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We consider a family of stochastic processes built from infinite sums of independent positive random functions on BBR+. Each of these functions increases linearly between two consecutive negative jumps, with the jump points following a Poisson point process on BBR+. The motivation for studying these processes stems from the fact that they constitute simplified models for TCP traffic. Such processes bear some analogy with Lévy processes, but are more complex since their increments are neither stationary nor independent. In the work of Barral and Lévy Véhel, the Hausdorff multifractal spectrum of these processes was computed. We are interested here in their Large Deviation and Legendre multifractal spectra. These “statistical” spectra are seen to give, in this case, a richer information than the “geometrical” Hausdorff spectrum. In addition, our results provide a firm theoretical basis for the empirical discovery of the multifractal nature of TCP traffic.