Robust design of service systems with immobile servers, general arrival and service patterns, and demand uncertainty

This paper addresses the problem of robustly designing a service system consisting of immobile servers, each modelled as a G/G/1 queuing system, when the arrival rates are not known with certainty. The problem involves locating service centers, determining their capacities and assigning customers to them to minimize the total cost, which includes the setup, access and waiting costs. Besides the nominal problem, two robust problems with budget and ball uncertainty sets are considered. A piecewise-linear approximation is applied to handle the nonlinear waiting cost, which enables all the problems to be tightly approximated as mixed-integer quadratic programs. We also propose a Lagrangian approach that is capable of finding high-quality solutions and strong bounds for instances of practical sizes. Numerical experiments were conducted to validate the proposed models and solution methods and to study the effect of the problem parameters, the uncertainty set size and the objective function approximations on the optimal solution.