The burning number of a graph was introduced as a simple model of spreading memes or other kinds of social influence. The smaller the burning number is, the faster an influence can be spread in the network. We may add randomness to the problem by burning vertices via a probabilistic process, or by studying burning on random graphs. When playing Cops and Robbers with exactly c(G)-many cops in a graph G, the length of the game is longest. It is natural to consider Overprescribed Cops and Robbers: as more cops are added, the capture time monotonically decreases. This chapter examines this effect on the length of the game temporal speed up, and provides a sequence of game lengths that are a function of the number of cops. It also provides results on temporal speed up in hypercubes and random graphs. For the case of hypercubes, probabilistic methods are key to understanding the dynamics of Overprescribed Cops and Robbers.