Serving ML prediction pipelines spanning multiple models and hardware accelerators is a key challenge in production machine learning. Optimally configuring these pipelines to meet tight end-to-end latency goals is complicated by the interaction between model batch size, the choice of hardware accelerator, and variation in the query arrival process. In this paper we introduce InferLine, a system which provisions and manages the individual stages of prediction pipelines to meet end-to-end tail latency constraints while minimizing cost. InferLine consists of a low-frequency combinatorial planner and a high-frequency auto-scaling tuner. The low-frequency planner leverages stage-wise profiling, discrete event simulation, and constrained combinatorial search to automatically select hardware type, replication, and batching parameters for each stage in the pipeline. The high-frequency tuner uses network calculus to auto-scale each stage to meet tail latency goals in response to changes in the query arrival process. We demonstrate that InferLine outperforms existing approaches by up to 7.6x in cost while achieving up to 34.5x lower latency SLO miss rate on realistic workloads and generalizes across state-of-the-art model serving frameworks.