The behavior of ann-component/n-1 cold standby system is analyzed over a horizon whose duration is assumed to be random. Under certain stochastic assumptions concerning the individual components, system performance measures are computed which are applicable for a system that is observed over a random interval of time. Two measures of system performance, expected accumulated uptime and random interval reliability, are used to demonstrate the computational tractability of the modeling perspective proposed in the paper. These measures may be used to optimally choose the number of standby components in the system. Optimization examples are included to illustrate the application of the theory developed.