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 g_i(t) is the density function for T_i, the time to failure of i^th component in a standby system with three independent and identical components and is given by g_i(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 λ_i, 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 Φ^D.
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, T_2,… , T_n 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 Y_1:n, Y_2::n,…,Y_n:n corresponding to n independent random variables are associated. (10 marks)

Examine whether Gamma distribution G (λ, α) is IFR or DFR. (10 marks)

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