## 2 Mathematical Modeling
### The Analytical Solution  
Consider a [PERSON] assembly line (Figure 2) composed of \(M\) machines of the Bernoulli reliability, \(p_{i}\)_-Bernoulli_(\(p_{i}\)), \(i=1\), \(2\),..., \(M\), and \(M-1\) buffers of capacity \(N_{i}\in\) N\({}_{0}\), \(i=1\), \(2\),..., \(M-1\), where machine \(\text{m}_{A}\) performs the assembly of parts delivered through \(R\) primary flows, and transfers the assembled product for further processing to the secondary flow. Each primary flow is composed of \(K_{i}\), \(i=1\), \(2\),..., \(R\), machines and buffers, while their sum, \(\sum\limits_{i=1}^{R}K_{i}\), equals \(A-1\). This assembly system can be mathematically modeled as an ergodic Markov chain over the system's state space, composed of all the combinations of the occupancies of buffers [5; 44]. However, additional assumptions have to be taken into account, in other words,  
Figure 1: Mathematical modeling approaches and the workflow of the methodology. * Each machine is in state {up} with probability \(p_{i}\), conversely it is in state {down} with probability \(1-p_{i}\);
* The state of _ith_ machine does not depend on the state of other machines and is determined at the beginning of each time instance;
* All machines have the same cycle time;
* The system obeys the mass conservation law, thus parts can enter the system only through the first machine at each primary flow, and can exit the only through machine \(\text{m}_{M}\);
* The first machines in the primary flow cannot be starved and machine \(\text{m}_{M}\) cannot be blocked. The steady-state response of a system at-hand can thus be evaluated using the well-known Chapman-Kolmogorov equations, [44; 45], relating probabilities \(P_{i}\) and \(P_{j}\) of a system being in the state \(i\), respectively \(j\), in other words,  
\[P_{i}=\sum_{j=1}^{S}P_{ij}P_{j},\;\sum_{i=1}^{S}P_{i}=1 \tag{1}\]  
where \(S\) is the total number of the system's states, and \(P_{ij}\) is the transition matrix. In the case when the Markov chain is irreducible and aperiodic, the Chapman-Kolmogorov equation, Equation (1), yields a unique stationary (limiting) solution, in turn reflecting the steady-state response of a stochastic system. One possibility to retrieve this solution is to employ the eigenvalue problem [3; 14],  
\[\text{det}([P]-\Lambda[I])=0, \tag{2}\]  
where \([P]\) is the transition matrix; \([I]\) the unit matrix; and \(\Lambda\) is the eigenvalue. Given the fact that the transition matrix is also stochastic, its first eigenvalue, \(\Lambda_{1}\), takes a unit value and is attributed to the first eigenvector \([P_{1}]\), representing the normalized form of a stationary solution of the problem. Although this may seem as a quite elegant and straightforward way for the evaluation of the steady-state probability distribution of stochastic systems, it may be far from the truth when realistic systems, composed of large-scale and transition-rich states, are considered. This problem manifests in a 2-fold way, as the inability to formulate the transition matrix and as significant and sometimes unrealistic CPU (central processing unit) resource and time requirements. The former is addressed here in the context of the analytical solution formulation, while the latter is tackled through the formulation of finite state elements. The formulation of transition probabilities in the general case is a challenging task, particularly when large-scale production systems are considered. The scale of the system,  
Figure 2: A model of a Bernoulli assembly line. \(D\), is usually determined as the total number of combinations of buffer occupancy levels and equals  
\[D=\prod_{i}^{M-1}(N_{i}+1) \tag{3}\]  
quickly taking quite significant values. For example, in the case when \(M\) equals \(11\) and \(N_{i}=9\), the scale of the system, \(D\), takes a value of \(10\) billion, leading to the conclusion that the dimension of the associated transition matrix equals \(D^{2}\), and thus contains \(10^{20}\) entries.