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Efficient design of online power allocation policies relies strongly on convex-analytic and optimization-theoretic properties of the optimization problem on hand. In this context we study the optimization of power allocation in cellular networks with so-called best-effort traffic. Our results exhibit a specific role of link QoS parameters, for which the dependence on the corresponding link SINR is log-convex. In such case the region of achievable QoS vectors is shown to be convex, the considered problem is globally solvable and can be easily transformed into a favorable convex form.