Reliability of Load-Sharing Systems Subject to Proportional Hazards Model Rahamat Mohammad Victoria University Akhtar Kalam, Ph. D., Victoria University Suprasad V. Amari, PhD, Parametric Technology Corporation Key Words: phased-mission systems, proportional hazards model, k-out-of-n redundancy, reliability analysis SUMMARY & CONCLUSIONS This paper presents a new model for load-sharing systems using k-out-of-n structure. It is assumed that the failure distribution of each component at a baseline load follows a general failure time distribution. Hence, the model can be used for analyzing the systems where components’ failure times follow Weibull, Gamma, Extreeme Value, and Lognormal distributions. In a load-sharing system, the system components experience different loads at different time intervals due to the load-sharing policy. Therefore, to analyze the reliability of load-sharing systems, the failure rate of each component must be expressed in terms of the current load and the current age of the component. In this paper, the load-dependent time- varying failure rate of a component is expressed using Cox’s proportional hazards model (PHM). According to the PHM the effects of the load is mulitplicative in nature. In other words, the hazard (failure) rate of a component is the product of both a baseline hazard rate, which can be a function of time t, and a multiplicative factor which is function of the current load on the component. The load-sharing model also considers the switchover failures at the time of load redistribution. We first show that the model can be described using a non- homogeneous Markov chain. Therefore, for the non-identical component case, the system reliability can be evaluated using well established methods for non-homogenerous Markov chains. In addition, when all components are identical, the paper provides a closed-form expression for the system reliability even when the underlying baseline failure time distribution is non-exponential. The method is demonstrated using a numerical example with components following Weibull baseline failure time distribution. The numerical results from non-homogeneous Markov chains, closed-form expressions, and Monte Carlo simulation are compared. Acronyms AFTM Accelerated Failure Time Model CE Cumulative Exposure Model PHM Proportional Hazards Model PMS Phased-Mission System TFR Tampered Failure Rate (Model) i.i.d Independent & Identical Distributed LSS Load Sharing System Notation n Number of components in the system k Minimum number of components required for successful operation z i Load on each component when i components failed; z ୧ ൌ z ଴. ୬ ୬ଵ  ଴ ሺݐሻ Baseline failure rate of PHM ߣ ௜ ሺݐሻ Failure rate of each component ߜ ௜ Failure rate multiplication factor Λ(t) Transition rate matrix ݌ ଴ሺ௧ሻ Initial probability state vector R(t) System reliability 1 INTRODUCTION In reliability engineering, it is a widespread practice to use redundancy techniques to enhance system reliability. A standard form of redundancy is a k-out-of-n arrangement in which at least k-out-of-n components must work for the triumphant operation of the system. The k-out-of-n configuration redundancy finds capacious purpose in both industrial and military systems. Examples include the generators in power systems, cables in a bridge and the multi- engine system in an airplane. Several examples of k-out-of-n systems are available in [1]. In numerous cases, when investigating redundancy, autonomy is ascertained across the components within the system. In other words, it is assumed that the failure of a component does not alter the failure properties (failure rates) of the remaining components. In the real-world, however, numerous systems are load-sharing, where the conjecturing of independence is no longer accurate. In a load-sharing system, if a component breaks down, the same workload has to be shared by the remaining components, resulting in an increased load shared by each surviving component. In most circumstances, an aggrandized load induces a colossal component failure rate [1]. Many empirical studies of mechanical systems [2] and computer systems [3] have evinced that the workload strongly impinges the component failure rate. Applications of load-sharing systems include 978-1-4673-4711-2/13/$31.00 ©2013 IEEE