  We study directed last passage percolation on the first quadrant of the
planar square lattice whose weights have general distributions, or
equivalently, ./G/1 queues in series. The service time distributions of the
servers vary randomly which constitutes a random environment for the model.
Equivalently, each row of the last passage model has its own randomly chosen
weight distribution. We investigate 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 means attached to the
rows.
