Given the current predictions of B5G, the maturity of 5G technology, and proposed network services that are both compute-intensive and latency-sensitive, there are different levels of criticality that require flexibility and agility in decision making to support new services on the network. The current traditional cloud computing service model cannot handle the explosive growth and demands of such use cases as HTC (Holographic Type Communication) services. The purpose of edge computing (EC) is to effectively solve problems such as minimizing latency and optimizing network utilization. However, flexibility and agility are also important requirements for cost-effective and resource-efficient deployment of such services. Considering the NP-hard nature of the problem, an artificial intelligence technique based on a reinforcement learning (RL) algorithm is proposed to make an intelligent decision on the optimal serving tier and edge-site selection, while considering criticality levels and multiple conflicting costs. In addition, at the edge network tier, a heuristic algorithm is used to map stream tasks onto the optimally selected edge-sites using a ranking mechanism based on the available network and computational resources. All algorithms aim to minimize the cost in the edge-cloud system to increase the revenue of a mobile network operator (MNO) by encouraging more people to use such outstanding future services at a lower cost. The simulation results show that the performance of the proposed RL-based algorithm is close to the optimal solution in achieving the main objective within a margin of 7%, moreover RL outperforms other benchmarks in most of the conducted experiments.