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{T1,T2}, where T1 is the weak CSR threshold and T2
is the time after which the purely pseudoperiodic CSR terms start to dominate
in the expansion. Various bounds have been derived for T1 and T2,
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 T1, 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.Comment: 42 pages, 9 figure