Quality-of-service (QoS) is the term used to evaluate the overall performance of a service. In healthcare applications, efficient computation of QoS is one of the mandatory requirements during the processing of medical records through smart measurement methods. Medical services often involve the transmission of demanding information. Thus, there are stringent requirements for secure, intelligent, public-network quality-of-service. This paper contributes to three different aspects. First, we propose a novel metaheuristic approach for medical cost-efficient task schedules, where an intelligent scheduler manages the tasks, such as the rate of service schedule, and lists items utilized by users during the data processing and computation through the fog node. Second, the QoS efficient-computation algorithm, which effectively monitors performance according to the indicator (parameter) with the analysis mechanism of quality-of-experience (QoE), has been developed. Third, a framework of blockchain-distributed technology-enabled QoS (QoS-ledger) computation in healthcare applications is proposed in a permissionless public peer-to-peer (P2P) network, which stores medical processed information in a distributed ledger. We have designed and deployed smart contracts for secure medical-data transmission and processing in serverless peering networks and handled overall node-protected interactions and preserved logs in a blockchain distributed ledger. The simulation result shows that QoS is computed on the blockchain public network with transmission power = average of −10 to −17 dBm, jitter = 34 ms, delay = average of 87 to 95 ms, throughput = 185 bytes, duty cycle = 8%, route of delivery and response back variable. Thus, the proposed QoS-ledger is a potential candidate for the computation of quality-of-service that is not limited to e-healthcare distributed applications.