A proof of wavelength conversion not improving Lagrangian bounds of the sliding scheduled RWA problem

Extensive previous studies confirmed that wavelength conversion may only marginally improve the solutions to the static Routing and Wavelength Assignment (RWA) problem. This means that, for the static RWA problem, certain RWA schemes that do not use wavelength conversion can achieve a performance almost as good as the one from the best RWA scheme. Previous research work on sliding scheduled RWA problems, where a given set of lightpath demands are allowed to slide within their time windows, has also indicated in limited simulation results that the benefit of using wavelength conversion is marginal. However, the observation cannot be conclusive without the solid mathematical proof. We are thus motivated to investigate whether schedule sliding really requires wavelength conversion to achieve a better performance. In this paper, we prove that wavelength conversion does not improve the Lagrangian bound of the sliding scheduled RWA problem. In most test cases, this bound is very close to the best achieved objective function value. Our proof implies that, for those cases, the improvements achieved by making use of wavelength conversion are very marginal.