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Three-dimensional (3D)-stacked memories, such as the Hybrid Memory Cube (HMC), provide a promising solution for overcoming the bandwidth wall between processors and memory by integrating memory and logic dies in a single stack. Such memories also utilize a network-on-chip (NoC) to connect their internal structural elements and to enable scalability. This novel usage of NoCs enables numerous benefits such as high bandwidth and memory-level parallelism and creates future possibilities for efficient processing-in-memory techniques. However, the implications of such NoC integration on the performance characteristics of 3D-stacked memories in terms of memory access latency and bandwidth have not been fully explored. This paper addresses this knowledge gap (i) by characterizing an HMC prototype using Micron's AC-510 accelerator board and by revealing its access latency and bandwidth behaviors; and (ii) by investigating the implications of such behaviors on system- and software-level designs. Compared to traditional DDR-based memories, our examinations reveal the performance impacts of NoCs for current and future 3D-stacked memories and demonstrate how the packet-based protocol, internal queuing characteristics, traffic conditions, and other unique features of the HMC affects the performance of applications.