Suppose I have square lattice $\mathbb{Z}^2$ and I endow each site $x$ with a weight $W_x$, which is an independent exponential random variable of rate $\mu$. An oriented path between $(1, 1)$ and a point $(M, N)$, with $M,N\geq{}1$ is called increasing if it only ever goes in the North and East directions. Define the weight of an increasing path $\pi$ to be
$W(\pi) = \sum_{x\in\pi}W_x$
and the passage time, $T(M,N)$, from $(1, 1)$ to $(M, N)$ to be the maximum of $W(\pi)$ over the possible increasing paths. My question is, what is the distribution of $T(M,N)$? So far everything I can find only considers limits for large $M,N$ but I'm interested in the distribution for a fixed $M,N$.