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

Properties of the limit shape for some last passage growth models in random environments (Dissertation)