Lead time distribution in a small manufacturing system with unreliable machines and finite buffers

Lead time variability is a critical source of risk for manufacturing firms. Delivery commitments, quality management, and effective line design all depend on the full lead time distribution, not merely the average lead time. This paper derives the steady-state lead time distribution for a three-machine, two-buffer production line with unreliable machines and finite buffers. This is the simplest system in which interactions between finite buffers create a coupled stochastic process. Extensive validation by comparison with discrete-event simulation confirms its accuracy across a wide range of system configurations. The full distribution reveals insights unobtainable from mean lead times alone or from two-machine line analyses: the lead time distribution can be multi-modal; it can exhibit a pronounced heavy tail depending on bottleneck location; its variance and tail percentiles exhibit a non-monotonic relationship with buffer capacity; and bottleneck placement can shift the mean lead time by nearly an order of magnitude while substantially increasing dispersion. These findings provide actionable guidelines for buffer sizing, delivery date quoting, and maintenance prioritization. The exact results for three-machine, two-buffer systems establish an essential foundation for extending lead time analysis to longer production lines.