Queueing modeling and cost analysis of M/G/1/K fault-tolerance machining system with general vacation under N-policy for industrial applications

This investigation focuses on modeling the performance of an M/G/1/K fault-tolerance machining system (FTMS) that incorporates the general vacation feature under threshold N-policy, along with handling failure recovery and reboot processes. In the system design, the repairman has the flexibility to take a vacation only when no failed machines are present in the system. The repairman returns from vacation and resumes repair operations when the number of failed machines reaches the specified threshold value N. In cases where faults in the system are not successfully detected, the system reconfigures itself by initiating a reboot process. To develop the mathematical model, supplementary variables representing the remaining repair and vacation times are employed to formulate the steady-state Chapman–Kolmogorov equations. These equations are the foundation for deriving explicit formulae for the probability distributions using the Laplace-Stieltjes transform (LST) and the recursive method. The resulting system state probabilities are then utilized to establish various performance measures. The study conducts a sensitivity analysis to explore the influence of different parameters on performance measures. Furthermore, we construct the expected cost function and formulate an optimization problem to determine the minimum cost. The particle swarm optimization (PSO) is employed to solve the optimization problem, and the results are compared with those obtained by implementing a sine cosine algorithm (SCA) to find the optimum cost. The proposed model demonstrates its practical usefulness in industrial settings, especially in the electrical sector for the power supply system (PSS) and the production sector for the plastic bottle processing hub (PBPH).