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We are entering a rapidly unfolding future driven by the delivery of real-time computation services, such as industrial automation and augmented reality, collectively referred to as AgI services, over highly distributed cloud/edge computing networks. The interaction intensive nature of AgI services is accelerating the need for networking solutions that provide strict latency guarantees. In contrast to most existing studies that can only characterize average delay performance, we focus on the critical goal of delivering AgI services ahead of corresponding deadlines on a per-packet basis, while minimizing overall cloud network operational cost. To this end, we design a novel queuing system able to track data packets' lifetime and formalize the delay-constrained least-cost dynamic network control problem. To address this challenging problem, we first study the setting with average capacity (or resource budget) constraints, for which we characterize the delay-constrained stability region and design a near-optimal control policy leveraging Lyapunov optimization theory on an equivalent virtual network. Guided by the same principle, we tackle the peak capacity constrained scenario by developing the reliable cloud network control (RCNC) algorithm, which employs a two-way optimization method to make actual and virtual network flow solutions converge in an iterative manner. Extensive numerical results show the superior performance of the proposed control policy compared with the state-of-the-art cloud network control algorithm, and the value of guaranteeing strict end-to-end deadlines for the delivery of next-generation AgI services.