9. Spectral Analysis
Society for Industrial and Applied Mathematics eBooks · 1999
The maximal eigenvalue η of R plays a very special role. We already saw in section 8.2 that it determines the speed of convergence of the linear progression algorithm. We show in this chapter that it also provides us with useful information about the dynamic behavior of the stochastic process itself. We give two illustrative examples in section 9.2. It is interesting to note that one may in principle construct the matrices R and G by computing from scratch their Jordan canonical form. We show in the second part of this chapter how this can be done for the matrix G (like in Chapter 8, we shall take the view that this matrix suffices to determine the whole distribution). We show in section 9.3 how the eigenvalues of G may be determined by using the polynomial matrix . The eigenvectors and generalized eigenvectors are characterized in section 9.4, and we show in section 9.5 how this might be used to actually compute the Jordan normal form of G. An entirely similar approach is directly applicable to the matrix R. The practical aspects are mentioned briefly at the end of the chapter. As in the rest of the book, in this chapter also, we concentrate mainly on QBDs, although the spectral approach applies to more general models. For instance, a thorough analysis of the spectral properties of M/G/1-type Markov chains may be found in Gail, Hantler, and Taylor [23].