All things considered, in my view, a Mandamus is possible. But the chances of success are remote at least in DC circuit. Typically, in delay cases, courts do not interfere with the executive branch of the govt. At the very least, lawsuits draw attention to difficult issues. I think a petition to the govt. should be the first step. May be the worthy President can help. Although, speedy processing in this environment is certainly not a politically popular issue, but we have all followed the laws and the promise of the laws.