The Justice Department will begin transferring thousands of inmates out of federal prisons this week as part of a sweeping criminal justice overhaul signed by President Donald Trump more than three years ago. The department, in a rule being published Thursday in the Federal Register, is spelling out how “time credits” for prisoners will work. The bipartisan law is intended to encourage inmates to participate in programs aimed at reducing recidivism, which could let them out of prison earlier. It also eases mandatory minimum sentences and gives judges more discretion in sentencing. While the transfers are expected to begin this week, it isn’t clear how many inmates will be released. The department would only say that “thousands” of inmates are being affected. Under the law signed in December 2018, inmates are eligible to earn time credits — 10 days to 15 days of credit for every 30 days they participate in prison programs to reduce recidivism. The programs range from anger management and drug treatment to educational, work and social skills classes. The announcement of a finalized rule being published comes about two months after the department’s inspector general sounded an alarm that the Bureau of Prisons had not applied the earned time credits to about 60,000 federal inmates who had completed the programs. It also comes a week after an announcement that the director of the prison agency, Michael Carvajal, will resign from his position in the face of mounting criticism over his leadership. The Biden administration has faced increased pressure from both Democratic and Republican lawmakers to do more to put in place additional aspects of the First Step Act, and the bureau has been accused of dragging its feet. There has been a significant staffing shortage at the bureau for years, and that has pressed teachers, cooks, nurses and other workers into service as correctional officers. Employees have long argued that pulling them away from their other duties to guard inmates slows action on the First Step Act because they have less time to teach classes, review release paperwork and provide inmate services. The Justice Department says the number of eligible programs has increased and inmates will not be punished if they cannot participate for reasons beyond their control. The department has been working for months to try to increase bureau staffing.