Matrix structures in queuing models

Abstract

Matrix structures are ubiquitous in linear algebra problems stemming from scientific computing, engineering and from any mathematical models of the real world. They translate, in matrix language, the specific properties of the physical problem.Often, structuredmatrices reveal themselves in a clear formand apparently seem to show immediately all their properties. Very often, structures are hidden, difficult to discover, and their properties seem to be hardly exploitable. In this note, we rely on the research area of queueing models and Markov chains to present, motivate, and analyze from different points of view the main matrix structures encountered in the applications. We give an overview of the main tools from the structured matrix technology including Toeplitz matrices—with their asymptotic spectral properties and their interplay with polynomials—circulant matrices and other trigonometric algebras, preconditioning techniques, the properties of displacement operators and of displacement rank, fast and superfast Toeplitz solvers, Cauchy-like matrices, and more. Among the hidden structures, besides the class of Toeplitz-like matrices, we just recall some properties of rank-structured matrices like quasi-separable matrices. Then we focus our attention to finite dimensional and to infinite dimensional Markov chains which model queueing problems. These chains, which involve block Hessenberg block Toeplitzmatrices of finite and infinite size, are efficiently solved by using the tools of the structured matrix technology introduced in the first part. In this note we also provide pointers to some related recent results and to the current research.