Random packet CDMA (RP-CDMA) is a recently proposed random transmission scheme which has been designed from the beginning as a cross-layer method to overcome the restrictive nature of the Aloha protocol. Herein, we more precisely model its performance and investigate throughput and network stability. In contrast to previous works, we adopt the spread Aloha model for header transmission, and the performance of different joint detection methods for the payload data is investigated. Furthermore, we introduce performance measures for multiple access systems based on the diagonal elements of a modified multipacket reception matrix, and show that our measures describe the upper limit of the vector of stable arrival rates for a finite number of users. Finally, we simulate queue sizes and throughput characteristics of RP-CDMA with various receiver structures and compare them to spread Aloha.