Dynamic matching markets are an ubiquitous object of study with applications in health, labor, or dating. We study a model proposed by Akbarpour et al. [2020] and Anderson et al. [2017], where homogeneous agents arrive at random according to a Poisson process and possess a random compatibility with other agents as in the Erdős-Rényi model. Agents leave according to a certain departure distribution and may leave early by forming a pair with a compatible agent. If agents are not matched until the end of their sojourn time, they perish and have to leave the market unmatched.