  In this paper, we develop an analytical framework for the statistical
analysis of the battery recharging time (BRT) in reconfigurable intelligent
surfaces (RISs) aided wireless power transfer (WPT) systems. Specifically, we
derive novel closed-form expressions for the probability density function
(PDF), cumulative distribution function, and moments of the BRT of the radio
frequency energy harvesting wireless nodes. Moreover, closed-form expressions
of the the PDF of the BRT is obtained for two special cases: i) when the RIS is
equipped with one reflecting element (RE), ii) when the RIS consists of a large
number of REs. Capitalizing on the derived expressions, we offer a
comprehensive treatment for the statistical characterization of the BRT and
study the impact of the system and battery parameters on its performance. Our
results reveal that the proposed statistical models are analytically tractable,
accurate, and efficient in assessing the sustainability of RIS-assisted WPT
networks and in providing key design insights for large-scale future wireless
applications. For example, we demonstrate that a 4-fold reduction in the mean
time of the BRT can be achieved by doubling the number of RIS elements. Monte
Carlo simulation results corroborate the accuracy of the proposed theoretical
framework.
