LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 53

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.

1. Define the terms: (a) Reliability function R(t)

(b) Hazard function r(t)

1. In the usual notation, show that MTBF = R^*(0)
2. If n components functioning independently and having equal reliabilities are operating in parallel, find the reliability of the entire system.
3. Comment on the following statement: Series and parallel systems are particular cases of an (m, n) system.
4. What are (a) parallel-series and (b) series-parallel systems?
5. Define a coherent structure and give two examples.
6. Define (a) Minimal path vector (b) Minimal cut vector.
7. What do you mean by (i) the number of critical path vectors of component i and (ii) relative importance of component i?
8. Give an example of a set of random variables that are not associated.
9. What is a cumulative damage shock model?

Section-B (5×8=40 marks)

Answer any FIVE questions

1. Obtain the reliability function, hazard rate and the system MTBF for Weibull   distribution with the parameters λ and α.
2. 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.
3. 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.
4. 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.

1. 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.
2. 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?

1. 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.
2. 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)

1. 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)

1. 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)

1. 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)

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

 

 

Go To Main page

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

AC 57

M.Sc. DEGREE EXAMINATION – STATISTICS

FOURTH SEMESTER – APRIL 2007

ST 4955 – RELIABILITY THEORY

 

 

 

Date & Time: 25/04/2007 / 9:00 – 12:00      Dept. No.                                       Max. : 100 Marks

 

 

SECTION-A (10 × 2 = 20 marks)

Answer ALL the questions. Each question carries TWO marks

1. Define the structure function of the system. Write down the structure function for k  out of  n structure.

 

1. If x is a path vector and y ≥ x, show that y is also a path vector.
2. If X₁, X₂, …, X_n are associated binary random variables, show that

(1-X₁),(1-X₂),…,(1-X_n) are  also  associated binary random variables.

1. Show that F is IFRA if and only if (αt) ≥ ^α(t) for all 0<α<1 and t≥0.
2. What  do you mean by

• the number of critical path vectors of component i and
• relative importance of component i?

 

1. Define the terms:

• System Reliability
• Steady state availability

 

1. If r(t) = λ; λ,t>0, obtain the corresponding probability distribution of time to failure.

 

1. In the usual notation, show that MTBF = R*(0).

 

1. Obtain the reliability of a parallel system consisting of n components, when the reliability of each component is known. Assume that the units are non-repairable.

 

1. Explain in detail an n unit standby system.

 

SECTION- B (5 × 8=40marks)

Answer any FIVE questions. Each question carries EIGHT marks

 

1. Consider 2 out of 3 system. Determine the number of critical path vectors of each component. Also determine the relative importance of each component. Are all the three components equally important?

 

1. 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.

Also show that the equality holds if and only if the system is parallel.

 

1. Suppose T₁, T₂, T₃,…,T_n  are the random variables that are associated. Show that

1. any subset of the associated random variables is also associated
2. A set consisting of a single random variable is associated.

 

 

 

 

 

 

 

1. If X₁, X₂, …, X_n are associated binary random variables show that

n                 n

1. a) P [ Π X_i=1] ≥ Π P [ X_i=1]

i=1              i=1

 

n                 n

1. b) P [ Ц X_i=1] ≤   Ц P [ X_i=1]

i=1              i=1

 

1. Obtain the reliability function, hazard rate and system MTBF for PH-distribution with representation (α, T).

 

1. 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,…,n. Find the system reliability.

 

1. 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 two independent components and is given by

g_i(t) = λ_i e^{– λit}, i=1,2; λ₁≠ λ₂.

Obtain the system failure time density function and hence find its expected value.

 

1. Find the mean life time of a (2,3) system of independent components, when the component lifetimes are uniformly distributed on ( 0, i ), i =1,2,3.

 

SECTION –C (2 × 20=40)

Answer any TWO questions. Each question carries TWENTY marks

 

1.  a) Show that increasing functions of associated random variables are associated.

1. b) Show that order statistics Y_1:n,Y_2:n,…,Y_n:n corresponding to n independent

random variables are associated.

1. c) Give an example of a set of random variables that are not associated. (6+8+6)

1.  a) If the probability density function of F exists, show that F is an IFR

distribution iff  r(t)↑t

1. b) Show that Wiebull distribution is a DFR distribution. Hence or otherwise,

establish that exponential distribution is both IFR and DFR.      (10+10)

1.  a) Obtain the reliability function, hazard rate and the system MTBF for Gamma

distribution with the parameters λ and p.

1. b) Suppose that g_i(t) is the density function for the T_i, the time to failure of i ^th

component in a standby system with two independent and identical components

and is given by g_i(t) = λe^{– λt}, i=1,2; λ,t>0. Obtain the system failure time

density function and hence find its expected value.                      (12+8)

1. a)  Define the terms: (i) System availability.

(ii) Hazard rate                                                  (4 marks)

1. 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)

 

 

Go To Main Page

 

      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 40

SECOND SEMESTER – April 2009

ST 2957 / ST 2955 – RELIABILITY THEORY

 

 

 

Date & Time: 27/04/2009 / 1:00 – 4:00   Dept. No.                                                   Max. : 100 Marks

 

 

SECTION –A (10 x 2 = 20)         

Answer any TEN questions. Each question carries TWO marks

 

1. Define the following: (i) Mean time before failure(MTBF)

(ii) Steady state availabilty

2. If the hazard function r(t)=3t², t>0, obtain the corresponding probability distribution of

time to failure

3. Obtain the reliability of a parallel system consisting of n components, when the

reliability of each component is known. Assume that the components are non-repairable.

