EIGENVALUES OF THE TRANSITION RATE MATRICES IN A GI/E_k/m QUEUEING SYSTEM

Abstract

In this paper, the eigenvalues of the transition rate matrices in a GI/E_k/m queueing system are analytically obtained for any k and m. First, it is supposed that each channel is distinguishable from others, as a semi-homogeneous queueing system. Here, a transition rate matrix S_m(θ) and the eigenvalues of it are easily found by the mathematical induction on m, for any fixed k, where 6 is a complex parameter. It can be shown that the matrix. S_m(θ) is similar to a diagonal matrix, and that an eigenvalue of S_m(θ) takes the form of a m-sum of d(j)'s, where d(j) is the eigenvalue of S_l(θ). On the other hand, the transition rate matrix T_m(θ) in a homogeneous queueing system is different from S_m(θ) in appearance. But T_m(θ) can be made from S_m(θ) by using an equivalence relation. Then it can be shown that the matrix T_m(θ) is similar to a diagonal matrix, and the matrices T_m(θ) and. S_m(θ) have the same eigenvalues except the multiplicity. Finally, to clarify the description, an example (k = 3 and m = 3) is shown.