to continue doing sth, especially working or staying in prison, for a fixed period of time that has been set
serve sth out
to continue doing sth, especially working or staying in prison, for a fixed period of time that has been set
He has three more years in prison before he's served out his sentence.
serve sth outBritish
to share food or drink between a number of people
I went around the guests serving out drinks.
Definitions from the Oxford Advanced Learner's Dictionary, 9th edition