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Fractional Brownian motion (fBm) became known as a useful model for Internet traffic incorporating its self-similar and long-range dependent properties. In this paper we derive end-to-end performance bounds for a through flow in a network of tandem queues under fBm cross traffic. We build on a previously derived sample path envelope for fBm, which possesses a Weibullian decay of overflow probabilities. We employ the sample path envelope and the concept of leftover service curves to model the remaining service after scheduling fBm cross traffic at a system. Using composition results for tandem systems from the stochastic network calculus we derive end-to-end statistical performance bounds for individual flows in networks under fBm cross traffic. We discover that these bounds grow in O(n(log n)1/2-2H) for n systems in series where H is the Hurst parameter of the fBm cross traffic. We show numerical results on the impact of the variability and the correlation of fBm traffic on network performance.