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The crucial role of networking in Cloud computing requires a holistic vision of both networking and computing resources that leads to a convergence of network and Cloud service provisioning. Therefore, networking and Cloud computing systems need to be modeled and analyzed as a converged service provisioning system in order to obtain thorough understanding about service performance. This paper proposes a new approach to modeling service capabilities of converged network-Cloud service provisioning systems and develops the analysis techniques for evaluating delay performance of converged network-Cloud services. The modeling and analysis techniques developed in this paper are general and agnostic to the implementations of network and Cloud infrastructure; thus are applicable to heterogeneous networking and Cloud computing systems for converged service provisioning.