4. Explain in detail an n unit standby system.
5. What is meant by reliability allocation?
6. Define a coherent structure and give two examples.
7. If x is a path vector and y ≥ x, show that y is also a path vector.
8. Write down the structure function for a two out of three system.
9. Let h ( p ) be the system reliability of a coherent structure. Show that h(p)is strictly

increasing in each p_i, whenever 0<p_i<1, i=1,2,3,…,n .

10. Give an example of a distribution, which is IFR as well as DFR.

 

SECTION-B (5×8 =40 marks)

Answer any FIVE questions. Each question carries EIGHT marks

 

11. Obtain the reliability function, hazard rate and the system MTBF for the following

failure time density function

f(t) = 12 exp(-4 t³)t²,  t>0.

12. 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 the parameter λ_i, i= 1,2,…,n. Find the system reliability.

13. Find the system MTBF for a (k,n) system, when the lifetime distribution is

exponential with the parameter λ. Assume that the components are non-repairable.

14. Assuming that the components are non-repairable and the components have identical

constant failure rate λ, obtain the MTBF of the series-parallel system.

15. Let Φ be a coherent structure. Show that

Φ(x .y ) ≤ Φ(x ) Φ(y )

Show that the equality holds for all x and y if and only if the structure is series.

16. 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.

Also, show that the equality holds if and only if the system is parallel.

17. If two sets of associated random variables are independent, show that their union is

the set of associated random variables.

18. Show that Wiebull distribution is a DFR distribution.

 

 

SECTION-C (2X20=40 marks)

Answer any two questions. Each question carries TWENTY marks

 

19. a) Obtain the reliability function, hazard rate and the system MTBF for exponential

failure time distribution with the parameter λ.                                                 (8 marks)

1. b) Obtain the system failure time density function for a (m, n) system. Assume that

the components are non-repairable.                                                      (12 marks)

20.a) Define the terms: (i) Hazard rate  and (ii) Interval reliability                (4 marks)

1. b) For a simple 1 out of 2 system with constant failure rate λ and constant repair rate

μ, obtain the system of  differential-difference equations. Also, obtain

the system reliability and system MTBF.                                              (16 marks)

21.a) Define: (i) Dual of a structure (ii) Minimal path vector and (iii) Minimal cut vector

(6 marks)

1. b) Let h be the reliability function of a coherent system. Show that

h( p . p‘) ≤ h( p ) . h( p‘) for all 0  ≤ p , p‘ ≤ 1.                        (10 marks)

Also show that the equality holds if and only if the system is series.

1. c) If X₁, X₂, …, X_n are associated binary random variables, show that

(1-X₁), (1-X₂),…,(1-X_n) are  also  associated binary random variables.(4 marks)

22.a) If the probability density function of F exists, show that F is an IFR

distribution iff  r(t)↑t.                                                                            (10 marks)

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

 

 

Go To Main page

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2012

ST 2957 – RELIABILITY THEORY

 

 

Date : 24-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

 

Answer all the questions                                                                                                             (10×2=20)

1. Define Reliability R(t) of a system
2. Define Hazard function r(t)

3.

R_A=0.87, R_B=0.85, R_C=0.89. Determine system reliability

4. Define Parallel-Series system
5. Define MTBF
6. Define a (k,n) system
7. Define Standby system
8. R(t) = e^{– 0.2t} determine the warranty period for a reliability of 0.9
9. An equipment has a hazard function r(t) = 6×10^-8t². The equipment is required to operate a 100

hours. What is the reliability at 100 hours?

10. Define a) DFRD b)IFRD

 

SECTION-B

Answer any five questions                                                                                                          (5X8=40)

 

11. Obtain the system reliability function R(t) and hazard function r(t) when the system failure time distribution follows Weibull distribution
12. Establish the following (3+3+2)
13. i) ii) If R^*(s) = LT{R(t)} then MTBF = R^*(0) iii)If T~Exponential distribution then MTBF=1/λ
14. Obtain system failure time density function for a (k,n) system
15. Define Series-Parallel system. Obtain system hazard function r(t) and MTBF for a

Series-Parallel System

15. Consider a series system consisting of two components with first component following a

exponential failure time distribution with λ=1/10,000 and second component following a

weibull with parameters β=6 and η=10,000. i)Obtain system reliability ii)Obtain system’s cdf

and pdf  iii) Given that the system has performed 500 hrs what is the reliability of the system

for an additional 1000hr mission  iv)Obtain the system failure rate v)What should be the

warranty period for a system reliability of 90%

16. Explain the methods of obtaining the reliability of a Complex system

17 Establish the following

1. F is IFR ó on [0,∞)
2. F is IFR ó
3. i) Establish: r(t) is a conditional probability function but not a conditional pdf
4. ii) Establish: r(t)↓t ó F is DFRD

 

SECTION-C

 

Answer any two questions                                                                                                         (2X20=40)

 

19. Obtain the reliability function R(t) and hazard function r(t) for the following failure time

distributions  i) Exponential   ii) Gamma

20. Obtain MTBF for the case when failure time(T) of a system is distributed as i) Exponential

ii)Weibull  iii) Gamma

21. Consider a Standby system of order 3 with T_i ~ Exponential(λ_i), i=1,2,3 . obtain the system

failure time density function and hence obtain the reliability function R(t) for the case when

λ₁= λ₂= λ₃ and λ₁≠ λ₂≠ λ₃(20)

22. Obtain system mean time between failure (MTBF) for a (k,n) system

 

 

Go To Main Page