Development of models for the failure process of a repairable system has been an interesting research topic for decades. There have been many models developed in the literature. However, more research is still needed to deal with various difficulties raised in the practical applications, which includes the cases that most real systems are composed of more than one component and that the failure process may not be stochastically monotone. For such cases, we have recently developed three models, which are briefly discussed in this paper

Wu, Shaomin

Kent Academic Repository

Kent Academic RepositoryFull text document (pdf)Copyright & reuseContent in the Kent Academic Repository is made available for research purposes. Unless otherwise stated allcontent is protected by copyright and in the absence of an open licence (eg Creative Commons), permissions for further reuse of content should be sought from the publisher, author or other copyright holder. Versions of researchThe version in the Kent Academic Repository may differ from the final published version. Users are advised to check http://kar.kent.ac.uk for the status of the paper. Users should always cite the published version of record.EnquiriesFor any further enquiries regarding the licence status of this document, please contact: researchsupport@kent.ac.ukIf you believe this document infringes copyright then please contact the KAR admin team with the take-down information provided at http://kar.kent.ac.uk/contact.htmlCitation for published versionWu, Shaomin  (2017) Three models for the failure process of a repairable system.    In: The 10thinternational conference on mathematical methods in reliability, July 3-6, 2017, Grenoble, France.DOILink to record in KARhttp://kar.kent.ac.uk/61622/Document VersionAuthor's Accepted ManuscriptTHREE MODELS FOR THE FAILURE PROCESS OF AREPAIRABLE SYSTEMSHAOMIN WUUniversity of KentCanterbury CT2 7FS, UKemail: s.m.wu@kent.ac.ukDevelopment of models for the failure process of a repairable system has beenan interesting research topic for decades. There have been many models de-veloped in the literature. However, more research is still needed to deal withvarious difficulties raised in the practical applications, which includes the casesthat most real systems are composed of more than one component and that thefailure process may not be stochastically monotone. For such cases, we haverecently developed three models, which are briefly discussed in this paper.Keywords: Non-homogeneous Poisson process, renewal process, repair,doubly geometric process, semi-geometric process.1. IntroductionModelling the failure process of a repairable system is needed in many ap-plications such as optimisation of maintenance policy and lifecycle costing.For this purpose, there are many models that have been developed in theliterature. If we denote times-between-failures of a repairable system asa sequence of random variables {Xk, k = 1, 2, . . . }, models in the existingliterature can be categorised into two classes, as shown in Figure 1. TypeI models include the renewal process, the geometric process (GP) and itsextensions [1,2,3,4]; type II models include the non-homogeneous Poissonprocess (NHPP) and its various extensions such as the virtual age model[5], the reduction of intensity model [6] and the hybrid intensity models [7].Both type I and type II models have limitations: most of those models aredesigned for the failure process of a single component system and type Imodels have restrictive assumptions. However, most real systems are com-posed of more than one component and the failure process may not needthe restrictive assumptions.While those models developed in pre-2017 have been widely studied, thispaper mainly focuses on three models we have developed and published inFigure 1. A taxonomy of models for the failure process of a repairable system2017. Those three models are: two extensions of the GP [4,3] and a hybrid-intensity model [7] and briefly discussion them.2. Two extensions of the GPThe definition of the GP is given by Lam [1].Definition 2.1. [1] Given a sequence of non-negative random variables{Xk, k = 1, 2, . . . }, if they are independent and the cdf of Xk is given byF (ak−1x) for k = 1, 2, . . . , where a is a positive constant, then {Xk, k =1, 2, · · · } is called a geometric process (GP).The GP has been widely applied in optimisation of maintenance policy sinceits introduction in 1988, which may be mainly due to its elegant expressionof the time-to-failure distribution after a repair and little consideration hasbeen devoted onto the