Technical Appendix for: Sensitivity of Optimal Capacity to Customer Impatience in an Unobservable M/M/S Queue (Why You Shouldn’t Shout at the DMV)

function of μ) satisfies sample path convexity (a term that has been introduced by Shaked and Shanthikumar (1988)). Specifically, let 0 ≤ μ1 ≤ μ2 ≤ μ3 ≤ μ4 be four service rates such that μ1 + μ4 = μ2 + μ3, and fix λ, β(·) and η(·). Suppose that there exist Y1, ..., Y4, which are versions of the original head-count processes (Yi has service rate of μi) that satisfy the following two properties for all t ≥ 0: 1. Y1(t) + Y4(t) ≥ Y2(t) + Y3(t), a.s. 2. Y1(t) ≥ max{Y2(t), Y3(t), Y4(t)}, a.s.