A relay network is one of the techniques to improve performances of wireless networks. A relay network uses one or more relay nodes to forward data under relaying scheme such as Decode-and-Forward (DF) scheme. Most of previous analytic studies on performances of relay networks assume that there are always enough data in the source node to be transmitted to the destination node. But this assumption is not always valid because the source node does not provide transmissions when the queue at the source node is empty. Therefore, in this thesis, we analyze performances of a DF relay network with a single relay under the unsaturated traffic condition. The packet arrival process is assumed to be a Markov modulated Poisson process, which well represents a bursty traffic such as multimedia traffic. We consider Rayleigh fading channels to describe fading channels. The performances in terms of delay is derived. To verify the validity of our analysis, we simulate the DF relay network and compare the simulation results with the analytical results.