We study the sojourn times in an open star-shaped queueing network, with a central processor-sharing (PS) node and multiple multi-server First-Come-First Served (FCFS) nodes. Each customer alternatingly visits the central node and one of the other nodes, before departing from the system. For this model, exact expressions for the mean sojourn time can be easily obtained, but an exact analysis of the variance is not possible. Therefore, we propose a method for deriving simple but accurate approximations for the variance of the sojourn times. Extensive simulations demonstrate that the approximations are extremely accurate for a wide range of parameter values.