Despite a large body of literature and methods devoted to the traffic matrix estimation problem, the inference of traffic flows volume from aggregated data represents a key subject facing the evolution of next generation networks. This is a particular problem in large-scale carrier networks, for which efficient, accurate and stable methods for traffic matrix modeling and estimation are vital and challenging to conceive. In the short-term, estimation methods must be efficient and stable to allow crucial real-time tasks such as on-line traffic monitoring. In the long-term, methods must provide an accurate picture of the traffic matrix to tackle problems such as network planning, design, and dimensioning. In this paper we present and compare two efficient methods for on-line traffic matrix estimation. Based on an original parsimonious linear model for traffic flows in large-scale networks, we present a simple approach to compute an accurate traffic matrix from easily available link traffic measurements. We further extend the validation of this parsimonious model to three operational backbone networks. We analyze in depth a method to recursively estimate the traffic matrix, studying the drawbacks and omissions of the former algorithm and proposing new extensions to solve these problems. We finally perform a comparative analysis of the performance of both methods in two operational backbone networks, taking into account significant aspects such as accuracy, stability, scalability, and on-line applicability.