  We study directed last-passage percolation on the planar square lattice whose
weights have general distributions, or equivalently, queues in series with
general service distributions. Each row of the last passage model has its own
randomly chosen weight distribution. We first show the existence of the
limiting time constant and list its properties. Next we study the problem for
models with Bernoulli and exponential weights, for which we already have more
precise results. We then present some universality results about the limiting
time constant close to the boundary of the quadrant. Close to the y-axis, where
the number of random distributions averaged over stays large, the limiting time
constant takes the same universal form as in the homogeneous model. But close
to the x-axis we see the effect of the tail of the distribution of the random
environment. In particular we will give some estimates of the upper bound in
this case.
