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Important problems in wireless networks can often be solved by computingfixed points of standard or contractive interference mappings, and theconventional fixed point algorithm is widely used for this purpose. Knowingthat the mapping used in the algorithm is not only standard but alsocontractive (or only contractive) is valuable information because we obtain aguarantee of geometric convergence rate, and the rate is related to a propertyof the mapping called modulus of contraction. To date, contractive mappings andtheir moduli of contraction have been identified with case-by-case approachesthat can be difficult to generalize. To address this limitation of existingapproaches, we show in this study that the spectral radii of asymptoticmappings can be used to identify an important subclass of contractive mappingsand also to estimate their moduli of contraction. In addition, if the fixedpoint algorithm is applied to compute fixed points of positive concavemappings, we show that the spectral radii of asymptotic mappings provide uswith simple lower bounds for the estimation error of the iterates. An immediateapplication of this result proves that a known algorithm for load estimation inwireless networks becomes slower with increasing traffic.

Spectral radii of asymptotic mappings and the convergence speed of the standard fixed point algorithm