Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 31 / 4 / d / Solution

Take $0<x\le B$ and a finite-cost burst rate $v>C$. For the constant-rate burst path just constructed, let $t_0=x/(v-C)$. On $0\le s\le t_0$ its net input is $z(s)=(v-C)s$, so the supremum is x and the infimum is zero. The finite-buffer formula evaluated at $t=t_0$ is therefore $\min (x,B)=x$. Since $\bar{q}\le q=x$, we conclude $\bar{q}=x$ exactly: the burst never needs more space than the permitted capacity.

That path's action bounds the contracted finite-buffer rate, and optimizing givesAt zero use the typical-rate path to obtain $\bar{J}(0)=J(0)=0$. If the positive infinite-buffer rate is infinite, the asserted upper inequality is automatic. Approximating burst rates suffice, so the argument does not rely on an unproved existence of an optimal duration. Consequently $\boxed{\bar{J}(x)\le J(x)\text{ for }0\le x\le B}$.