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Much work has focused on traffic grooming in WDM ring networks. Previous work has considered many aspects of traffic grooming, including minimizing the number of ADMs, minimizing the number of wavelengths, single hub and multiple hub architectures, switching capabilities, etc. We derive general and tighter bounds on the number of wavelengths in WDM ring networks that may be capable of switching streams across wavelengths. We also study traffic grooming in unswitched UPSRs (unidirectional path-switched rings) and derive a more general lower bound for the number of ADMs required in traffic grooming. This bound reduces to special cases first obtained by E. Modiano and A. Chiu (see IEEE J. of Light-wave Technology, 2000). A cost-effective multi-phase traffic grooming algorithm is proposed and studied for UPSRs. Our numerical results show that this algorithm outperforms existing traffic grooming algorithms, resulting in traffic grooming that uses a lower number of ADMs. Our algorithm in many cases also reaches the general lower bound derived.