This paper considers the stochastic modeling of a system of non-identical units in which one main unit is initially operative and the other unit (called duplicate unit) is taken as spare in cold standby. The concept of priority for operation and repair of the main unit over the duplicate unit is introduced for developing the model. The repairman is subjected to failure while conducting repair activities of the main unit and so termed as conditional failure of the repairman. There is no failure of the repairman during repair of the duplicate unit. The random variables associated with different time points follow some probability distributions such as exponential in case of failure rates of the units and arbitrary distributions are assumed for repair and treatment rates. The behavior of some signified reliability characteristics has been observed for particular cases of the parameters. The results are shown graphically to through light on the variation of reliability measures in a concrete way. The study of these measures has been carried out by adopting thewell known stochastic process approach such as semi-Markov process and regenerative point technique.