This paper considers the M/G/1 queueing system with randomized overhaul $\langle p,Y \rangle $-policy, in which when the system becomes empty, the system is overhauled with probability $p ( 0\le p\le 1 )$ and the length of overhauling time is a random variable with general distribution. Firstly, we analyze the embedded Markov chain of queue length, and obtain the probability generating function of its steady-state distribution. Secondly, the transient distribution of the queue size at any time $t$ is discussed, and the expressions of the Laplace transform of the transient queue length distribution with respect to time $t$ are presented. Meanwhile, based on the transient analysis of the queue length, the recursive formulas of the steady-state distribution of the queue length are obtained by employing L'Hospital rule. Furthermore, the stochastic decomposition structure of the steady-state queue size is presented. Finally, numerical examples are provided to determine the optimal overhaul policy for economizing the system cost under a given cost structure.