A batch arrival M^{［X］}／M／1 with start-up time and single working vacations is concerned. A model of three-dimensional Markov chain is established. Then, the PGF and the stochastic decomposition of stea-dy-state are given. The two parameters addition theorem of condision Erlang distribution is obtained, with which the upper and lower bounds of steadystate waiting time is given under Laplace transform order. The mean queue length, the mean waiting time of the upper and lower bounds, the mean sojourn time are also given. Finally, some numerical examples to verify the conclusions.