 Solution

Write $z(s)=a(-s,0]-Cs$. For each t the minimum in the finite-buffer formula is no greater than ${\sup }_{0\le s\le t}z(s)$. Taking the outer supremum gives $\bar{q}(a)\le {\sup }_{s\ge 0}z(s)=q(a)$. Also $z(0)=0$ ensures $0\le \bar{q}(a)\le B$: the choice $t=0$ gives zero, and each term is at most $B+{\inf }_{0\le s\le t}z(s)\le B$.

Every path in the fiber $\bar{q}(a)=x$ therefore lies in the set $q(a)\ge x$. Infima of its action giveThe last equality partitions that set by its queue workload fibers. It is a variational consequence of the pathwise order, without a questionable inference from probabilities of exact queue workload values.