The phrase "serve out one's time" refers to completing a specific duration of service or imprisonment. There are various synonyms for this phrase that can be used interchangeably in different contexts. One such synonym is "complete one's sentence," which is often used in legal and penal contexts. "Fulfil one's term" is another synonym, which implies that the service was a specific commitment that was successfully met. "Satisfy one's obligation" is also used for meeting contractual duties or judicial requirements. "Finish one's tenure" implies that the duration of service was fixed, and it has come to an end with successful completion. All of these phrases convey the idea of completing one's obligations, commitments, or tenure.