Reliability Analysis of k-out-of-n Systems with Phased- Mission Requirements

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1 International Journal of Performability Engineering, Vol. 7, No. 6, November 2011, pp RAMS Consultants Printed in India Reliability Analysis of k-out-of-n Systems with Phased- Mission Requirements SUPRASAD V. AMARI 1* and LIUDONG XING 2 1 Parametric Technology Corporation, Greensburg, PA USA 2 University of Massachusetts, Dartmouth, MA USA (Received on November 30, 2010, revised on March 08, 2011) Abstract: Many practical systems are phased-mission systems (PMS) where the mission consists of multiple, consecutive, and non-overlapping phases. An accurate reliability analysis of a PMS must consider the statistical dependencies of component states across phases as well as dynamics in system configuration, success criteria and component behavior. In this paper, we propose an efficient method for exact reliability evaluation of k-out-of-n systems with identical components subect to phased-mission requirements where the k values and failure time distributions can change with the phases. We also consider the time-varying and phase-dependent failure rates and associated cumulative damage effects. The proposed method is based on conditional probabilities and an efficient recursive formula to compute these probabilities. The main advantage of this method is that both its computational time and memory requirements are linear in terms of the system size. Keywords: phased-mission system, k-out-of-n redundancy, reliability analysis. 1. Introduction The operation of missions encountered in aerospace, nuclear power, and many other applications often involves several different tasks or phases that must be accomplished in sequence [1]. Systems used in these missions are usually called phased-mission systems (PMS). A classic example is an aircraft flight that involves taxi, take-off, ascent, levelflight, descent, and landing phases [2]. During each phase, the system has to accomplish a specified task and may be subect to different stresses and environmental conditions as well as different reliability requirements [3]. For example, in a twin-engine airplane, one engine is required during the taxi phase, but both engines are necessary during the take-off phase. In addition, the engines are more likely to fail during the take-off period because they are generally under enormous stress in this phase as compared to other phases of the flight profile. Thus, system configuration, success criteria, and component failure behavior may change from phase to phase [1-3]. The dynamic behavior of PMS usually requires a distinct model for each phase of the mission in the reliability analysis [1]. Further complicating the analysis are statistical dependencies across the phases for a given component. For example, the state of a component at the beginning of a new phase is identical to the state at the end of the previous phase [1-3]. Consideration of these dynamic dependencies poses unique challenges to existing reliability analysis methods [1, 2]. Considerable research efforts have been expended in the reliability analysis of PMS over the past four decades. *Corresponding author s samari@ptc.com 604

2 Suprasad V. Amari and Liudong Xing 605 However, even with the many advances in computing technology, only small-scale PMS problems can be solved accurately due to high computational complexity of the existing methods. A state-of-the-art review of PMS reliability modeling and analysis techniques is provided in [1]. Among many system structures, the k-out-of-n system structure has been a popular type of redundancy and it was introduced by Birnbaum, Esary, and Saunders [4] in The k-out-of-n: G system, or simply k-out-of-n system, consists of n components and it functions if and only if at least k of the n components function [5]. In other words, the system functions (fails) if and only if at most n-k (at least n-k+1) components fail. The k- out-of-n redundancy has found a wide range of applications in both industrial and military systems. Examples include cables in a bridge, a data processing system with multiple video displays, communication systems with multiple transmitters, and the multi-engine system in an airplane [6, 7]. Both series systems and parallel systems are special cases of the k-out-of-n systems. Although the k-out-of-n system has been studied extensively in the literature, the maor focus of these studies is on the binary state systems subect to single phase missions. However, most of the practical applications of k-out-of-n systems are subect to phased-mission requirements where