Applying optimization theory to study extremal GI/GI/1 transient mean waiting times

We study the tight upper bound of the transient mean waiting time in the classical GI/GI/1 queue over candidate interarrival-time distributions with finite support, given the first two moments of the interarrival time and the full service-time distribution. We formulate the problem as a non-convex nonlinear program. We derive the gradient of the transient mean waiting time and then show that a stationary point of the optimization can be characterized by a linear program. We develop and apply a stochastic variant of the Frank and Wolfe (Naval Res Logist Q 3:95-110, 1956) algorithm to find a stationary point for any given service-time distribution. We also establish necessary conditions and sufficient conditions for stationary points to be three-point distributions or special two-point distributions. We illustrate by applying simulation together with optimization to analyze several examples.