochastic matching model on hypergraphs

“…it does not contain any odd strong cycle. It is intuitively clear that the construction of stable stochastic matching models on hypergraphs is somewhat reminiscent of that of perfect matchings on a growing hypergraph that replicates the matching hypergraph a large number of times in the long run (in the case of graphs, see the discussion in Section Section 7 of [19]). This connection has a simple illustration in the next proposition, which provides a family of probability measures, naturally including the uniform measure on V, that cannot stabilize a matching model on the hypergraph H unless the latter satisfies Hall's condition.…”

“…The complete 3-uniform k-partite hypergraphs generalize the complete k-partite graphs introduced in [17, p. 4], also called separable graphs in [16] and [19], or blow-ups of the complete graph of order k in some other references. Roughly speaking, a complete 3-uniform k-partite hypergraph is a version of the complete 3-uniform graph of order k, in which the k nodes are replicated several times, each of the k sets of replicas forming an independent set I i , such that all replicas of the same set do not share any hyperedge with each other, but all share hyperedges of size 3 with all other pairs of replicas belonging to two different other sets of replicas.…”

“…Unmatched items are queued, waiting for a future compatible item, and leave the system in pairs as soon as they are matched. If the items enter the system individually, as in [16] and [19], and more recently in [2] and [18], we say that the system is a general stochastic matching model (GM), following the terminology of [16]. If the classes of items are partitioned into two subsets (say the classes of 'customers' and the classes of 'servers') and enter the system pairwise, as in the seminal papers [3,11] (which viewed such systems as generalizations of skill-based customer/server queueing systems) and then in [1], [2], [9], and [17], we say that the system is a bipartite stochastic matching model (BM).…”

A general stochastic matching model on multigraphs

“…On another hand, it is proven in Mairesse and Moyal (2016) that the matching policy 'Match the Longest' has a maximal stability region, that is, the latter necessary condition is also sufficient (we then say that the latter policy is maximal). However, Moyal and Perry (2017) shows that in fact, aside for a particular class of graphs, random policies are never maximal, and that there always exists a strict priority policy that isn't maximal either. Then, by adapting the dynamic reversibility argument of Adan et al (2018a) to the GM models, Moyal et al (2021) shows that the matching policy First Come, First Matched (FCFM) is also maximal, and derives the stationary probability in a product form.…”