Abstract-The growth of Internet video traffic imposes a severe capacity problem in today's Content Delivery Network (CDN). Rate-adaptive streaming technologies, such as the Dynamic Adaptive Streaming over HTTP (DASH) standard, reinforces this problem in the core CDN infrastructure since delivering one video means delivering multiple representations for an aggregated bit-rate that is commonly over 10 Mbps. In this paper, we explore better trade-offs between CDN infrastructure cost and Quality of Experience (QoE) of the end-users for live broadcast video streaming applications. We consider in particular underprovisioned CDN networks, our goal being to maximize the QoE for the population of heterogeneous end-users despite the lack of resources in the intermediate CDN equipments. We show that previous theoretical models based on elastic bit-rates do not fit for this context. We propose a user-centric discretized streaming model where the satisfaction of end-users is related to the context and where a stream has to be either delivered in its entirety, or not delivered at all. We first formulate an Integer Linear Program (ILP) that achieves the optimal delivery through a multi-tree delivery overlay. The evaluation of the ILP shows the benefits of this model. We then design a practical system by revisiting the three main algorithms implemented in CDN: userto-server assignment, content placement and content delivery. At last, we use a realistic trace-driven large-scale simulator to study the performances of our system. In particular, we show that the population of users is reasonably well served (three quarters of the population do not experience degradation) even when the CDN infrastructure experiences a severe underprovisioning (less than half of the required infrastructure).