Elastic Optical Network (EON) has been considered as a promising optical networking technology to architect the next-generation backbone networks. Routing and Spectrum Assignment (RSA) is the fundamental problem in EONs to realize service provisioning. Generally, the RSA is solved by routing the requests with lightpaths first and then assigning spectrum resources to the lightpaths to optimize the spectrum usage. Thus, the spectrum assignment explicitly decide the final spectrum usage of EONs. However, besides the spectrum assignment, there are three other factors, the network topology, traffic distribution and routing scheme, implicitly impact on the spectrum usage. Few related work involves in the implicit impact mechanism. In this paper, we aim to provide a thoroughly theoretical analysis on the impact of the three key factors on the spectrum usage. To this end, two theoretical chains are proposed: (1) The optimal spectrum usage can be measured by the chromatic number of the conflict graph, which is positively correlated to the intersecting probability, emph{i.e.}, the smaller the intersecting probability, the smaller the optimal spectrum usage; (2) The intersecting probability is decided by the network topology, traffic distribution and routing scheme via a quadratic programming parameterized with a matrix of conflict coefficients. The effectiveness of our theoretical analysis has been validated by extensive numerical results. Meanwhile, our theoretical deductions also permit to give several constant approximation ratios for RSA algorithms.
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