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Model-serving systems have become increasingly popular, especially in real-time web applications. In such systems, users send queries to the server and specify the desired performance metrics (e.g., desired accuracy, latency). The server maintains a set of models (model zoo) in the back-end and serves the queries based on the specified metrics. This paper examines the security, specifically robustness against model extraction attacks, of such systems. Existing black-box attacks assume a single model can be repeatedly selected for serving inference requests. Modern inference serving systems break this assumption. Thus, they cannot be directly applied to extract a victim model, as models are hidden behind a layer of abstraction exposed by the serving system. An attacker can no longer identify which model she is interacting with. To this end, we first propose a query-efficient fingerprinting algorithm to enable the attacker to trigger any desired model consistently. We show that by using our fingerprinting algorithm, model extraction can have fidelity and accuracy scores within 1% of the scores obtained when attacking a single, explicitly specified model, as well as up to 14.6% gain in accuracy and up to 7.7% gain in fidelity compared to the naive attack. Second, we counter the proposed attack with a noise-based defense mechanism that thwarts fingerprinting by adding noise to the specified performance metrics. The proposed defense strategy reduces the attack's accuracy and fidelity by up to 9.8% and 4.8%, respectively (on medium-sized model extraction). Third, we show that the proposed defense induces a fundamental trade-off between the level of protection and system goodput, achieving configurable and significant victim model extraction protection while maintaining acceptable goodput (>80%). We implement the proposed defense in a real system with plans to open source.