Stabilizing policies for probabilistic matching systems

Abstract

In this work, we introduce a novel queueing model with two classes of users, where users wait in the system to match with a candidate from the other class, instead of accessing a resource. This new model is useful for analyzing the traffic in web portals that match people who provide a service with people who demand the service, e.g. employment portals, matrimonial and dating sites and rental portals. We provide a Markov chain model for these systems and derive the probability distribution of the number of matches up to some finite time given the number of arrivals. We prove that if no control mechanism is employed these systems are unstable for any set of parameters, and suggest four different classes of control policies to assure stability. Contrary to the intuition that the rejection rate should decrease as the users get more likely to match, we show that for certain control policies the rejection rate is insensitive to the matching probability. Even more surprisingly, we show that for reasonable policies the rejection rate may be an increasing function of the matching probability. We also prove insensitivity results related to the average queue lengths and waiting times.

Fluid and diffusion approximations of probabilistic matching systems

“…In this work, we study probabilistic matching systems introduced in Büke and Chen [3], where two classes of users, indexed by i = 1, 2, arrive at the system to be matched with users of the other class. We assume that class-i users arrive according to a Poisson process with rate λ i .…”

“…The above model reduces to the one introduced in [3] if users do not abandon the system (γ = 0). It is sometimes useful in our analysis to express the queue length processes, X i (t), as the difference of counting processes.…”

“…In this work, our goal is to provide tractable methods to analyze probabilistic matching systems introduced in Büke and Chen [3] to study the web portals that B Hanyi Chen H.Chen-29@sms.ed.ac.uk Burak Büke B.Buke@ed.ac.uk 1 School of Mathematics, The University of Edinburgh, King's Buildings, James C. Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK serve as a meeting point for suppliers and customers of a specific product or service. The examples of such systems include employment and rental portals, matrimonial and dating Web sites and general purpose classified advertisement Web sites.…”

“…Hence, unlike the double-ended queues, users from different classes can coexist in the system when the matchings are probabilistic, which makes it essential to model the queueing system as a two-dimensional stochastic process. Büke and Chen [3] study the effects of the matching probability on the performance of these systems using an exact analysis and show that if uncontrolled these systems are unstable. Büke and Chen [3] also suggest stabilizing admission control policies to decide when to accept an arriving user into the system based on the system size and analyze some performance measures (for example, throughput and average queue length) under the suggested policies.…”

“…The resulting fluid and diffusion limits under this scaling involve differential equations which are not tractable analytically in the general case, although we can derive an analytical formula for the fluid limit when there are no abandonments. Büke and Chen [3] show that some performance measures, such as the difference between the average queue lengths of different classes, are insensitive to the matching probability under certain control policies. Despite not imposing any control policy, similar to the results in [3] we show that the difference between queue lengths for different classes is also insensitive to the matching probability in the fluid limit.…”