Fog-aided network architectures for 5G systems encompass wireless edge nodes, referred to as remote radio systems (RRSs), as well as remote cloud center (RCC) processors, which are connected to the RRSs via a fronthaul access network. RRSs and RCC are operated via network functions virtualization, enabling a flexible split of network functionalities that adapts to network parameters such as fronthaul latency and capacity. This paper focuses on uplink communications and investigates the cloud-edge allocation of two important network functions, namely, the control functionality of rate selection and the data-plane function of decoding. Three functional splits are considered: 1) distributed radio access network, in which both functions are implemented in a decentralized way at the RRSs; 2) cloud RAN, in which instead both functions are carried out centrally at the RCC; and 3) a new functional split, referred to as fog RAN (F-RAN), with separate decentralized edge control and centralized cloud data processing. The model under study consists of a time-varying uplink channel with fixed scheduling and cell association in which the RCC has global but delayed channel state information due to fronthaul latency, while the RRSs have local but more timely CSI. Using the adaptive sumrate as the performance criterion, it is concluded that the F-RAN architecture can provide significant gains in the presence of user mobility.