Abstract: We consider an M$^{X}$/M/1 queue with impatient customers and working vacations. Customs arrive in batches; during the vacation, the service is provided at a lower rate and the customer (except customers who are receiving service) may leave the system because the waiting time exceeds the expected waiting time. The model can provide some theoretical guidances for the decision maker of some service systems. Firstly, the steady-state equilibrium equation is established by building the model. Then, using the probability generating function method, the analytic expressions for the mean queue length in the normal and the working vacation are obtained. We further derive the analytic expressions for other performance measures such as: the mean sojourn time of customers; the mean sojourn time of customers arriving in the busy period; the rate of abandonment due to impatience. Finally, some numerical results are presented to demonstrate the effects of some parameters on the performance measures of the system.

Key words: queueing system, batch arrival, impatient customers, working vacation, probability generating function