In this review, we describe historical and recent developments towards tackling a century-long challenge in teletraffic theory, namely the evaluation of blocking probability in overflow systems with mutual overflow. Such systems have many applications in a variety of telecommunications and service systems, including wireless communications, cloud computing, intensive care, and emergency services, and various methods have been developed over the past century to address this challenge. In particular, the recent development of the Information Exchange Surrogate Approximation (IESA) provides significantly increased accuracy and robustness compared to previous approximation methods of its kind while also providing high computational efficiency not available via simulation or exact analysis. To the best of our knowledge, IESA is the first approximation method to combine high levels of accuracy, robustness, and computational efficiency when evaluating blocking probability in overflow systems with mutual overflow, and thus forms a major breakthrough in this century-long effort.