This paper considers an amplify-and-forward relay network with fading states. Amplify-and-forward scheme (along with its variations) is the core mechanism for enabling cooperative communication in wireless networks, and hence understanding the network stability region under amplify-andforward scheme is very important. However, in a relay network employing amplify-and-forward, the interaction between nodes is described in terms of real-valued "packets" (signals) instead of discrete packets (bits). This restrains the relay nodes from reencoding the packets at desired rates. Hence, the stability analysis for relay networks employing amplify-and-forward scheme is by no means a straightforward extension of that in packetbased networks. In this paper, the stability region of a four-node relay network is characterized, and a simple throughput optimal algorithm with joint scheduling and rate allocation is proposed.