  We analyze the asymptotic behavior of sequences of random variables defined
by an initial condition, a stationary and ergodic sequence of random matrices,
and an induction formula involving multiplication is the so-called max-plus
algebra. This type of recursive sequences are frequently used in applied
probability as they model many systems as some queueing networks, train and
computer networks, and production systems. We give a necessary condition for
the recursive sequences to satisfy a strong law of large numbers, which proves
to be sufficient when the matrices are i.i.d. Moreover, we construct a new
example, in which the sequence of matrices is strongly mixing, that condition
is satisfied, but the recursive sequence do not converges almost surely.
