8

BASED ON THE REPORT OF EXAMINER IIThe research scholar is very much grateful to you for scrutinizing the thesis and offering the most valuable suggestions which is very helpful not only for the improvement of the quality of the present work but also for pursuing further research after acquiring Ph.D, under your able scrutiny. This is a study undertaken by the research scholar in the area of reliability analysis of (k,n) system using the mathematical analysis. Hence the research scholar would like to humbly submit that the work leading to Ph.D is submitted under the faculty of Mathematics. As per the suggestions given by you, the scholar has carried out the corrections to the best of her knowledge. The scholar is thankful to you and appreciates the effort taken by you to ensure the clarity and authenticity in her work. Your kind observation is hereby solicited over the corrections carried out in the thesis.CLARIFICATIONSQuery 1. Many a time the author cites the work of Rau as for example on page 47; Rau in his book analyzed the reliability of (m,n) systems with parallel repair. Based on that, the reliability analysis of k-out-of-n system with parallel repair is as follows. So it is not clear what follows is taken from that book (in this case it should have been mentioned in the preliminaries of the first chapter) or if here the original work the author starts, I would think it is from the book as on page 49 it is written: then the following equations are derived(derived by who? Derived how?).CORRECTIONS CARRIED OUT: In Chapter 2, the reliability analysis of repairable k-out-of-n systems with sensor has been analyzed by the research scholar. This analysis starts from section 2.3, where the subtitle is given as reliability analysis of k-out-of-n systems with sensor. From that section onwards the derivation is being solved by the author with reference to the previous works carried out by Rau (1970) and Joseph and Manohar (1998). The results in Section 2.2 The reliability analysis of (k,n) system with parallel repair is the work executed by Rau and the author would like to humbly draw the attention of the esteemed examiner, to the citation made earlier in the beginning of the section 2.2. The citation referred in section 2.2 and the equations 2.1 to 2.11 are the original published work of Rau (1970). This reference has been quoted to draw the support of the already published work, to this new approach of the author, which the author perceives, that the results of Rau (1970) would give a fine platform for the authenticity of the present work. The examiner has asked that section 2.2 may be included in the introductory part. Since there are equations, expressions and methodology to be used for comparison, I included Raus work as a section in the corresponding chapter as a citation section.Query 2. Numerical results in Page No.60 in Section 2.4.2.: are these the results of Rau or not?CORRECTIONS CARRIED OUT: The author has tried to extensively study the transient analysis of (1,2) system with sensor, as reported in section 2.4.1.To validate the expression 2.51, the author has taken the reported work (1,2) system with and without repair and without sensor- of Rau (1970), for reference and comparison

Please refer pages 60 and 61:For the validity and the comparison, the reference of Rau (1970) is considered. According to his results

The MTSF of the (1, 2) system without repair is (2.52)

Also, the MTSF of the (1, 2) system with repair is (2.53)

Both the above results are for (1,2) system without sensor. As derived in the previous section, The MTSF of the (1, 2) system with repair and sensor is (2.54)These results along with the result of MTSF of (1,2) system with sensor derived in Section 2.4 are taken for numerical computation and the comparison is shown as in Table (2.1) and figure (2.1).

Query 3: In Page No.67 it is written: in state (i,A) the following equations are obtained; how are they obtained? CORRECTIONS CARRIED OUT: In state (i,A), there is only regular sensor and service facility. The expressions for the set of equations are obtained with the usual definition of transient analysis of system with repair and sensor as explained in the previous chapter with the revised notations. It has been explained and the correction is being carried out in the same section in the revised thesis.Please refer pages 67 and 68:

In state (i, A), there is only regular repair and sensor and service, the system balance equations for the transient analysis is as follows:

for

(3.2)

When the number of failed components becomes equal to the threshold value, the additional sensor and service facility becomes active and the system will be in state (i, B). Therefore, the balance equations in that state are the following:

for

and (3.3)Query 4: Also on the same page the index i is omitted, in the definition of the repair rate.

CORRECTIONS CARRIED OUT:

In page 67, in the definition for repair rate, it has been found that index i has been missing and research scholar noticed the serious error and carried out the necessary correction the revised version.

, a typographical error whichthe author regrets.

