The Cost of Impatience in Dynamic Matching: Scaling Laws and Operating Regimes

We study matching queues with abandonment. The simplest of these is the two-sided queue with servers on one side and customers on the other, both arriving dynamically over time and abandoning if not matched by the time their patience elapses. We identify nonasymptotic and universal scaling laws for the matching loss due to abandonment, which we refer to as the “cost of impatience.” The scaling laws characterize the way in which this cost depends on the arrival rates and the (possibly different) mean patience of servers and customers. Our characterization reveals four operating regimes identified by an operational measure of patience that brings together mean patience and utilization. The four regimes subsume the regimes that arise in asymptotic (heavy-traffic) approximations. The scaling laws, specialized to each regime, reveal the fundamental structure of the cost of impatience and show that its order of magnitude is fully determined by (i) a “winner-take-all” competition between customer impatience and utilization, and (ii) the ability to accumulate inventory on the server side. Practically important is that when servers are impatient, the cost of impatience is, up to an order of magnitude, given by an insightful expression where only the minimum of the two patience rates appears. Considering the trade-off between abandonment and capacity costs, we characterize the scaling of the optimal safety capacity as a function of costs, arrival rates, and patience parameters. We prove that the ability to hold inventory of servers means that the optimal safety capacity grows logarithmically in abandonment cost and, in turn, slower than the square-root growth in the single-sided queue.