Queueing models, block Hessenberg matrices and the method of Neuts

The authors introduce a numerical method to compute the stationary probability vector of queueing models whose infinitesimal generator is of block Hessenberg form. It is shown that the stationary probability vector is equal to the first column of the inverse of the coefficient matrix. Furthermore, it is shown that the first column of the inverse of an upper (or lower) Hessenberg matrix may be obtained in a relatively small number of operations. Together, these results allow us to define a powerful algorithm for solving certain queueing models. The efficiency of this algorithm is discussed and a comparison with the method of Neuts is undertaken. A relationship with the method of Gaussian elimination is established and used to develop some stability results.