For a Poisson process in R^2 with intensity 1, the distribution of the maximum number of points on an oriented path from (0,0) to (N,N) has been studied in detail, culminating in Baik-Deift-Johansson's celebrated Tracy-Widom fluctuation result. We consider a variant of the model where one adds, on the diagonal, additional points according to an independent one dimensional Poisson process with rate \lambda. The question of interest here is whether for all positive values of \lambda, this results in a change in the law of large numbers for the the number of points in the maximal path. A closely related question comes from a variant of Totally Asymmetric Exclusion Process, introduced by Janowsky and Lebowitz. Consider a TASEP in 1-dimension, where the bond at the origin rings at a slower rate r