Suppose you have two random processes that jump from state to state at different speeds and in different directions – maybe one is a model of a busy call centre and the other is a simpler, slower version. How can you be sure that the number of waiting customers in the first system never stochastically exceeds the second, even if you can’t solve either process exactly? This chapter gives a clean answer: by comparing their intensity matrices through a cumulative inequality, we can guarantee that one process stays smaller in a precise statistical sense, forever.