Minimizing SONET ADMs in Unidirectional WDM Rings with Grooming Ratio Seven

Abstract

In order to reduce the number of add-drop multiplexers (ADMs) in SONET/WDM networks using wavelength add-drop multiplexing, certain graph decompositions can be used to form a “grooming” that specifies the assignment of traffic to wavelengths. When traffic among nodes is all-to-all and uniform, the drop cost of such a decomposition is the sum, over all graphs in the decomposition, of the number of vertices of nonzero degree in the graph. The number of ADMs required is this drop cost. The existence of such decompositions with minimum cost, when every pair of sites employs no more than $\frac{1}{7}$ of the wavelength capacity, is determined within an additive constant. Indeed when the number n of sites satisfies $n \equiv 1$ (mod 3) and $n \neq 19$, the determination is exact; when $n \equiv 0$ (mod 3), $n \not\equiv 18$ (mod 24), and n is large enough, the determination is also exact; and when $n \equiv 2$ (mod 3) and n is large enough, the gap between the cost of the best construction and the cost of the lower bound is independent of n and does not exceed 4.