  We revisit the simple lattice random walk (P\'{o}lya walk) and the Sisyphus
random walk in $\mathbb{Z}$, in the presence of random restarts. We use a
relatively direct approach namely First passage under restart for discrete
space and time which was recently developed by us in PRE 103, 052129 (2021) and
rederive the first passage properties of these walks under the memoryless
geometric restart mechanism. Subsequently, we show how our method could be
generalized to arbitrary first passage process subject to more complex restart
mechanisms such as sharp, Poisson and Zeta distribution where the latter is
heavy tailed. We emphasize that the method is very useful to treat such
problems both analytically and numerically.
