A Finite State Method in the Performance Evaluation of the Bernoulli Serial Production Lines

Abstract

Research on the performance measure evaluation of Bernoulli serial production lines is presented in this paper. Important aspects of the modeling and analysis using transition systems within the Markovian framework are addressed, including analytical and approximation methods. The “dimensionality curse” problems of the large scale and dense transition systems in the production system engineering field are pointed out as one of the main research and development obstacles. In that respect, a new analytically-based finite state method is presented based on the proportionality property of the stationary probability distribution across the systems’ state space. Simple and differentiable expressions for the performance measures including the production rate, the work-in-process, and the probabilities of machine blockage and starvation are formulated. A finite state method’s accuracy and applicability are successfully validated by comparing the obtained results against the rigorous analytical solution.

Accuracy of Semi-Analytical and Numerical Approaches in the Evaluation of Serial Bernoulli Production Lines

“…The finite state method is a recently developed method that bypasses the demanding CPU problem of the analytical solution by discretizing the system's state space using the two-machine-one-buffer finite-state elements and the associated analytical solution. Here, we provide only a brief outline of the method as it is presented in detail in [9].…”

“…where, m < e < M − 1, p e+1 is the probability that the machine e + 1 is in the state {up}, and a e = p m (1 -p e + 1 )/p e + 1 (1 − p m ). Finally, the distribution of the steady-state probabilities for a complete system can be approximated using the intersection of independent events at each buffer [9], i.e.,…”

“…Unfortunately, such an approach requires significant processing time, particularly if more complex state spaces are considered. Finally, a new semi-analytical method called the finite state method (FSM) was presented recently in [9]. The FSM performs a discretization of the system's state space using finite-state elements and an analytical solution of the two-machines-singlebuffer problem.…”

“…On the other hand, the aggregation approach lumps a complete serial production line into a single machine using iterative forward and backward aggregation [8]. The recently developed finite-state method (FSM) discretizes the system's state space using an analytical solution of a well-known two-machine-single-buffer problem at the level of finite-state elements [9]. Finally, a closed-form analytical solution of the Chapman-Kolmogorov equation yields an exact solution to the problem using a concept of the generalized transition matrix.…”

“…Figure 2. Distribution of the finite-state elements along the serial Bernoulli production line (reproduced according to[9].…”