Cramér-Lundberg workload exponent (${\Lambda }_A({\theta }_{\ast })=c{\theta }_{\ast },{\theta }_{\ast }={\inf }_{t>0}t{I}_A(c+1/t)$)

For nondegenerate independent and identically distributed arrivals with mean below $c$, suppose the positive root is interior to the moment-generating function domain. Then ${\Lambda }_A^{\prime }({\theta }_{\ast })>c$. The Fenchel–Young inequality gives $t{I}_A(c+1/t)\ge {\theta }_{\ast }$ for every $t>0$, and equality occurs at $t=[{\Lambda }_A^{\prime }({\theta }_{\ast })-c{]}^{-1}$. The exponential workload bound from cumulative arrival moments gives the upper logarithmic tail bound for every $\theta <{\theta }_{\ast }$. 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.