In this paper, we study a continuous review $M/M/1$ queueing-inventory system with customer batch demand, lost sales and multiple working vacations, where the server takes multiple working vacations when the inventory is zero. Customers arrive according to a Poisson process. The service time, the lead time and the working vacation time are assumed to follow exponential distributions, and the batch size of demands is assumed to follow a general distribution. The steady-state condition and the steady-state probability distribution of the system are derived by using quasi-birth-and-death process and matrix geometry solution method, and then the computational formulae for the steady-state performance indicators of the system are obtained. The long-term average cost function of the system is established based on the performance measures, and the optimal inventory strategy and the optimal cost of the system are analyzed by numerical examples.