= Cramér-Lundberg workload exponent
{c}
{title2=$\Lambda_A(\theta_*)=c\theta_*,\quad\theta_* =\inf_{t>0}tI_A(c+1/t)$}
For nondegenerate <independent and identically distributed> arrivals with mean below $c$, suppose the positive root is interior to the <moment-generating function> domain. Then $\Lambda_A'(\theta_*)>c$. The <Fenchel–Young inequality> gives $tI_A(c+1/t)\ge\theta_*$ for every $t>0$, and equality occurs at $t=[\Lambda_A'(\theta_*)-c]^{-1}$. The <exponential workload bound from cumulative arrival moments> gives the upper logarithmic tail bound for every $\theta<\theta_*$. A horizon of the indicated length times buffer size gives the matching lower bound by <Cramér's theorem>. A positive interior root is an assumption; if arrivals never exceed capacity, positive workload cannot arise.
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