the number of working components required (k values) can change with the phases. An example of phased-mission k-out-of-n systems is the multi-engine system in an airplane. Applications of phased-mission k-outof-n systems can be found in a wide range of systems including space systems [8], airborne weapon systems [9], and distributed computing systems [3, 10]. In this paper, we propose an efficient method for exact reliability evaluation of phased-mission k-out-of-n systems with identical components, where the k values and failure time distributions can vary with the phases. The time-varying and phase-dependent failure rates and associated cumulative damage effects are also considered. The proposed method is based on conditional probabilities and an efficient recursive formula to compute these probabilities. The main advantage of this method is that both its computational time and memory requirements are linear in terms of the system size. 1. System Description and Assumptions The proposed method is based on the following system description and assumptions: 1. The system mission consists of M consecutive and non-overlapping phases. 2. The system has n identical components. 3. The components can have phase-dependent and time-varying failure rates. 4. The system uses a k-out-of-n active redundancy structure where the k values can change with the phases. In other words, the number of good components required can vary with the phases. 5. The system is not repairable during the mission. 6. The overall mission is considered to be failed, if the system fails in any one of the phases. In other words, for the mission to be a success, the system must operate successfully during each of the phases. Refer to [2] for the generalized combinatorial phase requirements where the failure of the mission is expressed as a logical combination of phase failures. 2. Component Reliabilities In the proposed method, reliabilities of individual components in each phase are first calculated. The concept of equivalent age associated with the cumulative exposure model (CEM) is used to account for effects of phase-dependent stress on the failure properties of

3 606 Suprasad V. Amari and Liudong Xing the components [11]. Let F (t) be the stress dependent cumulative failure probability distribution of a component in phase. If the life-stress relationship follows the accelerated failure time model (AFTM), then F (t) can be represented as [11-13]: F ( t) = F( α t) (1) where F is the baseline distribution and α is the acceleration factor during the phase. Let τ be the duration of phase, and Q be the cumulative failure probability of the component at the end of phase. According to the CEM, we have [13]: Q = F( α 1 τ 1 + L + α τ ) (2) The reliability of the component at the end of phase can be calculated as: P = 1-Q. Let f be the probability that the component first fails in phase. It can be calculated as: f Q Q (3) = 1 where, by definition: Q 0 =0. Let q be the conditional unreliability of the component in phase given that it is working at beginning of the phase. It can be calculated as: f q = (4) 1 Q Similarly, the conditional reliability of the component in phase can be calculated as: f 1 Q P p = 1 = = (5) 1 Q Q P 1 1 Note that even though AFTM and CEM models are commonly used for reliability analysis of dynamic systems where different stresses are applied at different ages or time intervals, there are several other alternative models [13]. 4. Phased-Mission Reliability Analysis The system has n components and it requires at least k working components in phase for its successful operation. Hence, the system is considered to be failed if there are at least m = (n-k +1) failed components during phase. In addition, the system is considered to be failed if it fails in any one of the phases. Let x be the number of components that have failed before the completion of phase, where =1, 2,, M. Hence, the system is considered to be successful if x <m for all values of (i.e., all phases). The system reliability can be calculated by summing the probabilities of all combinations of x values: (x 1, x 2,, x M ) where x <m for all values of. These individual probabilities can be calculated using the multinomial distribution. However, this method is computationally inefficient because the number of combinations increases exponentially. A similar computation is also involved in the reliability analysis of a generalized multi-state k-outof-n system (GMSS) model, and this model has been studied extensively by several researchers for more than a decade [14]. Recently, reference [14] proposed a fast and robust algorithm to analyze the GMSS