## 2 Mathematical Modeling
### Performance Measures  
Once steady-state distribution at the system level is formulated, it is possible to evaluate associated performance measures such as the production rate (\(PR\)), work-in-process at the \(i^{\text{th}}\) buffer (\(WIP_{i}\)), probabilities of blockage (\(BL\)), and probabilities of starvations (\(ST\)). The definitions of the production rate and work-in-process in the case of an assembly system remain the same as in the case of a typical Bernoulli serial line. On the other hand, defining the probabilities of blockage and starvation requires some modifications in the case of machines \(\text{m}_{\xi}\) (recalling that \(\xi=\{K_{1},K_{1}+K_{2},K_{1}+K_{2}+K_{3},\ldots,A-1\}\)) and \(\text{m}_{A}\) due to the structure of the assembly lines. Hence, the formulation of the production rate and work-in-process will be omitted here as it is well-known and available in the literature (e.g., [3; 5; 14]). The probability of the blockage of machine \(\text{m}_{\xi}\) can be formulated as the union of events [[\(\text{m}_{\xi}\) is up]\(\cap\)[\(\text{b}_{\xi}\) is full]\(\cap\)[\(\text{m}_{A}\) is down]], [[\(\text{m}_{\xi}\) is up]\(\cap\)[\(\text{b}_{\xi}\) is full]\(\cap\)[\(\text{m}_{A}\) is blocked]], and [[\(\text{m}_{\xi}\) is up]\(\cap\)[\(\text{b}_{\xi}\) is full]\(\cap\)[\(\text{m}_{A}\) is starved]], yielding  
\[BL_{\xi}=p_{\xi}\big{(}1-p_{A}+BL_{A}+ST_{A}^{\prime}\big{)}\sum_{h_{1}=0}^{N_{ 1}}\sum_{h_{2}=0}^{N_{2}}\ldots\sum_{h_{M-1}}^{N_{M-1}}P_{h_{1}h_{2}\ldots h_{ \xi}=N_{\xi}\ldots h_{M-1}}, \tag{14}\]  
where \(BL_{A}\) is the probability of blockage of machine \(\text{m}_{A}\). The probability \(ST_{A}^{\prime}\) takes into account the starvation of machine \(\text{m}_{A}\) due to the emptiness of buffers \(\text{b}_{\xi}\), other than a considered machine, in other words,  
\[ST_{A}^{\prime}=ST_{A}-\sum_{h_{1}=0}^{N_{1}}\sum_{h_{2}=0}^{N_{2}}\ldots\sum_ {h_{M-1}}^{N_{M-1}}P_{h_{1}h_{2}\ldots h_{\xi}=0\ldots h_{M-1}}, \tag{15}\]  
where \(ST_{A}\) is the probability of the starvation of machine \(\text{m}_{A}\), determined as a union of events [[one buffer from the set \(\xi\) is empty]\(\cap\)[\(\text{m}_{A}\) is up]], [[two buffers from the set \(\xi\) are empty]\(\cap\)[\(\text{m}_{A}\) is up]],..., [[\(\{R\) buffers from the set \(\xi\) are empty]\(\cap\)[\(\text{m}_{A}\) is up]], or as  
\[ST_{A}=p_{A}\left(\sum_{h_{1}=0}^{N_{1}}\sum_{h_{2}=0}^{N_{2}} \ldots\sum_{h_{M-1}}^{N_{M-1}}P_{\ldots h_{K_{1}}=0\ldots}+\sum_{h_{1}=0}^{N_{ 1}}\sum_{h_{2}=0}^{N_{2}}\ldots\sum_{h_{M-1}}^{N_{M-1}}P_{\ldots h_{K_{1}}=0 \ldots h_{K_{2}}=0\ldots}+\ldots\right.\] (16) \[\left.+\sum_{h_{1}=0}^{N_{1}}\sum_{h_{2}=0}^{N_{2}}\ldots\sum_{h_ {M-1}}^{N_{M-1}}P_{\ldots h  
Figure 5: Comparison of the performance measures in the case of line L3: (**a**) \(N_{1}=N_{2}=N\); (**b**) \(N_{1}=N\); \(N_{2}={}^{1}/{}_{2}N\). Figure 6: Comparison of the performance measures in the case of line L4: (**a**) \(N_{1}=N_{2}=N_{3}=N\); (**b**) \(N_{1}=N,N_{2}={}^{2}/_{3}N,N_{3}={}^{1}/_{3}N\). Figure 7: Comparison of the performance measures in the case of line L4B: (**a**) \(N_{1}=N_{2}=N_{3}=N_{4}\) (**b**) \(N_{1}=N\), \(N_{2}={}^{2}/3N\), \(N_{3}={}^{1}/_{3}N\). Figure 8: Comparison of the performance measures in the case of line L4C: (**a**) \(N_{1}=N_{2}=N_{3}=N_{3}\) (**b**) \(N_{1}=N\), \(N_{2}={}^{2}/_{3}N\), \(N_{3}={}^{1}/_{3}N\). Figure 9: Comparison of the performance measures in the case of line L5A: (**a**) \(N_{1}=N_{2}=N_{3}=N_{4}=N_{2}\) (**b**) \(N_{1}=N,N_{2}={}^{3}/{}_{4}N,N_{3}={}^{1}/{}_{2}N,N_{4}={}^{1}/{}_{4}N\).