In this paper, a simple and elegant geometric waterfilling (GWF) approach is proposed to solve the unweighted and weighted radio resource allocation problems. Unlike the conventional water-filling (CWF) algorithm, we eliminate the step to find the water level through solving a non-linear system from the Karush-Kuhn-Tucker conditions of the target problem. The proposed GWF requires less computation than the CWF algorithm, under the same memory requirement and sorted parameters. Furthermore, the proposed GWF avoids complicated derivation, such as derivative or gradient operations in conventional optimization methods, while provides insights to the problems and the exact solutions to the target problems. Most importantly, the GWF can be extended to solve a generalized form of radio resource allocation problem with more stringent constraints: (weighted) optimization problem with individual peak power constraints (GWFPP), and to include (weighted) group bounded power constraints (GWFGBP). On the other side, the CWF cannot solve these two general forms of the RRA problems, due to the difficulty to solve the non-linear system with multiple non-linear equations and inequalities in multiple dual variables. Optimality of the proposed water-filling solution is strictly proved for each of the proposed algorithms. Furthermore, numerical results show that the proposed approach is effective, efficient, easy to follow and insight-seeing.Index Terms-Water-filling, channel capacity, optimal radio resource allocation, multi-user MIMO (MU-MIMO), cognitive radio, optimization methods.

Abstract-With files proactively stored at base stations (BSs), mobile edge caching enables direct content delivery without remote file fetching, which can reduce the end-to-end delay while relieving backhaul pressure. To effectively utilize the limited cache size in practice, cooperative caching can be leveraged to exploit caching diversity, by allowing users served by multiple base stations under the emerging user-centric network architecture. This paper explores delay-optimal cooperative edge caching in large-scale user-centric mobile networks, where the content placement and cluster size are optimized based on the stochastic information of network topology, traffic distribution, channel quality, and file popularity. Specifically, a greedy content placement algorithm is proposed based on the optimal bandwidth allocation, which can achieve (1 − 1/e)-optimality with linear computational complexity. In addition, the optimal user-centric cluster size is studied, and a condition constraining the maximal cluster size is presented in explicit form, which reflects the tradeoff between caching diversity and spectrum efficiency. Extensive simulations are conducted for analysis validation and performance evaluation. Numerical results demonstrate that the proposed greedy content placement algorithm can reduce the average file transmission delay up to 45% compared with the non-cooperative and hit-ratio-maximal schemes. Furthermore, the optimal clustering is also discussed considering the influences of different system parameters.

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Providers should maintain a high index of suspicion for NS among patients with syphilis, particularly those with HIV infection.

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