Journal article: We analyze the dynamic assignment of objects to agents organized in a constant size waiting list. Applications include the assignment of social housing and organs for transplants. We analyze the optimal design of probabilistic queuing disciplines, punishment schemes, and information release. With private values, all agents prefer first-come first-served to the lottery, but waste is lower at the lottery. With common values, all agents prefer first-come first-served to any other mechanism, and waste is minimized at the lottery. Punishment schemes accelerate turnover in the queue and information release increases the value of agents at the top of the waiting list.