## 5 Conclusions  
Production has a significant role in modern society. This role is reflected in aspects of the global market, supply chains, transport systems, global and local development strategies, life quality, and others. Thus, any improvement or decline in production affects many ancillary services. However, each production system may be considered inherently stochastic including both external and internal factors. Therefore, the development and implementation of suitable mathematical models enabling the evaluation of associated key performance indicators as well as the overall equipment efficiency are necessary prerequisites for the development of successful company strategies oriented toward profit increase. Mathematical modeling of the Bernoulli assembly line was presented in this paper. In the first case, the analytical solution to the problem was developed within the Markovian framework including the formulation of the constitutive transition matrices through the separation of variables. Consecutive multiplication of the constitutive matrices resulted in a transition matrix containing transition probabilities associated with all possible states of the assembly system. This transition matrix was then employed within the eigenvalue problem to determine the probability distribution associated with the steady-state performance of the system and was employed to evaluate the key performance indicators such as the production rate, the work-in-process, and probabilities of blockage and starvation. Although this approach yielded accurate results, it suffered from dimensionality issues (memory storage and CPU requirement) when large-scale and transition-rich systems were considered. Hence, in the second case, the finite state method was developed to tackle issues of the problem of dimensionality and the CPU requirements. The finite state method employs the well-known analytical solution of the two-machine and one buffer problem at the level of finite state elements and reconstructs the system-leveled probability distribution by assuming the independence of events at each buffer. To validate the developed procedure, both methods were applied in several theoretical cases involving three, four, and five machines in different arrangements. A comparison of the obtained results proved the finite state method as a reliable yet CPU-efficient method yielding accurate results, except for some discrepancies attributed to the lowest range of the buffer capacities and induced due to a rather coarse system state discretization. Furthermore, it was concluded that the arrangement of machines in the assembly line does not affect the accuracy of the results. Hence, the FSM can be reliably employed in cases of assembly systems including multiple assembly machines and secondary flows. Following these findings, the application of the finite state method in the case of a realistic assembly line was presented including its validation against the factory floor measurements. When compared to similar methods available in the literature, FSM stands out as the only method operating in the complete system state space. For this reason, it enables the reconstruction of the steady-state probability distribution while keeping the functional relationship between the properties of an assembly line and the key performance indicators. This fact makes it quite suitable for purposes such as improvability and design considerations. Consequently, it yields predictive analytics suitable for long- and short-term decision-making in the context of production system management and the development of smarter and more efficient production policy. Finally, it has to be pointed out that the presented approach is still limited to synchronous production systems (serial, splitting, and assembly lines) only involving machines of the Bernoulli reliability.