A bulk GI/M/1 system is considered where the arrival group size is not necessarily bounded. We apply the method of matrix unfoldings to the ‘roots’ formulae and show that the stationary prearrival queue length in such a queue has a ‘general’ matrix-geometric distribution of dimension m when the distribution of the arrival group size has a rational generating function with m poles outside of the unit disk. The ‘ratio’ matrix R is shown to be a unique power bounded solution of a characteristic matrix equation (in fact, ‘geometric’ power bounded); a formula is given for R in terms of the coefficients of a polynomial related to the scalar characteristic equation. The formula is shown to be equal to the m-th power of the transposed companion matrix of that polynomial (the latter formula was derived independently by Gail, Hantler and Taylor for a different case), so the eigenvalues of R are the m-th powers of the roots of the scalar characteristic equation. Convergence of the standard iteration process for R is proved when the arrival group size distribution is ‘compound-geometric’.
