Abstract : This paper presents a stochastic modeling framework for analyzing the performance of a multi-machine production system subject to random failures, constrained by limited spare part availability and finite repair capacity. The system is modeled as a Quasi-Birth–Death (QBD) process, where failed machines may either wait for spares or repairs, and replenishment of spare parts follows an exponential distribution. A matrix-geometric solution is applied to derive steady-state probabilities and performance measures, including availability, expected queue length of failed machines, downtime probability, and throughput. Numerical illustrations are provided for systems with varying spare part inventories. Results show that availability improves with additional spares, but limited repair capacity emerges as the primary bottleneck. The study highlights the trade-offs between spare provisioning and repair resources, offering valuable insights into optimizing reliability under constrained operational environments.
Keywords : Reliability modeling; Spare part inventory; Machine repair system; Matrix-geometric method; Quasi-Birth–Death process (QBD); Availability; Throughput
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