## 2 Mathematical Modeling
### The Analytical Solution
Consequently, a case-by-case formulation of transition matrices is not a reasonable nor advisable strategy. Hence, in this paper, we propose an alternative way to formulate transition matrices by using the concept of the generalized transition matrix formulation. Each transition matrix, \([P]\), is a function of machine reliabilities, \(p_{i}\), and thus can be presented as \([P]=[P(p_{1},p_{2},\ldots,p_{A},\ldots,p_{M})]\), which upon the separation of its variables can be expressed as a consecutive multiplication of constitutive matrices, \([P(p_{i})]\), in other words,  
\[ \begin{split}&[P(p_{1},p_{2},\ldots,p_{A},\ldots p_{M})]=\\ &[P_{1}(p_{1})][P_{2}(p_{2})]\ldots[P_{A}(p_{A})]\ldots[P_{M}(p_{ M})]=\prod_{i}^{M}[P_{i}(p_{i})].\end{split}  \tag{4}\]  
The \(i\)th constitutive matrix, \([P_{i}(p_{i})]\), reflects the contribution of machine \(\text{m}_{i}\) to the movement of the stochastic system from state \(\left\langle H^{m}\right\rangle=\left\langle h_{1}^{n},h_{2}^{n},\ldots,h_{A} ^{n},\ldots h_{M-1}^{n}\right\rangle\) in the \(n\)th cycle, where \(h_{i}^{n}\), \(i=1\), \(2\),..., \(M-1\) is the state of the \(i\)th buffer in the \(n\)th cycle, to state \(\left\langle H^{n+1}\right\rangle=\left\langle h_{1}^{n+1},h_{2}^{n+1}\ldots, h_{A}^{n+1},\ldots h_{M-1}^{n+1}\right\rangle\), attributed to the cycle \(n+1\). In the considered case of the assembly production line, four different types of constitutive matrices can be formulated (i.e., \([P_{\xi}]\), \([P_{\xi}]\), \([P_{A}]\), and \([P_{M}]\)), where \(\zeta=\{1\), \(K_{1}+1\), \(K_{1}+K_{2}+1\),..., \(A-K_{R}\}\) and \(\overline{\zeta}=\{1\), \(2\),..., \(M-1\}\backslash\{\zeta,A\}\). The generic structure of each constitutive matrix contains four different entries, namely \(1-p_{i}\) in the case when the \(i\)th machine does not alter the state of the system due to a breakdown, \(p_{i}\), when the state of the system is altered due to the activity of machine \(\text{m}_{i}\), \(1\) in the case when the state of the system cannot be altered due to starvation or blockage of machine \(\text{m}_{i}\), and \(0\) for all other cases. Therefore, the constitutive matrix \([P_{\zeta}]\) is equal to  
\[\begin{split}&[P_{\zeta}]=\left\{\begin{array}{l}1-p_{\zeta}, \text{ if }h_{\psi}^{n+1}=h_{\psi}^{n},  
where \(\xi=\{K_{1},K_{1}+K_{2},K_{1}+K_{2}+K_{3},\ldots,A-1\}\) is a set of buffers placed immediately before the assembly machine, \(\mathrm{m}_{A}\), and \(\overline{\psi}=\psi\backslash\xi\). Finally, the constitutive matrix, associated with machine \(\mathrm{m}_{M}\), \([P_{M}]\), equals  
\[[P_{M}]=\left\{ \begin{array}{ll}1-p_{M},\text{ if }h_{\varphi}^{n+1}=h_{ \varphi}^{n},h_{M-1}^{n}>0\\ p_{M},\text{ if }h_{M-1}^{n+1}=h_{M-1}^{n}-1,h_{\overline{\psi}}^{n+1}=h_{ \overline{\psi}}^{n},h_{M-1}^{n}>0\\ 1,\text{ if }h_{\overline{\psi}}^{n+1}=h_{\overline{\psi}}^{n},h_{M-1}^{n}=0\\ 0,\text{ else,}\end{array} \right. \tag{8}\]  
where \(\overline{\psi}=\psi\backslash(M-1)\). The procedure for the transition matrix formulation is illustrated in Appendix A in the simple case of an assembly system composed of three machines and two buffers of capacity \(N_{1}=N_{2}=2\), while more complex cases are taken into account in Section 3.