The determination of the steady-state distribution of the vacation queuing network is very important in many applications. The steady-state distribution vector can be obtained by solving a singular linear system. However, it is difficult to solve this system directly due to its huge size and complicated structure. A GMRES iterative method with a block lower triangular matrix preconditioner is proposed in this paper to solve this system. The preconditioner GMRES method has the advantages of easy construction and rapid convergence. Numerical examples demonstrate the superiority of the proposed algorithm.