  We study two random processes on an $n$-vertex graph inspired by the internal
diffusion limited aggregation (IDLA) model. In both processes $n$ particles
start from an arbitrary but fixed origin. Each particle performs a simple
random walk until first encountering an unoccupied vertex, and at which point
the vertex becomes occupied and the random walk terminates. In one of the
processes, called \textit{Sequential-IDLA}, only one particle moves until
settling and only then does the next particle start whereas in the second
process, called \textit{Parallel-IDLA}, all unsettled particles move
simultaneously. Our main goal is to analyze the so-called dispersion time of
these processes, which is the maximum number of steps performed by any of the
$n$ particles.
  In order to compare the two processes, we develop a coupling which shows the
dispersion time of the Parallel-IDLA stochastically dominates that of the
Sequential-IDLA; however, the total number of steps performed by all particles
has the same distribution in both processes. This coupling also gives us that
dispersion time of Parallel-IDLA is bounded in expectation by dispersion time
of the Sequential-IDLA up to a multiplicative $\log n$ factor. Moreover, we
derive asymptotic upper and lower bound on the dispersion time for several
graph classes, such as cliques, cycles, binary trees, $d$-dimensional grids,
hypercubes and expanders. Most of our bounds are tight up to a multiplicative
constant.
