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In this paper, we consider the network power minimization problem in a downlink cloud radio access network (C-RAN), taking into account the power consumed at the baseband unit (BBU) for computation and the power consumed at the remote radio heads and fronthaul links for transmission. The power minimization problem for transmission is a fast time-scale issue, whereas the power minimization problem for computation is a slow time-scale issue. Therefore, the joint network power minimization problem is a mixed time-scale problem. To tackle the time-scale challenge, we introduce large system analysis to turn the original fast time-scale problem into a slow time-scale one that only depends on the statistical channel information. In addition, we propose a bound improving branch-and-bound algorithm and a combinational algorithm to find the optimal and suboptimal solutions to the power minimization problem …