A wireless sensor network (WSN) consists of a large number of sensor nodes deployed over a geographical area and have broad applications like environment monitoring, target tracking and surveillance. The unique characteristics of WSNs such as limited bandwidth, computing capacity, data delivery delay and severe energy constraints make their design more challenging. A critical issue in WSN is the limited availability of energy and hence optimizing energy is very important. This paper proposes an energy minimization technique based on queue threshold using N-policy M / M / 1 queueing model of a sensor node in a cluster acting as cluster member (CM) and cluster head (CH). We develop an analytical model of a cluster based sensor network based on queue threshold using N-policy M / M / 1 queueing model. The performance of the proposed model is analysed in terms of average energy consumption and mean delay when the node acts as CM and CH. We also derive the expression for the optimal value of queue threshold (N*) for which the node consumes minimum energy. Results show that the average energy consumption savings is high when the node acts as CH when compared to node acting as CM and the trade-off exist between the average energy consumption and mean delay is explored. We perform simulations and the results obtained show that the simulation results match with the analytical results thus validating the accuracy of the approach.