Machine-Type-Communication (MTC) is a key enabler for a variety of novel smart systems, such as smart grid, eHealth, Intelligent Transport System (ITS), and smart city, opening the area of the cyber physical systems. These systems may require the use of a huge number of MTC devices, which will put a great pressure on the whole network, i.e. Radio Access Network (RAN) and Core Network (CN) parts, resulting in the shape of congestion and system overload. Aiming at better evaluating the network performance under the existence of MTC traffic and also the effectiveness of the congestion control methods, the 3rd Generation Partnership Project (3GPP) group has proposed two traffic models: Uniform Distribution (over 60 s) and Beta Distribution (over 10 s). In this paper, a recursive operationbased analytical model, namely General Recursive Estimation (GRE), for modeling the performance of RACH procedure in the existence of MTC with Beta traffic is proposed. In order to show the effectiveness of our analytical model GRE, many metrics have been considered, such as the total number of MTC devices in each Random Access (RA) slot, the number of success MTC devices in each RA slot, and the Cumulative Distribution Function (CDF) of preamble transmission. Numerical results demonstrate the accuracy of GRE. Moreover, our model GRE could be used to model the performance of RACH procedure with any type of traffic.