The maximum line length problem

We consider a queueing system, which is constituted by a set of M/M/1 (sub-)systems, sharing the same scarce resources, but otherwise running independently. We analyse the nonlinear programming problem of minimizing the expectation of the maximum line length among the subsystems, with the service rates as the decision variables. Furthermore, we introduce three different nonlinear programming problems, which have natural interpretations with reference to the same queueing system and whose optimal solutions are useful to solve the original problem.