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Network slicing allows different applications and network services to be deployed on virtualized resources running on a common underlying physical infrastructure. Developing a scalable system for the orchestration of end-to-end (E2E) mobile network slices requires careful planning and very reliable algorithms. In this paper, we propose a novel E2E Network Slicing Orchestration System (NSOS) and a Dynamic Auto- Scaling Algorithm (DASA) for it. Our NSOS relies strongly on the foundation of a hierarchical architecture that incorporates dedicated entities per domain to manage every segment of the mobile network from the access, to the transport and core network part for a scalable orchestration of federated network slices. The DASA enables the NSOS to autonomously adapt its resources to changes in the demand for slice orchestration requests (SORs) while enforcing a given mean overall time taken by the NSOS to process any SOR. The proposed DASA includes both proactive and reactive resource provisioning techniques). The proposed resource dimensioning heuristic algorithm of the DASA is based on a queuing model for the NSOS, which consists of an open network of G/G/m queues. Finally, we validate the proper operation and evaluate the performance of our DASA solution for the NSOS by means of system-level simulations.