= Linear workload rate for a local convex action
{title2=$J(x)=x\inf_{v>C}g(v)/(v-C),\quad x>0$}
For a stable input with path action $I(a)=\int_{-\infty}^0g(\dot a_s)ds$, 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.
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