To the best of my knowledge, the question that first passage percolation tries to address is whether/when something (say a fluid) will reach a certain destination from some given source. A paper I was reading calls last passage percolation a 'small variation' of FPP; at the same time, it defines the last passage time as
$$L(u,v)=\max_{\Gamma:u\rightarrow v}T(\Gamma)$$ $T$ being the sum of the weights. If I were to think of the physical realisation of this, should I think of it as the longest possible time required for the fluid to reach the destination? If yes, why is this important? I have found little to no motivation in most place, I feel like I'm missing out on something.