Let {X-i(t), t >= 0}, 1 <= i <= n be independent centered stationary Gaussian processes with unit variance and almost surely continuous sample paths. For given positive constants u, T, define the set of conjunctions C-[0,C-T],C-u := {t is an element of [0, T] : min(1 <= i <= n) X-i(t) >= u}. Motivated by some applications in brain mapping and digital communication systems, we obtain exact asymptotic expansion of P {C-[0,C-T],C-u not equal phi}, as u -> infinity. Moreover, we establish the Berman sojourn limit theorem for the random process [min(1 <=iota <= n) X-i(t), t >= 0) and derive the tail asymptotics of the supremum of each order statistics process.