An analytical foundation for resource management in wireless communication

The maximization of a ratio of the form f(x)/x, with f some "S-curve", plays a central role in several important problems involving resource management for data communication over a wireless medium. This includes decentralized power control, power and data rate assignment for maximal network throughput in a 3G-CDMA context, and power and coding rate choice for multimedia files which have been scalably encoded, as with the JPEG-2000 and MPEG-4 standards. In this note, the ratio f(x)/x, where f is a real-valued, univariate "s-shaped" function, is shown to be quasiconcave, and to always have a unique global maximizer, which can be identified graphically. The analysis is strictly based on geometrical properties derived from the sigmoidal shape, imposing no specific algebraic functional form ("equation") on the function. Hence, it applies to a wide range of practical situations.