Finding equilibrium probabilities of QBD processes by spectral methods when eigenvalues vanish

AbstractIn this paper, we discuss the use of spectral or eigenvalue methods for finding the equilibrium probabilities of quasi-birth–death processes for the case where some eigenvalues are zero. Since this leads to multiple eigenvalues at zero, a difficult problem to analyze, we suggest to eliminate such eigenvalues. To accomplish this, the dimension of the largest Jordan block must be established, and some initial equations must be eliminated. The method is demonstrated by two examples, one dealing with a tandem queue, the other one with a shorter queue problem

Spectrum of certain tridiagonal matrices when their dimension goes to infinity

AbstractThis paper deals with the spectra of matrices similar to infinite tridiagonal Toeplitz matrices with perturbations and with positive off-diagonal elements. We will discuss the asymptotic behavior of the spectrum of such matrices and we use them to determine the values of a matrix function, for an entire function. In particular we determine the matrix powers and matrix exponentials

TWO-STATION QUEUEING NETWORKS WITH MOVING SERVERS, BLOCKING, AND CUSTOMER LOSS ∗

Abstract. This paper considers a rather generalmodelinvolving two exponentialservers, each having its own line. The first line is unlimited, whereas the second line can only accommodate a finite number of customers. Arrivals are Poisson, and they can join either line, and once finished, they can either leave the system, or they can join the other line. Since the space for the second line is limited, some rules are needed to decide what happens if line 2 is full. Two possibilities are considered here: either the customer leaves prematurely, or he blocks the first server. The model also has moving servers, that is, the server at either station, while idle, can move to help the server of the other station. This model will be solved by an eigenvalue method. These eigenvalue methods may also prove valuable in other contexts