  Inspired by blockchains, we introduce a dynamically growing model of rooted
Directed Acyclic Graphs (DAGs) referred to as the asynchronous composition
model, subject to i.i.d. random delays $(\xi_t)$ with finite mean. The new
vertex at time $t$ is connected to vertices chosen from the graph
$G_{(t-\xi_t)_+}$ according to a construction function $f$ and the graph is
updated by taking union with the graph $G_{t-1}$. This process corresponds to
adding new blocks in a blockchain, where the delays arise due to network
communication. The main question of interest is the end structure of the
asynchronous limit of the graph sequence as time increases to infinity.
  We consider the following construction functions of interest, a) Nakamoto
construction $f_{\mathrm{Nak}}$, in which a vertex is uniformly selected from
those furthest from the root, resulting in a tree, and b) mixture of
construction functions $(f_k)_{1\le k\le \infty}$, where in $f_k$ a random set
of $k$ leaves (all if there are less than $k$ in total) is chosen without
replacement. The main idea behind the analysis is decoupling the time-delay
process from the DAG process and constructing an appropriate regenerative
structure in the time-delay process giving rise to Markovian behavior for a
functional of the DAG process. We establish that the asynchronous limits for
$f_\mathrm{Nak}, (f_k)_{k \ge 2}$, and any non-trivial mixture $f$ are
one-ended, while the asynchronous limit for $f_1$ has infinitely many ends,
almost surely. We also study fundamental growth properties of the longest path
for the sequence of graphs for $f_\mathrm{Nak}$. In addition, we prove a phase
transition on the (time and sample-path dependent) probability of choosing
$f_1$ such that the asynchronous limit either has one or infinitely many ends.
Finally, we show that the construction $f_\infty$ is an appropriate limit of
the $(f_k)_k$.
