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We present a systematic treatment of efficient nonlinear optimizations of queuing systems. The suite of formulations uses the computational tool of convex optimization, with fast polynomial time algorithms to obtain the global optimum for these nonlinear problems under various constraints. We first show convexity structures of several queuing systems, including some surprising transition patterns, followed by formulating and showing numerical examples of several convex performance optimizations for both single queues and queuing networks. Blocking probability minimization and service rate allocation through the effective bandwidth approach is also presented.