Queueing theory employs mathematical analysis to establish effectiveness metrics. Optimization models are then formulated using significant and efficient measures, such as data, to ascertain system efficiency and requirements. Each queuing system represents a discrete event system problem, and simulating these systems aids in addressing challenges and conducting practical performance analysis. Blockchain offers various benefits, including redistribution, accessibility, durability, reliability, constancy, anonymity, auditability, and data security. Its applications span across cryptocurrencies, financial services, reputation management, the 'Internet of Things', the sharing economy, and social and community services. Notably, foundational theory is increasingly pertinent in the blockchain field. For instance, performance analysis and optimization of blockchain systems rely on mathematical models like Markov processes and queueing theory. In smart healthcare, blockchain technology enhances disease diagnosis, patient care, and overall quality of life. Due to the substantial patient data stored on blockchain in smart healthcare architectures, queueing models are indispensable for efficient data processing. This paper leverages Markov chains to establish queueing theory for blockchain systems and assess the performance of smart healthcare architecture. A "Markovian-batch-service" queueing framework is devised for this purpose, modeling input and processing parameters essential for reliable queuing network simulations.