  We study the transients of matrices in max-plus algebra. Our approach is
based on the weak CSR expansion. Using this expansion, the transient can be
expressed by $\max\{T_1,T_2\}$, where $T_1$ is the weak CSR threshold and $T_2$
is the time after which the purely pseudoperiodic CSR terms start to dominate
in the expansion. Various bounds have been derived for $T_1$ and $T_2$,
naturally leading to the question which matrices, if any, attain these bounds.
  In the present paper we characterize the matrices attaining two particular
bounds on $T_1$, which are generalizations of the bounds of Wielandt and
Dulmage-Mendelsohn on the indices of non-weighted digraphs. This also leads to
a characterization of tightness for the same bounds on the transients of
critical rows and columns. The characterizations themselves are generalizations
of those for the non-weighted case.