Query 5: Sometimes the name equation has been misrepresented as on page No.69, formula 3.8 is not an equation (there is not an unknown) but an expression. Similarly for expression 3.14, 3.20,

CORRECTIONS CARRIED OUT: The author is grateful for making to understand a new concept from your observation. Since there is an equal to sign in expressions 3.8, 3.14,3.20etc., the researcher thought that the word equation has to be used. But, you have explained that as there is no unknown, the term expression has to be used. The necessary correction is carried out.

Query 6: How are expressions 3.19, 3.24, 3.25 obtained?

CORRECTIONS CARRIED OUT:

It has been noticed that some steps have been omitted in the derivation of 3.19, 3.24, 3.25 All these have been included in the revised version to the best of the knowledge. The author pleads to go through them.

In the revised version, as the researcher included some more steps in some derivations the equations are renumbered. The equation number 3.19 of the old version of the thesis is now reconstructed and derived as follows.

The failure density function is

(3.20)

The expression for and are substituted from equation (3.17) and simplified. Thus, the expression for failure density function is

(3.21)Similarly, the previous equations 3.24 and 3.25 in the old edition are now equations 3.26 and 3.27 respectively.Let, P denote the revenue per unit time per operating system and C is the operating cost per system. Also, when regular repairmen are in service, the cost per system is C1. Therefore the expectation of the profit per unit time with Single Sensor Model (SSM)

= (Total Revenue-Cost) Expected number of components in operation in Single Sensor Model (SSM) Expected cost of regular sensor and repair facility = (3.26)If there is additional sensor and repair facility, the cost per unit time for the system is C2. Thus the expected profit per unit time with Additional Sensor Model (ASM) = (Total Revenue-Cost) Expected number of components in operation in Additional Sensor Model Expected cost of regular sensor and repair facility Expected cost of additional sensor and repair facility

= (3.27)Query 7: How are expressions 5.27, 5.29 obtained?The steps for obtaining expressions 5.27 and 5.29 are now included in the revised edition as follows.

CORRECTIONS CARRIED OUT:

From the definition of Tm given in the previous section and the equations (5.22) the following results are obtained. Integrating equations (5.22) over the interval using the results given in equation (5.9)

Thus, (5.27)

Similarly, integrating

and substituting (5.9), By, recursive substitution of these equations in (5.22), the following equation follows.

for From the results, by recursive substitution in equation (5.22) the following expressions are obtained.

For single repair facility for and

(5.28)

For multiple repair facility for

and (5.29)Query 8: I checked the calculation of the limit on page 106 and I found that it is not correct. So expression 4.85 is not correct.

CORRECTIONS CARRIED OUT: The author regrets the inadvertent mistake committed due to oversight. It is now rectified.

Query 9: What is the meaning of the first sentence of section 3.4?

CORRECTIONS CARRIED OUT:

The sentence was The definition of basic probability derived the following results.The sentence is written in order to explain that the expected number of components in operation for both the models is obtained using the definition of expectation of a random variable. i.e. for .Now the correction is incorporated.

Expected number of components in operation in Additional Sensor Model (ASM) is equal to

(3.30)

Expected number of components in operation in Single Sensor Model (SSM) is equal to (3.31)

Expected number of components waiting for or receiving repair in Additional Sensor Model (ASM) is equal to (3.32)

Expected number of components waiting for or receiving repair in Single Sensor Model (SSM) is equal to (3.33)

Here, is the probability that the system works with single sensor and repair facility. No additional facility is available for this state.Query 10: The author should state very clearly if the results are original and if yes how the results are obtained and preferably in the usual mathematical form of proposition and proof.The author swear says that all that is reported in the thesis are very much the original work done sincerely and it is the extension of the already reported works in the field. As the thesis is on the topic of applied mathematics, the author couldnt follow the usual mathematical analysis of theorem and proof method. Kindly accept the findings as they are based on the genuine and original study of the author.

