Lexicographic fairness in WDM optical cross-connects

We consider fair allocation of sessions at the outputs of optical cross-connects employing wavelength division multiplexing (WDM). Each session consists of traffic on one or more wavelengths (channels). We identify lexicographic fairness as the most appropriate fairness criterion that is relevant to this setting. Achieving a fair lexicographic solution, commonly referred to as lexicographic optimality (LEX), is trivial and polynomial-time computable when any incoming wavelength can be converted to any outgoing wavelength (full conversion). This is not apparent in the practical and realistic case of limited conversion. We prove that LEX is also polynomial-time computable for the limited conversion case by reducing the problem to a min-cost max-flow optimization objective in network flows. We also motivate, formulate and solve a stronger variant of lexicographic optimality that we refer to as worst-case fair lexicographic optimal (W-LEX). Although our effective setting is an optimization problem in bipartite graphs (the request graph is bipartite), the network-flow based algorithms are applicable to unit capacity graphs in general. Further, we provide fast polynomial-time algorithms that furnish solutions for LEX and the W-LEX optimality problems for arbitrary bipartite graphs (i.e. arbitrary wavelength-conversion rules) and are computationally less expensive than network-flow methods. Finally we report simulation results to validate our findings.