Minimizing the externalities variance in a LCFS-PR $M\mathrm{/}G\mathrm{/}1$ queue under various constraints

Consider a LCFS-PR $M\mathrm{/}G\mathrm{/}1$ queue and assume that at time $t=0$, there are $n+2$ customers $c_1,c_2,\mathrm{.}\mathrm{.}\mathrm{.},c_{n+1},c$ who arrived in that order such that $t=0$ is the arrival time of $c$. Then, the externalities which are generated by $c$ is the total waiting time that would be saved by $c_1,c_2,\mathrm{.}\mathrm{.}\mathrm{.},c_{n+1}$ if $c$ reduced his service requirement to zero. Motivated by some applications, this work is about the minimization of the externalities variance under various constraints.