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Ensuring the conformance of a service system's end-to-end delay to service level agreement (SLA) constraints is a challenging task that requires statistical measures beyond the average delay. In this paper, we study the real-time prediction of the end-to-end delay distribution in systems with composite services such as service function chains. In order to have a general framework, we use queueing theory to model service systems, while also adopting a statistical learning approach to avoid the limitations of queueing-theoretic methods such as stationarity assumptions or other approximations that are often used to make the analysis mathematically tractable. Specifically, we use deep mixture density networks (MDN) to predict the end-to-end distribution of the delay given the network's state. As a result, our method is sufficiently general to be applied in different contexts and applications. Our evaluations show a good match between the learned distributions and the simulations, which suggest that the proposed method is a good candidate for providing probabilistic bounds on the end-to-end delay of more complex systems where simulations or theoretical methods are not applicable.