The researcher regrets the errors in the thesis due to the inadvertent mistakes committed during the preparation and proofreading of the report. As per the instructions cited in the examiners evaluation report, the author has keenly observed the areas and carried out the corrections as far as possible. Hence, the author humbly pleads your goodself to reconsider of the above mentioned areas and offer a positive comment for the recommendation of her thesis. BASED ON THE REPORT OF EXAMINER I

Query: 1. In page xvi-xviii, at time t is missing in Pj(t),,, pj(t), I have made the above corrections in pages xvi, xvii and xviii.Query: 2.In page xvii, C-Operating cost per system. I assume that, one system consideration I have made the above correction in page xvii.Query: 3.In page 3, last statement replace set by state. I have made the above correction in page 3. A Markov process whose state space is discrete is called a Markov chain.Query: 4.In page 5, big O is used instead of small o. It is a mistake. Author is requested to correct all the places in the thesis wherever big O is typed. I regret for the mistake. I have done the necessary corrections in all the places in the thesis wherever big O is used.Query: 5. In page 7, the following statement is wrong. but the time spent in the state before the transition occurs, is random. It is still random, but it may follow exponential distributionI have made the above correction in page 7.Query: 6. In page 17, Poisson distribution is described. It has probability mass function not probability density function. The second paragraph discusses Poisson process with the started sentence Poisson distribution. I have made the above correction in page 17.

Another commonly used distribution is Poisson distribution which is a limiting case of a binomial distribution with parameters as and probability of success is very small such that =np. The probability mass function of the Poisson distribution is stated as

. (1.7)The Poisson Process is used generally to model the random event such as random failures in non-overlapping intervals of times. The probability of the occurrence of the random event in intervals of the same length is identical and does not depend on the location. Query: 7. In page 18, instead of variance standard deviation is written.

I have made the above correction in page 18. The mean of the distribution is , also known as time to failure (MTTF) for constant hazard rate and variance is .Query: 8. In page 19, reference for Weibull (1951) is missing at the end. I have included the Reference. Wallodi Weibull. A Statistical Distribution Function of Wide Applicability, ASME Journal of Applied Mechanics, Transaction of the American Society of Mechanical Engineers, Vol. 18, No.3, pp. 293-297, Sept. 1951.Query: 9. In page 22, MTSF is referred as the mean time to first failure.I have made the above correction in page 22. MTSF is the mean time to system failure while MTBF is the mean time between two successive component failures.Query: 10. There is no citation for sensor in the whole section.I have made the above correction the citation is included.Query: 11. The reference for Rau (1970) is missing at the end. Author is requested to cross check all the references.The reference was there. All the references are cross checkedJohn, G. Rau. Optimization and Probability in systems Engineering, Van Nostrand Reinhold Company, 1970.Query: 12. In page 38 Krishnamoorthy et al. (2001,2002,2004) are wrong. Few articles are with two authors.I have made the correction in page 38. Query: 13. In page 42, replace Chapman-Kolmogorov forward equations with Kolmogorov forward equations.I have made the correction in page 42. Kolmogorov forward equations and Laplace transform techniques are used to derive, the steady state availability and MTTFF of the system.Query: 14. In many places author uses the word done. Author is requested to replace done with a better and correct word which suits to the corresponding statement.I have done the necessary corrections in all the places in the thesis.Query: 15. In page 49, to get steady state results, why Laplace Transform is needed?The method of solution of system of balance equations with the initial conditions is very tedious. Also, by using Laplace Transform simplification of these expressions is very easy. So to find the steady state solution Laplace Transform method is used. Query: 16. In page 55, Q matrix is not a symmetric matrix.

I regret for the error. It is a real tri-diagonal matrix. Equations (2.20) and (2.21) is expressed as , (2.22)

where is a real tri diagonal matrix. Query: 17. In page 81, replace Zhang and Zuo (2003) by Zhang et al. (2003).I have done the necessary corrections. As some more steps included in the thesis page number has been changed and it is in page 83. Zhang et al (2003) obtained the reliability indices for a circular system with a single repairman. They analyzed a repairable circular consecutive-2-out-of-n system with priority in repair.Query: 18. In page 92 and 100, j\in Q are wrong.I made the necessary corrections in page 94 and 102.Query: 19. Reference 87 is with Wiley, 2001.I made the necessary corrections. Trivedi, K.S. Probability and Statistics with Reliability, Queuing and Computer Science Applications, 2nd edition, John Wiley and Sons , 2001.

SUPERVISOR DOCTORAL COMMITTEE MEMBER IDOCTORAL COMMITTEE MEMBER II