“…We here instead focus on closed systems, where a fixed population of customers cyclically issues requests to the servers; in these models, the problem becomes even more difficult because, besides remaining non-convex, a single evaluation of the objective function is often much more expensive [17,24,38]. This holds because non-iterative expressions for the stationary performance indexes are lacking for multiclass closed models, thus most formulations require to use gradient-hill methods in combination with some model evaluation technique [10,17,24], e.g., approximate mean-value analysis (MVA) [15,33,43]. These formulations are popular in applications [5,16,31,39] but, as we show in the paper, they often fail to converge to a local optimum at the short timescales of minutes at which online capacity management systems operate today.…”