Real eigenvalues of certain tridiagonal matrix polynomials, with queueing applications Original Research Article

Many queueing problems lead to tridiagonal lambda-matrices containing polynomials that have, except for the diagonal, non-negative coefficients. This paper deals with the question, addressed in the literature only for special cases, whether the eigenvalues corresponding to such lambda-matrices are real. In most cases, they are, as the theorems of this paper show, but sometimes, complex eigenvalues occur. Our results are derived by using Sturm sequences. In addition to simplifying the proofs of our theorems, Sturm sequences are also valuable to verify whether or not a given interval contains eigenvalues.