The IEEE 802.11ah standard, marketed as Wi-Fi Halow, introduces a new channel access mechanism called the Restricted Access Window (RAW), aiming to provide connectivity for the Internet of Things (IoT) applications over broad areas. RAW aspires to alleviate the contention by splitting the channel access into periods and allocating each period to a given group of stations. This paper develops an analytical framework based on Probability and Renewal theories for modeling and evaluating an IEEE 802.11ah-based network implementing the RAW mechanism. We consider a Rayleigh-fading channel with the presence of the capture effect: a realistic scenario for IoT networks deployed in dense urban environments. Considering a single-hop scenario of stations randomly distributed around an Access Point (AP) and the power attenuation of transmitted packets, we model the channel access under capture awareness. As the RAW mechanism presents a time-limited contention for channel access, we develop a counting process that tracks transmissions up to the end of the contention time interval. Henceforth, we evaluate the network performance in terms of throughput. We meticulously validate the derived analytical results through extensive campaigns of discrete-event simulations. Our study evaluates the impact of different parameters on the overall performance, including the contention time, the number of stations, the number of groups, and the capture threshold. We henceforth study the impact of the capture effect on enhancing the network performance under the grouping feature introduced by the RAW mechanism. This work contributes to developing an analytical modeling framework to evaluate the performance of time-limited random access mechanisms accurately and can be an excellent basis for proposing practical scheduling algorithms to configure the RAW mechanism under non-ideal channel conditions.