In the current work, we perform reliability analysis of a repairable data transmission system with the use of the elaborated mathematical model of a closed heterogeneous redundant cold standby system. The system consists of a single repair unit with two different data sources with an exponential distribution function (DF) of uptime and a general independent DF of the repair time of its components. We study the system- level reliability, defined as the steady-state probability (SSP) of failure-free system operation. The proposed analytical methodology made it possible to evaluate the reliability of the entire system in case of failure of its components. Explicit analytic expressions and asymptotic estimates are obtained for the SSP of the system and the SSP of failure-free system operation. The obtained results which allow analyzing other operational characteristics of the system with respect to the performance of the backup components. The problem of analyzing the sensitivity of the reliability characteristics of the system under consideration to the types of repair time distributions was also studied. Obtained formulas showed that there is a clear dependence of these characteristics on the DF types of the recovery time of system's components. However, numerical studies and analysis of the constructed graphs showed that this dependence becomes vanishingly small with a quick restoration of system components.