Finding equilibrium probabilities of QBD processes by spectral methods when eigenvalues vanish
Consider spectral or eigenvalue methods for finding the equilibrium probabilities of quasi-birth-death processes (QBD processes) for the case where some eigenvalues are zero. This leads to the rather difficult problem of the analysis of continuous time Markov chains with block-structured infinitesimal generator matrices \(Q\) with multiple eigenvalues at zero. The author suggests to eliminate such eigenvalues by block elimination methods. Some of the reported resulting numerical difficulties such as instabilities can be bypassed by his algorithm. To accomplish this, the dimension of the largest Jordan block must be established, and some initial equations must be eliminated. The author also discusses the geometric and algebraic multiplicities of related eigenvalues in order to deal with the initial conditions. The theory is applied to two examples, one is dealing with a tandem queue, the other one with a shorter queue problem. Questions of numerical accuracy and computational complexity are addressed too.
scientific article; zbMATH DE number 2081263
scientific article