  In this paper we focus on concurrent processes built on synchronization by
means of futures. This concept is an abstraction for processes based on a main
execution thread but allowing to delay some computations. The structure of a
general concurrent process with futures is more or less a directed acyclic
graph. Since the quantitative study of such increasingly labeled graphs
(directly related to processes) seems out of reach, we restrict ourselves to
the study of arch processes, a simplistic model of processes with futures. They
are based on two parameters related to their sizes and their numbers of arches.
The increasingly labeled structures seems not to be specifiable in the sense of
Analytic Combinatorics, but we manage to derive a recurrence equation for the
enumeration. For this model we first exhibit an exact and an asymptotic formula
for the number of runs of a given process. The second main contribution is
composed of an uniform random sampler algorithm and an unranking one that allow
efficient generation and exhaustive enumeration of the runs of a given arch
process.
