Assessing the accuracy of externalities prediction in a LCFS-PR M/G/1 queue under partial information

Consider a LCFS-PR $M/G/1$ queue and assume that at time $t=0$, there are $n+1$ customers $c_1,c_2,...,c_{n+1}$ who arrived in that order. In addition, at time $t=0$ there is an additional customer c with service requirement $x>0$ who makes an admission request. At time $t=0$, the system's manager should decide whether to let c join the system or not. To this end, the manager wants to evaluate the externalities generated by c, i.e., the additional waiting time that $c_1,c_2,\dots ,c_{n+1}$ will suffer as a consequence of the admission of c. We assume that at the decision epoch the manager knows only n, x, the remaining service time of $c_{n+1}$ (who was getting service just before c had made his admission request) and the total workload at $t=0$. In a previous work by Jacobovic, Levering and Boxma (2023), it was shown that the manager can compute the expected externalities (i.e., the natural predictor for the externalities value) but not their variance (i.e., the conventional measure of the predictor's accuracy). Motivated by this problem, in the current work, we study a convex piecewise-linear program which yields the spectrum of variance values which are consistent with the manager's information.