The present study is confined on the stochastic analysis of a 2-out-of-2: G system be considering one identical unit as spare in cold standby. The faults of the system have been rectified by a single server who takes some time to arrival at the system. The failed unit is replaced by new one if its repair is not possible by the server in a given pre-specific time (called maximum repair time). The random variables associated with failure time, the time to which failed unit undergoes for replacement, repair and replacement time of the unit and arrival time of the server follow Weibull distribution with different probability density functions. The semi- Markov process and regenerative point techniques are used to derive the expressions for some important performance measures of the system model. The graphical behavior of these measures has been observed for arbitrary values of various parameters and costs. The particular cases of Weibull distribution have also been discussed to make the study more significant and applicable in idustries.