  Functional equations (FE) arise quite naturally in the analysis of stochastic
systems of different kinds : queueing and telecommunication networks, random
walks, enumeration of planar lattice walks, etc. Frequently, the object is to
determine the probability generating function of some positive random vector in
$Z\_n^+$. Although the situation n = 1 is more classical, we quote an
interesting non local functional equation which appeared in modelling a divide
and conquer protocol for a muti-access broadcast channel. As for n = 2, we
outline the theory reducing these linear FEs to boundary value problems of
Riemann-Hilbert-Carleman type, with closed form integral solutions. Typical
queueing examples analyzed over the last 45 years are sketched. Furthermore, it
is also sometimes possible to determine the nature of the functions (e.g.,
rational, algebraic, holonomic), as illustrated in a combinatorial context,
where asymptotics are briefly tackled. For general situations (e.g., big jumps,
or n $\ge$ 3), only prospective comments are made, because then no concrete
theory exists.
