This paper considers the transient reliability measures of a $k$ out of $n$ warm standby system with repairman's multiple vacations and replaceable repair facility where the standby time, the operating time and the repair time of components all follow exponential distributions. Employing the Markov process theory and the Laplace transform tool, we deduce the analytical expressions of the transient system availability, the rate of occurrence of failures at time $t$, the probability that the repairman is busy with broken components and the probability that the repair facility is being replaced. Further, we derive the system reliability and the function of the busy period of the repairman with the probabilistic properties of the phase type distribution. Based on this, we discuss the time-dependent behavior of the system performance measures under different initial system states, and validate the correctness of our results through some special cases.