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This paper proposes an effective and novel multiagent deep reinforcement learning (MADRL)-based method for solving the joint virtual network function (VNF) placement and routing (P&R), where multiple service requests with differentiated demands are delivered at the same time. The differentiated demands of the service requests are reflected by their delay- and cost-sensitive factors. We first construct a VNF P&R problem to jointly minimize a weighted sum of service delay and resource consumption cost, which is NP-complete. Then, the joint VNF P&R problem is decoupled into two iterative subtasks: placement subtask and routing subtask. Each subtask consists of multiple concurrent parallel sequential decision processes. By invoking the deep deterministic policy gradient method and multi-agent technique, an MADRL-P&R framework is designed to perform the two subtasks. The new joint reward and internal rewards mechanism is proposed to match the goals and constraints of the placement and routing subtasks. We also propose the parameter migration-based model-retraining method to deal with changing network topologies. Corroborated by experiments, the proposed MADRL-P&R framework is superior to its alternatives in terms of service cost and delay, and offers higher flexibility for personalized service demands. The parameter migration-based model-retraining method can efficiently accelerate convergence under moderate network topology changes.