An application of exact linear algebra to capacity planning models

This note illustrates a novel application of exact linear algebra to performance evaluation and stochastic modeling. We focus on queueing network models, which are high-level abstractions of Markov chains used in capacity planning of computer and communication systems [6, 7]. Until recently, it was prohibitively expensive to compute exact solutions for these models when they describe hundreds or thousands of users interacting with a network of servers, a case of large practical application when sizing web architectures. Here, we overview a new approach, which we have recently proposed [3, 4], that overcomes this limitation by means of a linear matrix difference equation that strictly requires exact linear algebra to be evaluated. Exact linear algebra is required in our method because of uncontrollable numerical instabilities that arise if round-off errors are introduced in the recursive evaluation of the matrix difference equation. This difficulty is also exasperated by the “astronomical” growth of the number of digits of the operands, which can be as large as 101000−1010000. The application presented in this paper shows a case where accepting the computational costs of exact algebra to stabilize the numerical evaluation leads to massive computational savings compared to established approaches based on standard (inexact) linear algebra. The remainder of this work is as follows. In Section 2, we give minimal background about queueing network models and explain how they can be solved recursively. We point to textbooks such as [7] for extensive background on queueing networks. In Section 3, we discuss the new solution approach based on a matrix difference equation. The numerical properties of the method are discussed in Section 3.1, where we argue that exact algebra is the only viable approach to prevent numerical instability. For the reader interested in experimenting with the problem, we report an example in Section 3.2. Finally, we draw conclusions in Section 4. Additional material, including a MAPLE implementation of the matrix difference equation approach to queueing networks, can be obtained by contacting the author or from his homepage.