Reward maximization in general dynamic matching systems

Abstract

We consider a matching system with random arrivals of items of different types. The items wait in queues -one per each item type -until they are "matched." Each matching requires certain quantities of items of different types; after a matching is activated, the associated items leave the system. There exists a finite set of possible matchings, each producing a certain amount of "reward". This model has a broad range of important applications, including assemble-to-order systems, Internet advertising, matching web portals, etc.We propose an optimal matching scheme in the sense that it asymptotically maximizes the long-term average matching reward, while keeping the queues stable. The scheme makes matching decisions in a specially constructed virtual system, which in turn control decisions in the physical system. The key feature of the virtual system is that, unlike the physical one, it allows the queues to become negative. The matchings in the virtual system are controlled by an extended version of the greedy primal-dual (GPD) algorithm, which we prove to be asymptotically optimal -this in turn implies the asymptotic optimality of the entire scheme. The scheme is real-time, at any time it uses simple rules based on the current state of virtual and physical queues. It is very robust in that it does not require any knowledge of the item arrival rates, and automatically adapts to changing rates.The extended GPD algorithm and its asymptotic optimality apply to a quite general queueing network framework, not limited to matching problems, and therefore is of independent interest.

A product form for the general stochastic matching model

“…Applying fluid instability arguments to a continuous-time version of the GM model, [17] shows that, aside from a particular class of graphs, whenever G is not complete k -partite there always exists a policy of the strict priority type that does not have a maximal stability region, and that the ‘uniform’ random policy (natural in the case where no information is available to the entering items on the state of the system) never has a maximal stability region, thereby providing a partial converse of the result in [16]. Related models are studied in [14, 19], proposing optimization schemes for models of various matching structures (general graphs and hypergraphs), in which matchings are associated to a cost or a reward. Notice that in both references the matching policies are possibly retarded (or idling ), i.e.…”

A stochastic matching model on hypergraphs

“…Other references address specific models for designated applications: [5] on kidney transplants, [22] on housing allocations systems, and [21] on ride-sharing models. In another line of research, such stochastic matching architectures are addressed from the point of view of stochastic optimization in [10], [13], and [20], among others.…”

A general stochastic matching model on multigraphs

“…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. More recently, following the work of Nazari and Stolyar (2019), matching policies of the broader Max-Weight type (including 'Match the Longest') are shown to be maximal, and drift inequalities allow to bound the speed of convergence to the equilibrium, and the first two moments of the stationary state. Variants of the GM model to the case of (i) hypergraphical structures (i.e.…”

“…Variants of the GM model to the case of (i) hypergraphical structures (i.e. matching items by groups of two or more) and (ii) graphical systems with reneging are investigated, respectively in Gurvich and Ward (2014); Nazari and Stolyar (2019); Rahmé and Moyal (2019) and Jonckheere et al (2020) (see also Boxma et al (2011)).…”