model by utilizing the properties of an embedded Markov chain associated with the sequence of x values: (x 1, x 2,, x M ). The speed and efficiency of the algorithm in [14] is compared with the existing methods for the GMSS model using several published benchmark problems. For small-scale problems, this algorithm is 150 times faster than the existing methods. For large-scale problems, it is 841,000 times faster. This enormous efficiency improvement motivated us to apply similar computations based on the Markov property to solve the PMS problems. Let Z,i be the probability of the system state such that x =i and x l <m l for all l<. That is, Z, i = Pr{ x = i; x 1 < m 1 ; L ; x 1 < m 1 } (6) 1 1 1

4 Reliability Analysis of k-out-of-n System with Phased-Mission Requirements 607 Using the Markov property of the x sequence [14], equation (6) can be calculated as: m 1 1 a= 0 Z = Z (7), i ( 1), a Pr{ x = i x 1 = a} where, n i n i Z1, i = Pr { x1 = i} = ( q ) ( p ) i 1 1 (8) 0 if i < a Pr{ x = i x = a} = n a (9) 1 i a n i ( q ) ( p ) if i a i a where q and p are defined in (4) and (5). Equation (7) forms the basic recursion for system reliability calculations. To improve the efficiency of the calculations and reduce the storage requirements, we use the following recursive relationships: n i + 1 q Pr{ x = i x 1 = 0} = Pr{ x = i 1 x 1 = 0} (10) i p i a { x i x = a} = Pr{ x = i x = a 1} Pr = 1 1 (11) n a + 1 q { x = 0 x = } = ( p ) n 0 Pr 1 (12) Once we calculate Z M,i values using the recursive formulas, we can calculate the system reliability R as: m M 1 Z i= 0 M, i R = (13) Similar to the recursive method of [14], the computational complexity of the recursive method for the reliability analysis of phased mission k-out-of-n systems proposed in this section is O(nmM),where m is the mean value of the vector m = [m 1, m 2,, m M ]. The MATLAB script provided in [14] has been modified according to the proposed method to compute reliability of phased mission k-out-of-n systems. 5. Numerical Examples In this section, we first demonstrate the step-by-step procedure of the proposed method using a simple system. Next, we demonstrate the efficiency of the proposed method using a large system Example 1: A Simple System Consider a phased-mission k-out-of-n system with 4 phases. The system has five components. Hence, n = 5. The baseline failure time distribution of each component is Weibull with η = 1000 and β = 2. The cumulative distribution function for the Weibull distribution is shown in (14). β t F( t; η, β ) = 1 exp (14) η The duration of phases and the phase-dependent system parameters (k and α values) are shown in Table 1.

5 608 Suprasad V. Amari and Liudong Xing Table 1: Phase-Dependent Requirements and Parameters Phase Phase 1 Phase 2 Phase 3 Phase 4 Duration k α In the proposed method, we first calculate the conditional reliabilities of components in all phases using equations (1)-(5): p = [p 1, p 2, p 3, p 4 ] = [0.9996, , , ]. From Table 1, we have: k = [k 1, k 2, k 3, k 4 ] = [2, 4, 3, 2]. Further, we have: n = 5 and M = 4. Using the recursive method in Section 4, in particular, (7), we calculate Z,i at the end of each phase as shown in Table 2. According to (13), we obtain the mission reliability of the system as: R PMS = Z 4,0 + Z 4,1 + Z 4,2 + Z 4,3 = The CPU time for solving this problem is 5.8E-5 seconds. Refer to [14] for a method to calculate these small CPU times accurately. Table 2: Z,i Values Z,0 Z,1 Z,2 Z,3 Z,4 Z, e e e e e e e e e e Example 2: A Large System In this section, we consider a large-scale PMS problem to demonstrate the efficiency of the proposed method. The inputs are fixed values and can be reproduced exactly for the verification and future research comparisons. We used modulo operator (mod), i.e., remainder, to generate non-monotonic inputs. The system has 100 components. Hence, n = 100. The baseline failure time distribution of each component is Weibull with η = 5000 and β = 2. The mission has 200 phases. Hence, M = 200. The duration of phase is: τ = 1 + mod(, 10). Hence, the total mission duration is: t =1100. The k value for phase is: k = 10 + mod(, 75). Further, m = n-k +1. The acceleration factors during phase are: α = mod(, 10). Using the proposed method, the mission reliability is: R PMS = , mission unreliability is 3.56E-4, and the CPU time is seconds. 