  Let $G = (V,E)$ be a connected directed graph on $n$ vertices. Assign values
from the set $\{1,2,\dots,n\}$ to the vertices of $G$ and update the values
according to the following rule: uniformly at random choose a vertex and update
its value to the maximum of the values in its neighbourhood. The value at this
vertex can potentially decrease. This random process is called the asynchronous
maximum model. Repeating this process we show that for a strongly connected
directed graph eventually all vertices have the same value and the model is
said to have \textit{converged}. In the undirected case the expected
convergence time is shown to be asymptotically (as $n\to \infty$) in
$\Omega(n\log n)$ and $O(n^2)$ and these bounds are tight. We further
characterise the convergence time in $O(\frac{n}{\phi}\log n)$ where $\phi$ is
the vertex expansion of $G$. This provides a better upper bound for a large
class of graphs. Further, we show the number of rounds until convergence is in
$O((\frac{n}{\phi}\log n)g(n))$ with high probability, where $g(n)$ satisfies
$\frac{1}{g^2(n)} \to 0$ as $n \to \infty$. For a strongly connected directed
graph the convergence time is shown to be in $O(nb^2 + \frac{n}{\phi'}\log n)$
where $b$ is a parameter measuring directed cycle length and $\phi'$ is a
parameter measuring vertex expansion.
