Spectral expansion solution for a class of Markov models: application and comparison with the matrix-geometric method

Generalized QBD Processes, Spectral Expansion and Performance Modeling Applications

“…It is confirmed by a number of works that the spectral expansion method is better than the matrix geometric method in a number of aspects [5,32,33,39]. It is observed that the spectral expansion method is proved to be a mature technique for the performance analysis of various problems [5-17, 20-25, 27, 28, 31, 32, 38, 39, 49-51, 53, 54, 52, 58].…”

“…A QBD process is a Markov process on a twodimensional lattice, finite in one dimension (finite or infinite in the other). A state is described by two integer-valued random variables: the one in the finite dimension is the phase and the other is the level [37,39,41]. Transitions in a QBD process are possible within the same level or between adjacent levels.…”

“…It can also be shown that for certain parameter values the computation requirements are uncertain and formidably large. The fourth method is known as spectral expansion method [5,6,39]. It is based on expressing the invariant vector of the process in terms of eigenvalues and left eigenvectors of a certain matrix polynomial.…”

Spectral Expansion Solutions for Markov-Modulated Queues

“…The idea of the spectral expansion solution method has been known for some time (e.g., see Neuts [18]), but there are rather few examples of its application in the performance evaluation literature. Some instances where that solution has proved useful are reported in Elwalid et al [3], and Mitrani and Mitra [17]; a more detailed treatment, including numerical results, is presented in Mitrani and Chakka [16]. More recently, Grassmann [7] has discussed models where the eigenvalues can be isolated and determined very efficiently.…”

“…More recently, Grassmann [7] has discussed models where the eigenvalues can be isolated and determined very efficiently. Some comparisons between the spectral expansion and the matrix-geometric solutions can be found in [16] and in Haverkort and Ost [8]. The available evidence suggests that, where both methods are applicable, spectral expansion is faster even if the matrix R is computed by the most efficient algorithm.…”