## 2 Mathematical Modeling
### The Finite State Method  
The analytical solution of the assembly systems' steady-state response reveals the complexity of the problem, particularly as the formulation of the transition matrix becomes rather cumbersome and impractical, even for the simplest cases, not to mention the issues of the consecutive multiplication of large-scale matrices and the solution of the eigenvalue problem. Therefore, a more convenient way to evaluate performance measures of the assembly systems, especially when design issues are in question, is to consider some of the approximative procedures, for example, the finite state method. This method has already been successfully applied in the case of Bernoulli serial and splitting lines [3,6], and will thus be further extended here in the case of assembly systems. The central idea of the finite state method is to model complex transition relationships of an assembly system in the general case by using the simple and analytical solution of the two-machine-one buffer problem. In this way, an intricate mathematical problem can be significantly simplified, while keeping all of the necessary relationships between the system properties and key performance indicators. The analytical solution of the two machine-one buffer problem can be retrieved in several ways, however, its formulation, using a recursive procedure, is readily available in the literature [5], and equals  
\[P_{0}=\frac{(1-p_{L})(1-\alpha)}{1-\frac{p_{L}}{p_{R}}\alpha^{N}},\ P_{i}=\frac {\alpha^{i}}{1-p_{R}}P_{0},\ i=1,2,\ldots,N,\ \alpha=\frac{p_{L}(1-p_{R})}{p_{R}(1-p_{L})}, \tag{9}\]  
where \(P_{i}\), \(i=0\), \(1\), \(2\), \(\ldots\), \(N\), is the probability of a system being in state \(I\); \(N\) is the capacity of the buffer; and \(p_{L}\) and \(p_{R}\) are the reliabilities of machines positioned, respectively, at the left and right hand of the buffer. The same solution can be applied when formulating finite state elements of the assembly production line concerning the weakest machine, \(\mathrm{m}_{m}\), where \(m\) is such that  
\[p_{m}=\min\{p_{1},p_{2},\ldots,p_{M}\} \tag{10}\]  
An example of the structural scheme of finite state elements is presented in Figure 3 including the primary flows and a secondary flow. In this case, the total number of \(M-1\) elements can be formulated, with each composed of only two machines and one buffer placed between them. Depending on the relative position of machine \(\mathrm{m}_{m}\), finite state elements can be formulated and characterized as upstream or downstream. In the case when \(m\leq K_{1}\), machine \(\mathrm{m}_{m}\) is placed in the first primary flow (\(r=1\)), while it is placed in the second primary flow (\(r=2\)) when \(K_{1}<m\leq K_{1}+K_{2}\). In the more general case, machine \(\mathrm{m}_{m}\) is a component of the \(r^{\text{th}}\) primary flow if \(m\) falls within range \(\left\langle\sum\limits_{i=1}^{K_{-1}}i,\sum\limits_{i=1}^{K_{i}}i\right\rangle\), \(r=2\), \(3\), \(\ldots\), \(R\). In a special case when \(r=1\), this range is simplified to the form \(\left\langle 1,\sum\limits_{i=1}^{K_{1}}i\right\rangle\). Additionally, when \(m\geq A\), machine \(\mathrm{m}_{m}\) is placed in the secondary flow. Consequently, the assembly system (Figure 3) can be decomposed into a set of serial lines while extending flows not containing the weakest machine with the additional (imaginary) machine to the right in the case of primary flows and to the left in the case of the secondary flow. This fictive machine is presented in Figure 3b using a dashed line. Thus, upstream elements are defined in the range \(e=[1,m-1]\) and \(e=[K_{1},K_{1}+K_{2}-1]\). Similarly, downstream elements range within \(e=[m,K_{1}]\) and \(e=[A-1,M-1]\). In the case of the upstream elements, the left-hand side machine is associated with a reliability of \(p_{L}=p_{e}\), \(e=[1,m-1]\) and \(e=[K_{1},K_{1}+K_{2}-1]\), while the reliability of the right-hand side machine always equals \(p_{R}=p_{m}\).