Linear workload rate for a local convex action ($J(x)=x{\inf }_{v>C}g(v)/(v-C),x>0$)

For a stable input with path action $I(a)={\int }_{-\infty }^{0}g({\dot{a}}_s)ds$, a workload x requires some interval of length t with average input near $C+x/t$. Jensen inequality bounds its cost below by $tg(C+x/t)$; taking the supremum workload and then the infimum over t yields the displayed lower bound. Constant-rate burst paths give the matching upper bound. At zero the typical-rate path has zero cost; negative workloads have infinite rate. If no finite-cost rate exceeds C, every positive workload has infinite rate.