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Providing differentiated services to meet the unique requirements of different use cases is a major goal of the fifth generation (5G) telecommunication networks and will be even more critical for future 6G systems. Fulfilling this goal requires the ability to assure quality of service (QoS) end to end (E2E), which remains a challenge. A key factor that makes E2E QoS assurance difficult in a telecommunication system is that access networks (ANs) and core networks (CNs) manage their resources autonomously. So far, few results have been available that can ensure E2E QoS over autonomously managed ANs and CNs. Existing techniques rely predominately on each subsystem to meet static local QoS budgets with no recourse in case any subsystem fails to meet its local budgets and, hence will have difficulty delivering E2E assurance. Moreover, most existing distributed optimization techniques that can be applied to assure E2E QoS over autonomous subsystems require the subsystems to exchange sensitive information such as their local decision variables. This paper presents a novel framework and a distributed algorithm that can enable ANs and CNs to autonomously "cooperate" with each other to dynamically negotiate their local QoS budgets and to collectively meet E2E QoS goals by sharing only their estimates of the global constraint functions, without disclosing their local decision variables. We prove that this new distributed algorithm converges to an optimal solution almost surely, and also present numerical results to demonstrate that the convergence occurs quickly even with measurement noise.