Is the prison they are being assigned where they have to do their time?

Yes, the Department of Corrections or the Bureau of Prisons determine where an convicted offender will serve their time. There is a point system that is based on the criminal history, violence in their past, length of sentence and likelihood of escape which arrives at a number which slots the inmate into security level. The location selected reflects the points, whether it is high/maximum, medium, low or minimum - and it is this designation and it cannot be challenged. Inmates may get transferred from their designated location by keeping their record clean. No incident reports will give the inmate an option to make a lateral transfer. After an inmate has done significant chunks of their sentence, the staff can change their designation and points, therefore the security level of the prison they are to be held going forward. The change could be from medium to low, or low to minimum. Inmates call this "working their way down".