Reliability evaluation and spare parts inventory optimization of <i>k</i> / <i>n</i> : G systems based on retrial mechanism of failure information

Optimizing spare parts inventory is critical for balancing system availability and resource efficiency in digitalized environments. Under the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> inventory control policy, this paper develops a joint optimization model of reliability and spares inventory control for a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">/</mml:mo> <mml:mi>n</mml:mi> <mml:mo>:</mml:mo> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> </mml:math> retrial system with the Bernoulli vacation mechanism. This model holds significant reference value and application prospects in the design of cloud computing systems. Through probabilistic analysis and Markov process methodology, we analyze the state probability and derive key performance and reliability indices of the system. Based on the important cost elements and performance indices, we provide the mathematical expectation of the total system cost per time unit. Then, we establish a cost-minimization inventory policy optimization model constrained by joint system availability and spare parts availability, using an Adaptive Hybrid Branch-and-Bound (AHBB) algorithm to determine the optimal inventory policy. Numerical experiments are provided to demonstrate the newly proposed model and analyze the effects of different parameters on the system reliability indices. Next, we give the optimal inventory <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>*</mml:mo> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>*</mml:mo> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and the corresponding minimum cost for different <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo stretchy="false">/</mml:mo> <mml:mi>n</mml:mi> <mml:mo>:</mml:mo> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> </mml:math> systems under the joint reliability constraint of spare parts and system. Finally, the cost-effectiveness of our scale-adaptive <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>s</mml:mi> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> policy and the efficiency of the proposed AHBB algorithm through comparative experiments is demonstrated.