This paper considers a realistic multi-service system design problem in which each service type is a stochastic sequence of services provided by different units of facilities where each facility is modeled as a set of open Jackson queueing networks. The problem is first formulated as a mixed-integer nonlinear programming model, which is further simplified to a model with a smaller number of constraints. Three exact solution methods are applied to solve the amended model. The first one is a cutting-plane method, which is based on a piecewise-linear approximation. The second is based on a mixed-integer linear programming formulation, which is enhanced by valid inequalities. The third is to use mixed-integer second-order cone programming. The methods are compared using a numerical study. Finally, an online pharmacy is considered as an example to illustrate the applicability of the problem, and some managerial insights are provided.