Exploring correctness and accuracy of solutions to matrix polynomial equations in queues

Abstract

Spectral expansion and matrix analytic methods are important solution mechanisms for matrix polynomial equations. These equations are encountered in the steady-state analysis of Markov chains with a semi-finite or finite two dimensional lattice of states, which describe a significant class of finite and infinite queues. We prove that the limited size of the eigenspectrum of the matrix geometric representation used in matrix analytic solution mechanisms confines its applicability to systems with a number of eigenvalues less than or equal to the dimension of the matrices used to form the solution. As well as proving this limitation, we relate our experience of a practical queue with generalized exponential traffic whose steady state cannot be represented using one or two rate matrices. We also provide an explanation for the numerical issues creating difficulty in finding matrix geometric solutions for finite queues. While we have not found a solution to these numerical issues, we do outline the steps required to enable complete matrix geometric solutions with larger eigenspectra, but which may not be efficient. On the other hand, we identify a case where care must be taken when using spectral expansion. Essentially, the eigensystem of a finite queue degenerates at saturation. We therefore formulate an enhanced spectral expansion method using generalized eigenvectors, which we prove gives a complete solution, even at saturation. We conclude that the state of the art requires the use of efficient matrix analytic methods where applicable, but correct solution in the general case is currently only guaranteed using generalized spectral expansion. We suggest that use of matrix analytic tools directed toward efficiency for solving a given queueing system should be preceded by an analysis of the eigensystem of the solution through spectral expansion, whether algebraic or numerical, to verify that the solutions produced by such tools are correct.