The exponential growth of the Internet of things (IoT) necessitates long-lasting, low-cost, and sustainable green IoT network designs. In this paper, relay deployment is proposed to increase the battery lifetime of IoT devices in smart homes. Then, the effect of this deployment on the battery lifetime of IoT sensors is analyzed and evaluated in large scales using tools from stochastic geometry. Such analysis is more realistic as it captures real-world spatial correlation of IoT devices and reveals better insights to network design. Further, a closed-form expression of the battery consumption model is presented, and its cumulative density function is derived analytically. Numerical and analytic results match and confirm that relay deployment is a promising opportunity that can extend smart homes' IoT battery lifetime from about one year to several years.