In this paper, we study a matching market model on a bipartite network where agents on each side arrive and depart stochastically by a Poisson process. For such a dynamic model, we design a mechanism that decides not only which agents to match, but also when to match them, to minimize the expected number of unmatched agents. The main contribution of this paper is to achieve theoretical bounds 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 over a bipartite network. We remark that the Patient algorithm requires the planner to identify agents who are about to leave the market, and, under the requirement, the Patient algorithm is shown to be an optimal algorithm. We also show that, without the requirement, the Greedy algorithm is almost optimal. In addition, we consider the $\textit{1-sided algorithms}$ where only an agent on one side can attempt to match. This models a practical matching market such as a freight exchange market and a labor market where only agents on one side can make a decision. For this setting, we prove that the Greedy and Patient algorithms admit the same performance, that is, waiting to thicken the market is not valuable. This conclusion is in contrast to the case where agents on both sides can make a decision and the non-bipartite case by [Akbarpour et al.,$~\textit{Journal of Political Economy}$, 2020].