For a linear operator Q from R(sup)d into R(sup)d and 0<b<1, the (Q,b)-semi-stability and the strict (Q,b)-semi-stability of probability measures on R(sup)d are defined. The (Q,b)-semi-stability is an extension of operator stability with exponent Q on one hand and of semi-stability with index $alpha$ and parameter b on the other. characterization of strictly (Q,b)-semi-stable distributions among (Q,b)-semi-stable distributions is made. Existence of (Q,b)-semi-stable distributions which are not translation of strictly (Q,b)-semi-stable distribution is discussed.