### Brief Literature Review
However, the obtained system of equations turned out to be complex to the point that its closed-form solution could not be retrieved for the general case. Therefore, the problem was reduced to two random variables, and an associated analytical solution was formulated [31]. As the reach of such a solution is well below practical needs, many researchers (mostly originating from the field of operations' research and bioinformatics) have developed alternative methods and algorithms through tackling large-scale and dense transition systems. Examples include the decomposition method, aggregation procedure, external memory storage (storing complete transition matrix), selective matrix criteria, state lumping, and sparse matrix approximation [6]. The decomposition method assumes that a large-scale random system may be represented by the decomposition of its state space into multiple communicating classes, with characteristic probability distributions summarizing the behavior of the original system [32]. The applicability of the method was demonstrated in several production lines by addressing various performance measures. The need for more rigorous verification using analytical results was pointed out in [33; 34; 35]. An opposite approach to the problem of dense and large-scale random systems was applied in the case of the aggregation procedure through the reduction in the system's state space into a single class of states and the evaluation of the associated probability distribution. This procedure was first introduced in [36] in the research on the simple and analytical modeling of traditional production lines. The method has demonstrated accuracy and validity for application in modern mass-production industry and its main feature is the modeling simplicity and low computational burden. Apart from operations research, the issue of state-and transition-rich random processes was discussed in [37], following [38] in the case of bioinformatics. Several approaches were suggested, namely the external storage of the complete transition matrix, selective matrix criteria, and matrix approximation through state lumping or by neglecting some of its elements. In the first case, a dedicated parallel computing algorithm has to be employed in order to enable communication between the local and global memory stored on separate devices [39]. Although this approach may solve memory storage issues, it can affect the computing resources, though only to a limited extent. This approach is based on the assumption that the transition matrix is known and readily available, easily far from the truth in most cases. Alternatively to storing a complete matrix, only its selected part may be formulated and used when evaluating the state probability distributions, as an eigenvalue problem, using an algorithm such as the implicitly restarted Arnoldi method [40]. As in the previous case, this approach does not solve the fundamental question of transition matrix formulation, but rather aims for the reduction in the memory requirements and processing times of a computer. A similar effect is achieved if the transition matrix is approximated to its sparse form by neglecting some of the transitional probabilities, or if the scale of the transition matrix is reduced through state lumping by neglecting some of the system's states. All of the presented approaches, procedures, and approximations are based on the assumption that the transition matrix is reachable and formulated. However, this may be an issue in numerous realistic cases, particularly those involving myriads of states, interrelated through rich and diverse transition mechanisms. In addition, these approaches do not enable the formulation of complete joint probability distribution at the level of a random system, nor do they yield functional relationships between the system's properties and key performance indicators. This, in turn, is of importance concerning post-processing, interpretation, and the further evaluation of the results.