A computational method for the boundary vector of a BMAP/G/1 queue

For calculations of the boundary vector arising in a BMAP/G/1 queue, we consider a spectral method based on eigenvalues and eigenvectors without assuming a structure of a BMAP. We define a nonlinear function of the determinant of a matrix function. It is proved that there are M zeros of the nonlinear function on a disk in a complex plane, where M is the size of rate matrices of a BMAP. And for the calculation of all the zeros, we propose a modification of the Durand–Kerner method which is known as an iterative method for calculating all zeros of a polynomial simultaneously. Spectral methods for calculating the stationary probabilities just after service completion epochs and at arbitrary time are given.