The throughput of a packet switch is a major switch property, and therefore, of major interest to analyze it. An approximation of the throughput of a staged random selection algorithm with a single iteration under uniform for a three-stage Clos-network packet switch, also called a Space- Space-Space (S3) Clos-network packet switch, has been recently presented. However, the difference between this approximation and the actual throughput of the staged random selection algorithm is significant. To address this issue, this paper presents a theoretical throughput analysis of the staged random selection algorithm with a single iteration for a S3 Clos-network switch and show that the throughput is higher than that estimated by the existing approximation. Second, the paper extends the analysis to calculate the throughput of the staged random selection algorithm with multiple iterations by considering the analysis of the parallel iterative matching scheme, which is a random-based matching scheme for single-stage switches. The introduced derivation carefully considers the behavior of the selection algorithm at the switching modules in all three stages of the switch. The probability that a request reaches the third-stage modules is affected by the matching results at the second-stage modules. Numerical evaluations of the analytical formulas are performed. The results show that the staged random selection algorithm with multiple iterations for a S3 Clos-network switch without internal expansion can achieve 100% throughput under uniform traffic.