Source Clarifications

Source: Mendelson and Whang, Optimal Incentive-Compatible Priority Pricing for the M/M/1 Queue.

Positive-flow and interior reading

• Theorem 1, pp. 873–874: marginal externality identity for every class → a selected class with $\lambda_i^*>0$. Other optimal flows remain nonnegative. The identity is $$p_i=\sum_jv_j\lambda_j^*\partial_iW_j(\lambda^*),$$ where $v_j$ is delay cost, $W_j$ mean system time, and $p_i$ class $i$’s price. For welfare $F(\lambda)=\sum_jV_j(\lambda_j)-\sum_jv_j\lambda_jW_j(\lambda)$, an interior optimum has $\partial_iF=0$; the source demand equality $V_i'=p_i+v_iW_i$ gives the formula. At zero flow only one-sided optimality follows. This proof establishes the identity for participating classes, without proving that every boundary class violates it.

Effect on the theorem conclusions

• Theorems 2–4, pp. 876–881: nonnegative flows → positive flow for every class for strict self-selection and strict cheating-penalty comparisons, retaining the source’s stable stationary domain and strict delay-cost order. Theorem 2 gives $C_i(i)<C_i(k)$ for every other priority $k$, where $C_i(k)$ is class $i$’s total cost at $k$. Theorems 3–4 give positive penalties off the assigned priority and strict increase in both directions. Adjacent comparison formulas contain a class-arrival-rate factor; unused levels can make it zero and permit indifference. This narrower strict domain does not refute weak incentive compatibility at zero flows, and the observation is not a full necessity proof for every strict conclusion.