# A Finite State Method in the Performance Evaluation of the Bernoulli Serial Production Lines
## 1 Introduction  
The manufacturing industry is of great importance to the national and global economies. Many historical examples, starting with the Industrial Revolution, prove that it is a path towards development and prosperity. More specifically, we have seen many times that national power correlates to control over the global manufacturing industry and means of production, like industry sectors in the US, Japan, Germany, Korea, or China. These examples prove that the growth of the manufacturing output and technological improvement boost long-term economic growth. Additionally, it is estimated that, via the multiplier effect, each manufacturing job supports 5-7 other employments in the economy across the global trade of goods and services [1]. Consequently, an efficient and sustainable economy depends on efficient and sustainable manufacturing. Given that, production system engineering (PSE) stands as a path to rationalization and improvement of the existing production systems through successful production planning and control [2]. PSE relies on production system modeling and analysis using transition system theory at different scales, and particularly on discrete timed models describing steady-state (time-invariant) and transient production system responses. Such models have been predominantly used for the performance evaluation of various serial and assembly production systems focusing on throughput, reliability, sensitivity, lead time, and bottleneck analysis, or other design and optimization problems. In addition, similar applications can be found in fields of molecular biology, evolution, healthcare, city traffic, communication services, computer algorithms, money flow, network structures, etc. [3]. The application of the stochastic modeling in the case of production systems intensified significantly in the last two decades. Currently, significant research efforts are being put into a deeper understandingof the production system behavior in the context of the internet of things [4], still strongly relying on the application of the operation research in the industrial context [5]. Therefore, a systematic approach to production system modeling using stochastic processes has great potential to contribute to current research dealing with Industry 4.0 [6] and advanced production system concepts like digital twinning [7]. In addition to that, stochastic modeling has been successfully applied in cases like tool wear condition monitoring [8], prediction of cutting force during micro-machining [9], or grinding wheel topography modeling [10]. More generally, the importance of the Markovian framework, its background, and broad application cases is proven by a quite significant number of research papers dedicated to that topic [11; 12]. Concerning production system modeling, a variety of problems has been addressed in the literature, such as the performance evaluation of serial lines, assembly lines, job shops, flexible manufacturing cells, or other specific types of production. A more detailed review of different modeling approaches, including applications of the Markov processes, semi-Markov processes, queuing networks, stochastic automata networks, Petri nets, performance algebra, and diodic algebraic models, was presented recently by [PERSON] et al. [3]. It was pointed out that the majority of the research body is still dominated by the application of the Markovian framework in cases of serial production lines--\"the working horse of production systems\". Given that, the primary goal of this paper is to present the application of the transition system theory in the case of the serial production line composed of machines and buffers of arbitrary storage capacity. Such an approach has great potential for enhancing the competitiveness and profitability of the production facilities. However, it is presently faced with a significant computational burden. This issue is bypassed by this research using Markovian modeling and the developed finite state method validated against the rigorous analytical solution to the problem.