This paper considers the Stochastic Queue Center problem, which seeks to locate a single facility with a center-type objective in an M/G/1 queue operating environment. The objective function that we consider is to minimize a positive weighted linear function of the square of the average response time and the variance of the response time to a call. The Stochastic Queue Center problem is discussed on both a discrete and a network location topology. When potential facility locations are restricted to a finite set of discrete points, an efficient algorithm is developed to solve for the optimal facility location parametrically in the arrival rate. By exploiting convexity properties of the objective function, we develop an efficient finite-step algorithm to find the Stochastic Queue Center on a network. The major conclusion of this work is that incorporating the variance term in the objective function has a major impact on the choice of the optimal location. We illustrate the results with an example drawn from a potential application of the model for locating an emergency transport center serving different municipalities in Camden County, NJ.
