In a distributed multihop mesh/relay network (e.g., wireless ad hoc/sensor network, cellular multihop network), each node acts as a relay node to forward data packets from other nodes. These nodes are often energy-limited and also have limited buffer space. Therefore, efficient power saving mechanisms (e.g., sleeping mechanisms) are required so that the lifetime of these nodes can be extended while at the same time the quality of service (QoS) requirements (e.g., packet delay and packet loss rate) for the relayed packets can be satisfied. In this paper, we present a novel queueing analytical framework to study the tradeoff between the energy saving and the QoS at a relay node. Specifically, by modeling the bursty traffic arrival process as a MAP (Markovian Arrival Process) and the packet service process as having a phase-type (PH) distribution, we model each node as a MAP/PH/1 nonpreemptive priority queue. Here, the relayed packets and the node's own packets form two priority classes and the medium access control (MAC)/physical (PHY) layer protocol in the transmission protocol stack acts as the server process. Moreover, we use a phase-type vacation model for the energy-saving mechanism in a node when the MAC/PHY protocol refrains from transmitting in order to save battery power. Two different power saving mechanisms due to the standard exhaustive and the number-limited exhaustive vacation models (both in multiple vacation cases) are analyzed to study the tradeoff between the QoS performance of the relayed packets and the energy saving at a relay node. Also, an optimization formulation is presented to design an optimal wakeup strategy for the server process under QoS constraints. We use matrix-geometric method to obtain the stationary probability distribution for the system states from which the performance metrics are derived. Using phase-type distribution for both the service and the vacation processes and combining the priority queueing model with the vacation queueing model make the analysis very general and comprehensive.