okay ...if you refer to the ingame term "fatigue", the correct german translation is "Erschöpfung", not Müdigkeit.
Müdigkeit is best translated with tiredness. tired = müde, although the english language seems to also use fatigue for that one. but the meaning of fatigue in Arma best refers to Erschöpfung...
exhausted translates best to "ausgelaugt" and might be the right one if you say "ins Bett gehen, weil ich fertig/ erledigt bin"... the noun for exhausted is "exhaustion"
hope that helps