Fluid and diffusion approximations of probabilistic matching systems

Abstract

This paper focuses on probabilistic matching systems where two classes of users arrive at the system to match with users from the other class. The users are selective and the matchings occur probabilistically. Recently, Markov chain models were proposed to analyze these systems; however, an exact analysis of these models to completely characterize the performance is not possible due to the probabilistic matching structure. In this work, we propose approximation methods based on fluid and diffusion limits using different scalings. We analyze the basic properties of these approximations and show that some performance measures are insensitive to the matching probability, agreeing with the existing results. We also perform numerical experiments with our approximations to gain insight into probabilistic matching systems.

On spatial matchings: The first-in-first-match case

“…Bušić et al [7] generalize the bipartite matching model by dropping independence of arriving types and considering other matching policies. Büke and Chen [6] study a model where the matching policy is probabilistic. In their model, when a customer (or server) arrives in a system, it looks at the possible matches and, independently of everything else, selects one using a probability distribution.…”

“…There is also a positive probability of not finding any suitable server (customer), in which case it waits for a compatible server (customer). Büke and Chen [6] also consider models where the users are impatient and may depart if they are not matched by a certain time. An exact analysis of these models becomes quite intractable; Büke and Chen [6] study the fluid and diffusive scaling approximations of these systems.…”

“…Büke and Chen [6] also consider models where the users are impatient and may depart if they are not matched by a certain time. An exact analysis of these models becomes quite intractable; Büke and Chen [6] study the fluid and diffusive scaling approximations of these systems.…”

A stochastic matching model on hypergraphs

“…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). Stabilizing policies and fluid/diffusion approximations of two-sided systems are obtained respectively in [7,8]. Other references address specific models for designated applications: [5] on kidney transplants, [22] on housing allocations systems, and [21] on ride-sharing models.…”

A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation

“…Typically, a customary rescaling procedure allows one to approximate the queue length process by a diffusion process, as indicated in Giorno et al [26]. Examples of diffusion models arising from heavy-traffic approximations of double-ended queues and of similar matching systems can be found in Liu et al [27] and Büke and Chen [28], respectively. In the case of queueing systems subject to catastrophes, a customary approach leads to jump-diffusion approximating processes (see, for instance, Di Crescenzo et al [29] and Dharmaraja et al [30]).…”