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Executive Summary of the research work done by CHACKO V M for the period of 10-02-2011 to 31-03-2014 MRP (S)-835/10-11/KLCA019/UGC-SWRO dated 10-02-2011 Sanction Letter Title: Total time on test transform ordering of semi-Markov process with application to multistate system reliability theory Reliability is a critical measure of performance of engineering systems such as power generators, space crafts, telecommunication networks, control systems, nuclear reactors, oil and gas pipelines etc. Conventional reliability theory is built on a framework in which both the systems and its components can be in one of only two possible states: ‘working ‘ or ‘failed’. Consequently, the system structure function is a binary function of binary variables. However the binary feature of these reliability models places serious limitations on its utility, because most of the engineering systems and their components exhibit many levels of performance between two extremes of ‘working’ and ‘failed’. Multistate system reliability models allow both system and components to assume more than two levels of performance. First passage times of appropriate stochastic process have often been used to represent times to failure of devices or systems which are subject to shocks and wear, random repair time and random interruptions during their operations. The life distribution properties of these processes have therefore been widely investigated in Multistate system reliability and maintenance literature. The life distributions involved in devices or systems have several interesting properties such as increasing mean residual life (IMRL), decreasing mean residual life (DMRL), etc. The total time on test (TTT) transform is used as a tool for identification of failure distribution model in binary system. Marshell and Shaked (1983), (1986) and Shantikumar (1984) considered processes with new better than used (NBU) first passage times. Belzunce et al. (2002) derived, for the uniformizible, continuous time Markov process, conditions in terms of discrete uniformized Markov chain for the second order NBU and NBU based on laplace transformation classes. Lam (1992) studied NBUE and new worse than used in expectation (NWUE) properties of increasing Markov processes and Markov Chains and properties of Markov renewal processes. considered the NBUE and NWUE Use of TTT transform for the identification of failure rate models is discussed by Barlow and Campo (1975). Later, Klefsjo (1982) presented some relationship between the TTT transform and other ageing properties (with their duals) of random variable, eg. decreasing mean residual life (DMRL), NBU, NBUE, harmonically new better than used in expectation (HNBUE) and heavy tailedness. It further discussed properties of scaled TTT transform for some criteria of the mean residual life such as decreasing mean residual life average (DMRLA), decreasing harmonic mean residual life average (DHMRLA), new better than used harmonic mean residual life average (NBUHMRLA), and new better than used mean residual life average (NBUMRLA). But when we consider a complex system whose performance process is Markov or semiMarkov, we need the knowledge of DMRLA properties or other relevant ageing properties for applying suitable maintenance and repair/replacement policies. The identification of failure rate model of a system whose performance process is Markov/semi-Markov will be helpful to the engineers and designers for applying suitable maintenance and repair or replacement policies, since identification of failure rate model using TTT describes new methods for analyzing nonnegative observations. Chacko et.al (2010) discussed use of TTT transform in identifying failure rate model of semi-Markov reliability system. I consider a semi-Markov process whose first passage time distribution is IFR or IFRA or NBU or NBUE or DMRL (with their duals) or DMRL or DMRLA or NBUHMRL or NBUMRL (with their duals). We consider the reliability function based on the transition probability function in the upstates. The conditions of identification using transition probability function provide equivalent information drawn from failure rate function of first passage time variable. I introduced some sufficient conditions for the TTT ordering of the semi-Markov process based on MRL criteria built from transition probability function. This result has been presented at Annual Conference of Kerala Statistical Association held at University of Calicut during 1517 March 2012 and published in an International journal Reliability: Theory and Application, 01 (28) (vol.8) 53-64, 2013, march The identification of the failure rate model of first passage time distribution of a semiMarkov process is important. The results are applicable to systems like power generation system whose performance is measured in terms of productivity or capacity and having more than two levels of performance. Preventive or corrective maintenance can be applied to the MSS if we have the knowledge regarding its failure behavior, since type of the failure rate is an important parameter for the maintenance and replacement policies. Total time on test transform orders based on transition probability function of semimarkov system are studied. Connections among the new orders and other common stochastic orders are examined and investigated. To establish the implication between other ordering of two processes with TTT ordering. The advantage of the proposed ordering over the others in case of censored data is analyzed. The application to engineering areas are given. This results are published in Reliability: Theory And Applications, Title of the publication is-Total Time on Test Transforms ordering of semi-Markov system, 03 (34) (vol.9) 57-63, 2014, September . A survey has been conducted. It shows that the many engineering systems like computers, electric power generation systems, flight engine system, oil transportation systems, signal transmission and communication systems etc can be effectively modeled as Multistate systems. So that reliability calculation and importance and joint importance measure of components can be computed easily using MSS techniques. Using exponential assumption for life in a state, we could model the computers as MSS, being the failure occur only because of virus, voltage variation, jerking, overload etc but not because of ageing, and obtained the reliability as 0.98 for one year of mission time in the branded items ACER, SONY, LENEVO and COMPAQ. SONY is found to be stochastically better than other items, based on TTT ordering. We have developed a theory on Large multistate semi-Markov system and it is communicated for the publication. We have data from oil transportation companies and applied our results to the transportation network system. A wide area for the application comes under this results. Reliability analysis of more complex systems with large number of components and configuration make the system as multistate. As soon as a single component fails from a system of large components, the load is shared by other components and a reliability diminishing occur. It calls for MSS modeling and TTT ordering for the comparison. Research publication: papers published/ Accepted for publication/ Communicated for 1. Mean residual life criteria of first passage time of semi-markov process based on total time on test transforms, Reliability; Theory and Applications, 01 (28) (vol.8) 53-64, 2013, march 2. Presented at Annual Conference and National Seminar of Kerala Statistical Association, University of Calicut, Kerala. 3. Total Time on Test Transforms ordering of semi-markov system, Reliability: Theory And Applications (International Journal of World Reliability GroupGnedenko E-forum), 03 (34) (vol.9) 57-63, 2014, September 4. Total Time on Test Transform ordering in Large Multistate semi-Markov System (Communicated) 5. P-Birnbaum Distributions: Applications to Reliability and Banking Habits (To Appear) Reliability: Theory And Applications, March 2015 Presented at National Level Conference on Interdisciplinary Realms of Statistics, Sree Kerala Varma College, Thrissur, Jan 9-10, 2015 6. A Generalization of Birnbaum Saunders Distribution, Presented at National Seminar on Recent Advances in Statistical Theory and its Applications, 2526 Feb, 2015, St.Thomas College, Thrissur 7. Association in Time of Markov Process, Presented at National Conference on Applied Mathematics, 27-27, Feb, 2015, St.Thomas College, Thrissur References 1. Barlow, R. E. and Wu, A. S. (1978) Coherent system with multistate components, Math.Oper. Res., 3, 275-281. 2. Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing, Holt, Rinehart and Winston, New York. 3. R. E. Barlow and Campo, R. A. 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(1961) Markov renewal processes: definitions and preliminary properties, Ann. Math.Stat., 32, 1231-1242. 20. Unnikrishnan. N., P. G. Sankaran and B. Vinesh Kumar (2008) TTT transforms of order n and their implications in reliability analysis, Cochin University of Science and Technology 18