We propose and analyse a medium access control (MAC) protocol for low energy critical infrastructure monitoring (LECIM) networks. As the packet drop probability plays crucial role in LECIM applications, we propose framed slotted aloha based MAC for LECIM using linearly increased contention window size to reduce the packet drop probability. We present a mathematical model of our proposed MAC under both saturated and nonsaturated traffic scenarios. We use probabilistic approach to find the performance metrics such as collision probability, packet drop probability, throughput, and energy consumption. Also, we obtain the probability-generating function of the head-of-line (HoL) delay of packet. The analytical results match with simulations. Our results can be used in the design of a system by providing the optimum system parameters for endpoints satisfying the given quality of service requirements on packet drop probability, energy consumption, and delay.