LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
|
FOURTH SEMESTER – APRIL 2006
ST 4955 – RELIABILITY THEORY
Date & Time : 29-04-2006/9.00-12.00 Dept. No. Max. : 100 Marks
Section-A (10×2=20 marks)
Answer ALL the questions. Each question carries TWO marks.
- Define the terms: (a) Reliability function R(t)
(b) Hazard function r(t)
- In the usual notation, show that MTBF = R*(0)
- If n components functioning independently and having equal reliabilities are operating in parallel, find the reliability of the entire system.
- Comment on the following statement: Series and parallel systems are particular cases of an (m, n) system.
- What are (a) parallel-series and (b) series-parallel systems?
- Define a coherent structure and give two examples.
- Define (a) Minimal path vector (b) Minimal cut vector.
- What do you mean by (i) the number of critical path vectors of component i and (ii) relative importance of component i?
- Give an example of a set of random variables that are not associated.
- What is a cumulative damage shock model?
Section-B (5×8=40 marks)
Answer any FIVE questions
- Obtain the reliability function, hazard rate and the system MTBF for Weibull distribution with the parameters λ and α.
- Suppose that gi(t) is the density function for Ti, the time to failure of ith component in a standby system with three independent and identical components and is given by gi(t) = λ e-λt, i = 1, 2, 3; t>0. Obtain the system failure time density function and hence find its expected value.
- What is a series system? Obtain the system failure time density function for a series system with n independent components. Suppose each of the n independent components has an exponential failure time distribution with constant failure rate λi, i = 1, 2, 3, …, n. Find the system reliability.
- Let Φ be a coherent structure. Show that
Φ(x Ц y) ≥ Φ(x) Ц Φ(y)
Further, show that the equality holds for all x and y if and only if the structure is parallel.
- Given the structure Φ, define the dual of the structure Φ. Also, show that the minimal path sets for Φ are the minimal cut sets for ΦD.
- Consider a coherent system with three components having the structure function Φ(x1, x2, x3) = x1. (x2 Ц x3)
Determine the number of critical path vectors of each component. Also determine the relative importance of each component. Are components 2 and 3 equally important?
- When do you say that a set of random variables T1, T2,… , Tn are associated? Show that a set consisting of a single random variable is associated.
- Let the density of exist. Show that F is DFR if and only if r(t) is decreasing in t.
Section-C (2×20 = 40 marks)
Answer any TWO questions. Each carries TWENTY marks
19.a. What is a series- parallel system of order (m, n)? Write down the system reliability and system failure rate of the same. (10 marks)
- Assuming that the components have identical constant failure rate λ, obtain MTBF of the series- parallel system. (10 marks)
20.a. Define the terms. (i) System availability.
(ii) Steady state availability. (4 marks)
- A system consists of a single unit, whose lifetime X and repair time Y are independent random variables with probability density functions f (.) and g (.) respectively. Assume that initially at time zero, the unit just begins to operate. Determine the reliability, availability and steady state availability of the system. (4+6+6 marks)
21.a. Let Φ be a coherent structure. Show that
Φ(x .y) ≤ Φ(x) .Φ(y)
Also, show that the equality holds for all x and y if and only if the structure is series. (10 marks)
- Let h be the reliability function of a coherent system. Show that h (p Ц p’) ≥ h (p) Ц h (p’) for all 0 ≤ p, p’ ≤ 1 (10 marks)
22.a. Show that the order statistics Y1:n, Y2::n,…,Yn:n corresponding to n independent random variables are associated. (10 marks)
- Examine whether Gamma distribution G (λ, α) is IFR or DFR. (10 marks)