To evaluate the performance of multistage sleep telecommunication systems, this paper presents a bounded multi-vacation queue model. The energy consumption predicted by this model, shows an average error rate of 0.0177 and the delay (predicted by the same model) shows an average error rate of 0.0655. Both error rates were calculated over 99 instances. An optimisation algorithm is also proposed to minimise the energy consumption whilst not violating the delay bound. Based on the model, numerical results,which quantify the performance of current state-of-art multistage sleep 5G base stations, are presented.