Matrix-geometric solutions for bulk GI/M/1 systems with unbounded arrival groups
Dukhovny Alexander · Stochastic Models · 1999
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” formulas and show that the stationary pre-arrival 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”