In this paper, we are motivated to establish a generalization of $_pR_q(\alpha,\beta;z)$ function in terms of incomplete exponential functions and obtain some properties of the incomplete exponential $_pR_q(\alpha,\beta;z)$ function. Further we give generating relations for incomplete exponential $_pR_q(\alpha,\beta;z)$ function. Some applications related to ground water pumping (Hydrology) modelling and probability theory have also been discussed.