  We introduce a probability distribution on the set of states in a generalized
box-ball system associated with Kirillov-Reshetikhin (KR) crystals of type
$A^{(1)}_n$. Their conserved quantities induce $n$-tuple of random Young
diagrams in the rigged configurations. We determine their limit shape as the
system gets large by analyzing the Fermionic formula by thermodynamic Bethe
ansatz. The result is expressed as a logarithmic derivative of a deformed
character of the KR modules and agrees with the stationary local energy of the
associated Markov process of carriers.
