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In this paper, we study a matching market model on a bipartite network whereagents on each side arrive and depart stochastically by a Poisson process. Forsuch a dynamic model, we design a mechanism that decides not only which agentsto match, but also when to match them, to minimize the expected number ofunmatched agents. The main contribution of this paper is to achieve theoreticalbounds on the performance of local mechanisms with different timing properties.We show that an algorithm that waits to thicken the market, called the$\textit{Patient}$ algorithm, is exponentially better than the$\textit{Greedy}$ algorithm, i.e., an algorithm that matches agents greedily.This means that waiting has substantial benefits on maximizing a matching overa bipartite network. We remark that the Patient algorithm requires the plannerto identify agents who are about to leave the market, and, under therequirement, the Patient algorithm is shown to be an optimal algorithm. We alsoshow that, without the requirement, the Greedy algorithm is almost optimal. Inaddition, we consider the $\textit{1-sided algorithms}$ where only an agent onone side can attempt to match. This models a practical matching market such asa freight exchange market and a labor market where only agents on one side canmake a decision. For this setting, we prove that the Greedy and Patientalgorithms admit the same performance, that is, waiting to thicken the marketis not valuable. This conclusion is in contrast to the case where agents onboth sides can make a decision and the non-bipartite case by [Akbarpour etal.,$~\textit{Journal of Political Economy}$, 2020].

Dynamic Bipartite Matching Market with Arrivals and Departures