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Many storage cache allocation methods use the miss ratio curve (MRC) to improve cache efficiency. However, they have focused only on single-tier cache architectures and require the whole MRC as input for cache management, while modern datacenters embrace hierarchical caching architectures to maximize resource utilization. Generating the MRC for multi-tier caches—we call it the miss ratio function—is far more challenging due to different eviction policies and capacities in each cache tier. We introduce eMRC, a multi-dimensional miss ratio approximation technique, to enable efficient MRC generation for multi-tier caching. Our approach uses a novel multi-dimensional performance cliff removal method and convex hull approximation technique to efficiently generate a multi-dimensional MRC without cliffs using a small number of sampling points. To demonstrate the benefits of eMRC, we designed ORCA, a multi-tier cache management framework that orchestrates caches residing in different hierarchies through eMRC and provides efficient multi-tier cache configurations to cloud tenants with diverse service level objectives. We evaluate the performance of our eMRC approximation technique and ORCA with real-world datacenter traces.