Analysis of Repairable Systems Availability with Lindley Failure and Repair Behavior

Maintainability analysis is a cornerstone of reliability engineering. While the Markov approach is the classical analytical foundation, its reliance on the exponential distribution for failure and repair times is a major and often unrealistic limitation. This paper directly overcomes this critical constraint by investigating and modeling system maintainability using the more flexible and versatile Lindley distribution, which is represented via phase-type distributions. We first present a comprehensive maintainability analysis of a single-component system, deriving precise closed-form expressions for its time-dependent and steady-state availability, as well as the mean time to repair. The core methodology is then systematically generalized to analyze common series and parallel system configurations with n independent and identically distributed components. A dedicated numerical study compares the system performance under the Lindley and exponential distributions, conclusively demonstrating the significant and practical impact of non-exponential repair times on key reliability metrics. Our work provides a versatile and more widely applicable analytical framework for accurate maintainability assessment that successfully relaxes the restrictive exponential assumption, thereby offering greater realism in reliability modeling.

💡 Research Summary

The paper tackles a fundamental limitation of classical reliability analysis, namely the reliance on exponential distributions for both failure and repair times in Markov models. By employing the Lindley distribution—a flexible mixture of an exponential component and a Gamma(2) component—the authors represent failure times as a phase‑type distribution, enabling the use of continuous‑time Markov chains while capturing non‑exponential behavior.

First, a single‑component repairable system is modeled. The Lindley failure mechanism is split into two parallel paths: an exponential path selected with probability p and a two‑stage gamma path selected with probability 1‑p. Each path leads to a failure (absorbing) state, after which repair occurs at a constant rate μ. Upon repair completion the system is re‑initialized according to the original mixture probabilities, preserving the Lindley structure. By constructing the infinitesimal generator matrix Q and solving the Kolmogorov forward equations via Laplace transforms, the authors derive closed‑form expressions for the time‑dependent availability A(t) and the steady‑state availability A∞. They prove analytically that for any non‑negative failure rate λ and repair rate μ, the Lindley‑based steady‑state availability exceeds that of the corresponding exponential model, highlighting the practical benefit of modeling non‑exponential repair times.

Second, the methodology is extended to an n‑component system with independent and identically distributed Lindley components. The state space grows exponentially (4ⁿ states for n components), but because phase‑type distributions are closed under minimum, maximum, and convex combinations, the long‑run system availability can be expressed simply in terms of the single‑component availability. For series configurations the system availability is the product of component availabilities, while for parallel configurations it follows the standard complement‑of‑product formula. This yields compact algebraic formulas for both series and parallel systems without the need to solve the full high‑dimensional Markov chain.

A numerical case study revisits a biomass‑fueled combined cooling, heating, and power (CCHP) plant previously analyzed with exponential assumptions. Using the same failure and repair rates for the gasification unit, internal combustion engine, and absorption chiller, the authors compute reliability and availability curves under both exponential and Lindley assumptions. The Lindley model shows an initial decline similar to the exponential case but quickly stabilizes at a higher steady‑state availability, reflecting the effect of the two‑stage repair process. Corresponding mean time to repair (MTTR) and mean time between failures (MTBF) are also larger, indicating a more realistic depiction of system performance.

Overall, the paper contributes a rigorous analytical framework that retains the computational tractability of Markov models while accommodating realistic, non‑exponential failure and repair behaviors. The derived formulas enable rapid assessment of maintenance policies, resource allocation, and reliability‑driven design decisions for both simple and complex engineered systems. Future work is suggested on discrete‑time phase‑type extensions, multiple repair crews, and dependent component interactions.