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We study settings where a set of identical, reusable resources must be allocated in an online fashion to arriving agents. Each arriving agent is patient and willing to wait for some period of time to be matched. When matched, each agent occupies a resource for a certain amount of time, and then releases it, gaining some utility from having done so. The goal of the system designer is to maximize overall utility given some prior knowledge of the distribution of arriving agents. We are particularly interested in settings where demand for the resources far outstrips supply, as is typical in the provision of social services, for example homelessness resources. We formulate this problem as online bipartite matching with reusable resources and patient agents. We develop new, efficient nonmyopic algorithms for this class of problems, and compare their performance with that of greedy algorithms in a variety of simulated settings, as well as in a setting calibrated to real-world data on household demand for homelessness services. We find substantial overall welfare benefits to using our nonmyopic algorithms, particularly in more extreme settings – those where agents are unwilling or unable to wait for resources, and where the ratio of resource demand to supply is particularly high.