In this paper, we study the transition matrix of M/G/1 type, The radius of conver-gence is discussed, conditions onf the a-classificaion of the states are obtained and expressions of the β-invariant measure are constructed. The censoring technique is generalized to deal with nonnegative matices which may be neither stochastic nor substochastic. This allows us to prove a factorization result for the discounted transition matrix. This factorization provides a unifiedalgorithmic approach for expressing theβ-invariant measure for transition matrices with a block-sturcture, including the matrix of M/G/1 type.