real-world application of this model. As can beseen, the GP has two restrictive assumptions: (1) {Xk, k = 1, 2, · · · } areindependent and their distributions and (2) Xk is stochastically monotone.Those assumptions may restrict it from wide applications. To relax thosetwo assumptions, we introduce the following two definitions, respectively.Definition 2.2. [3] Given a sequence of non-negative random variables{Xk, k = 1, 2, . . . }, if P{Xk < x|Xk−1 = xk−1, . . . , X1 = x1} = P{Xk <x|Xk−1 = xk−1} and the marginal distribution of Xk is given by P{Xk <x} = Fk(x)(≡ F (ak−1x)), where a is a positive constant, then {Xk, k =1, 2, . . . } is called a semi-geometric process (SGP).Definition 2.3. [4] Given a sequence of non-negative random variables{Xk, k = 1, 2, . . . }, if they are independent and the cdf of Xk is given byF (ak−1xh(k)) for k = 1, 2, . . . , where a is a positive constant, h(k) is afunction of k and the likelihood of the parameters in h(k) has a knownclosed form, and h(k) > 0 for k ∈ N, then {Xk, k = 1, 2, · · · } is called adoubly geometric process (DGP).As can be seen, the SGP removes the independence assumption of the GPand the DGP can model the cases where Xk is stochastically monotone ornon-monotone.[3,4] discuss the probabilistic properties of the SGP and the DGP, re-spectively and also describe parameter estimation methods. Numerical ex-amples on real-world data show that the SGP and DGP outperform theGP and other models.3. A hybrid intensity model[7] proposed two models for the failure process of a repairable series multi-component system. Assume a replacement is carried out upon system fail-ure. Then the failure process can be regarded as a process resulted fromthe maintenance activities of two subsystems: one subsystem that is main-tained with minimal repair and one subsystem, part of which is replacedupon system failure. [7] then proposes two models, Model I and Model II.On the basis of numerical examples, Model II outperforms model I. Denoteλ(t|Ht−) as the intensity function of the system. Then Model II gives thefollowing definition.λ(t|Ht−) =λ1(t) + λ2(t), if Nt = 0,λ1(t) +1m(Nt−1∑k=0λ2(t− TNt−k) + (m−Nt)λ2(t)), if 1 ≤ Nt < m,λ1(t) +1mm−1∑k=0λ2(t− TNt−k), if Nt ≥ m.where t ∈ (0,+∞), λ1(t) and λ2(t) are two intensity functions.The above model regards the failure process of a series system equivalentto that of a virtual system composed of two subsystems: subsystems 1and 2. Subsystem 1 contains one virtual component and subsystem 2 iscomposed of a pre-specified number, m, of virtual components. Broadlyspeaking, whenever the real system fails, minimal repair is assumed to beconducted on subsystem 1 and the oldest virtual component in subsystem2 is assumed to be replaced with a new identical virtual component.It is shown that Model II outperforms the four models (i.e, RP, GPNHPP, the virtual age model) on six artificially generated data and onsome real data [7].4. ConclusionThe three models discussed above may be useful in different applications.Our future work will be focused on investigation of their statistical proper-ties.AcknowledgementsThe author would like thank Professor Phil Scarf and Dr Guanjun Wangfor joint work.References[1] Y, Lam. Geometric processes and replacement problem. Acta MathematicaeApplicatae Sinica. 4, 366-377, 1988.[2] L. Bordes. and S. Mercier. Extended geometric processes: semiparamet-ric estimation and application to reliability imperfect repair, Markov re-newal equation, replacement policy. Journal of the Iranian Statistical Soci-ety, 12(1), 1-34, 2013.[3] S. Wu, and G. Wang. The semi-geometric process and some properties. IMAJournal of Management Mathematics. DOI: 10.1093/imaman/dpx002, 2017.[4] S. Wu. Doubly geometric processes and applications. Journal of the Opera-tional Research Society. DOI: 10.1057/s41274-017-0217-4, 2017.[5] M. Kijima, H. Morimura and Y. Periodical replacement problem withoutassuming minimal repair. European Journal of Operational Research 37(2),194-203, 1988.[6] L. Doyen. and G. Olivier. Classes of imperfect repair models based on reduc-tion of failure intensity or virtual age. Reliability Engineering and SystemSafety 84(1),45-56, 2004.[7] S. Wu, and P. Scarf. Two new stochastic models of the failure process ofa series system. European Journal of Operational Research 257(3), 763-772,2017.

Three models for the failure process of a repairable system

https://kar.kent.ac.uk/61622/1/Multi-component-system.pdf

Three models for the failure process of a repairable system

Abstract

Similar works

Full text

Available Versions

Kent Academic Repository