6. Conclusions We presented a new model for k-out-of-n systems with phased-mission requirements and proposed a recursive method for computing the mission reliability. The proposed method is computationally efficient and can be used to find the reliability of large-scale k- out-of-n systems subect to time-dependent and phase-dependent failure parameters. The method can easily be integrated with optimization algorithms to find the optimal configurations for phased mission systems [15]. Acknowledgement: Thanks are due to referees for their useful and helpful comments. References [1] Xing, L. and S. V. Amari. Reliability of Phased-Mission Systems. Chapter 23 in Handbook of Performability Engineering (Editor: K. B. Misra), Springer, 2008, pp [2] Xing, L. Reliability Importance Analysis of Generalized Phased-Mission Systems. International Journal of Performability Engineering 2007; 3(3): [3] Somani, A. K., J. A. Ritcey, and S. H. L. Au. Computationally Efficient Phased-Mission Reliability Analysis for Systems with Variable Configurations. IEEE Trans. on Reliability

6 Reliability Analysis of k-out-of-n System with Phased-Mission Requirements ; 41(4): [4] Birnbaum, Z. W., J. D. Esary, and S. C. Saunders. Multi-Component Systems and Structures and Their Reliability. Technometrics 1961; 3(1): [5] Misra, K. B. Reliability Analysis and Prediction: A Methodology Oriented Treatment. Elsevier, The Netherlands, [6] Amari, S. V. and H. Pham. A New Insight into k-out-of-n Warm Standby Model. International Journal of Performability Engineering 2010; 6(6): [7] Amari, S. V., M. J. Zuo, and G. Dill. O(kn) Algorithms for Analyzing Repairable and Non- Repairable k-out-of-n:g Systems. Chapter 21 in Handbook of Performability Engineering (Editor: K. B. Misra), Springer, 2008, pp [8] Pedar, A. and V. V. S. Sarma. Phased-Mission Analysis for Evaluating the Effectiveness of Aerospace Computing-Systems. IEEE Trans. on Reliability 1981; R-30(5): [9] Winokur, H. S. Jr. and L. J. Goldstein. Analysis of Mission-Oriented Systems. IEEE Trans. on Reliability 1969; R-18(4): [10] Bricker, J. L. A Unified Method for Analyzing Mission Reliability for Fault Tolerant Computer Systems. IEEE Trans. on Reliability 1973; R-22(2): [11] Amari, S. V. and R. Bergman. Reliability Analysis of k-out-of-n Load-Sharing Systems. Proc. of Annual Reliability & Maintainability Symp. 2008; pp [12] Amari, S. V. and G. Dill. A New Method for Reliability Analysis of Standby Systems. Proc. of Annual Reliability & Maintainability Symposium 2009; pp [13] Amari, S. V., K. B. Misra, and H. Pham. Tampered Failure Rate Load-Sharing Systems: Status and Perspectives. Chapter 20 in Handbook of Performability Engineering (Editor: K. B. Misra), Springer, 2008, pp [14] Amari, S. V., M. J. Zuo, and G. Dill. A Fast and Robust Reliability Evaluation Algorithm for Generalized Multi-State k-out-of-n Systems. IEEE Trans. on Reliability 2009; 58(1): [15] Astapenko, D. and L. M. Bartlett. Phased Mission System Design Optimisation Using Genetic Algorithms. International Journal of Performability Engineering 2009; 5(4): Suprasad V. Amari is a Senior Reliability Engineer at Parametric Technology Corporation, U.S.A. He pursued his Master s and Ph.D. degree in Reliability Engineering at the Reliability Engineering Centre of Indian Institute of Technology, Kharagpur, India. He has published over 50 research papers in reputed international ournals and conferences. He is on the editorial boards of the International Journal of Performability Engineering and International Journal of Reliability, Quality and Safety Engineering and also on a management committee of RAMS. He is a member of the US Technical Advisory Group (TAG) to the IEC Technical Committee on Dependability Standards (TC 56), advisory board member of several international conferences, and a reviewer for several ournals on reliability and safety. He received 2009 Stan Oftshun Award from SRE for the best RAMS paper. He is a senior member of ASQ, IEEE, and IIE; and a member of ACM and SRE. He is also an ASQ-certified Reliability Engineer. Liudong Xing received her M.S. and Ph.D. degrees in Electrical Engineering from the University of Virginia, Charlottesville, USA in 2000 and 2002, respectively. Dr. Xing is currently an Associate Professor with the Electrical and Computer Engineering Department, University of Massachusetts Dartmouth. Dr. Xing served as a program co-chair for IEEE DASC 2006, a program vice chair for ICESS 2007 and ICPADS 2008, and an associate guest editor for the Journal of Computer Science on a special issue of Reliability and Autonomic Management in She is the Assistant Editorin-Chief for International Journal of Performability Engineering. Dr. Xing received the IEEE Region 1 Technological Innovation (Academic) award in Her current research interests include dependable computing and networking, reliability engineering, and wireless sensor networks. She is a senior member of IEEE and a member of Eta Kappa Nu.