In this paper, we consider a continuous review $M/M/1$ queueing-inventory system with perishable items and $\left( {s,S}\right)$ policy, where the server takes multiple vacation which follows an exponential distribution when the on-hand inventory level is zero. Customers arrive according to a Poisson process, and the service time, the life time of each item and the lead time are assumed to be exponentially distributed. Firstly, the stability condition of the system is derived by using quasi-birth-and-death process theory. Then, we consider an $M/M/1$ inventory system with server's vacation where the service time is negligible. The steady-state distribution of the system is derived. Based on this distribution, we further analyze an $M/M/1$ queueing-inventory system with server's vacation and positive service time. The stationary joint distribution of the queue length, the on-hand inventory level and server status is obtained. Besides, some performance measures of the system and the expected cost function per unit time are calculated. Finally, the effect of the parameters on some important performance measures is analyzed by numerical examples. The optimal inventory policy and the optimal expected cost are calculated by using the genetic algorithm.