The paper characterizes the trajectory of the Stochastid Queue Median (SQM) location problem in a planar region with discrete demands and a general Lp travel metric (1<p<•). The location objective is to minimize expected response time to customers (i.e., travel time plus queue delay). The paper uses an ∈-perturbed version of the SQM objective function (to account for points of nondifferentiability) to show that for the ∈-perturbed problem the optimal SQM location occurs in a region bounded by the point minimizing the first and second moments of service time (s*•∈ and s2*•∈, respectively); all optimal locations can be characterized by a simple ratio condition relating the derivatives of the first and second moments of service time; and the trajectory as a function of the customer call rate moves monotonically along a path from s*•∈ toward s2*•∈, then turns and retraces the same path back to s*•∈. Finally, the paper establishes convergence of the ∈-optimal solution to an optimal SQM solution as ∈ approaches zero, as well as a general condition under which the SQM problem can be solved directly, with no perturbation.
