We consider the problem of traffic grooming in wavelength-division-multiplexing rings with all-to-all uniform traffic. Our objective is to minimize the total number of electronic add-drop multiplexers (ADMs) required. We derive explicit optimal solutions for two special cases: one with a traffic granularity of 4 and the other with 16. When the traffic granularity is equal to 4, we also show that the minimum number of ADMs can be achieved either with the minimum number of wavelengths, a result that was conjectured in earlier research [J. Lightwave Technol. 18, 2 (2000)], or with ADMs uniformly placed among nodes. Our results here are among the few in which analytically tractable optimal solutions are obtained for the traffic grooming problem. Our solutions provide insight as well as valuable tools for evaluating other approximate and heuristic algorithms.
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