Figure 6.35 presents a block diagram of a piping sys- tem made of three redundant parallel pumps: Pump1, Pump2, and Pump3.
6.5.1.1 Exponential Distributions of Components’ ttf,
Nonrepairable Components
All components in Fig. 6.35 are supposed to be not re- pairable and the probability distributions of the ttf ran- dom variables, assumed to be exponential, are based on the following assumptions:
• MTTFPump1 D10;000 h;
• MTTFPump2 D6;000 h;
• MTTFPump3 D7;000 h.
Pump1
Pump2
Pump3
Fig. 6.35 Block diagram parallel system, piping system
By Eq. 6.60 the reliabilityRS.t /of the parallel system is
RS.t /D an iD1
Ri.t /
D1Œ1RPump1.t /Œ1RPump2.t /Œ1RPump3.t / DR1.t /CR2.t /CR3.t /R1.t /R2.t /
R2.t /R3.t /R1.t /R3.t / Šexp
t
10;000
Cexp
t 6;000
Cexp
t 7;000
exp.2:67104t / exp.2:43104t /exp.3:10104t /:
Considering two values for the mission timeT,T D 4;000 h andT D 8;000 h, respectively, the values of the system reliability are
RS.T D4;000/Š0:737;
RS.T D8;000/Š0:687:
By Eq. 6.61 the unconditional failure ratefS.t /of the parallel system is
fS.t /D Xn iD1
fi.t /Y
j¤i
Œ1Rj.t /
Df1.t /Cf2.t /Cf3.t /Cf1.t /R2.t /R3.t / Cf2.t /R1.t /R3.t /Cf3.t /R1.t /R2.t / f1.t /ŒR2.t /CR3.t /
f2.t /ŒR3.t /CR1.t / f3.t /ŒR2.t /CR1.t /:
Similarly,S.t /is given by S.t /D
Xn iD1
i.t /Y
j¤i
Œ1Rj.t /
D1.t /C2.t /C3.t /C1.t /R2.t /R3.t / C2.t /R1.t /R3.t /C3.t /R1.t /R2.t / 1.t /ŒR2.t /CR3.t /
2.t /ŒR3.t /CR1.t / 3.t /ŒR2.t /CR1.t /:
Figure 6.36 presents the trend of the system’s proba- bility functionF .t /, reliabilityR.t /, density function
f .t /, and failure rate.t /compared with the trends of the components involved (i. e., three pumps).
The results illustrated in Fig. 6.36 and related to a parallel configuration of the system can be directly compared with those reported in Fig. 6.20 and related to the same components in a serial configuration.
As a consequence, comparing the results obtained, in terms of system reliability, the parallel configuration is much more reliable than the serial one.
Figure 6.37 presents the values of the reliabil- ity importanceIRi.t /for different values of t, while Fig. 6.38 presents the reliability importance for t D 4;000 h andt D10;000 h. Every graph shown in these figures was obtained with ReliaSoft® software:The most critical component is Pump1 because it is the most reliable one; in other words it is convenient to improve it and further increase the values of reliabil- ity.
6.5.1.2 Mix of Probability Distributions of Components’ ttf, Nonrepairable Components
Figures 6.39–6.41 illustrate the results obtained by as- suming the following distributions of the blocks’ ttf in Fig. 6.35:
• Pump1. Exponential distribution, MTTFPump1 D 10;000 h;
• Pump2. Normal distribution, MTTFPump2 D 6;000 h, and standard deviation of ttf 100 h.
• Pump3. Weibull distribution, scale parameter˛ D 7;000 h, and shape parameterˇD1:5.
Figure 6.39 presents the trend of the system’s proba- bility functionF .t /, reliabilityR.t /, density function f .t /, and failure rate.t /compared with the trends of the three components involved (i. e., blocks). Fig- ures 6.40 and 6.41 present the results of the reliability importance evaluation for the components of the par- allel block diagram.
6.5.1.3 Repairable Components and Exponential Distributions of ttf and ttr Random Variables
In this case every component in the parallel system in Fig. 6.35 is supposed to be repairable under correc- tive actions and the probability distributions of random
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Block Unreliability vs Time
Time, (t)
Unreliability, F(t)=1-R(t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
0.000 1.000
0.200 0.400 0.600 0.800
Unreliability Diagram1
System Pump 1 Pump 2 Pump 3
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Block Probability Density Function
Time, (t)
f(t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
-3.000E-20 2.000E-4
4.000E-5 8.000E-5 1.200E-4 1.600E-4
Pdf Diagram1
System Pump 1 Pump 2 Pump 3
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Block Failure Rate vs Time
Time, (t)
Failure Rate, f(t)/R(t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
-3.000E-20 2.000E-4
4.000E-5 8.000E-5 1.200E-4 1.600E-4
Failure Rate Diagram1
System Pump 1 Pump 2 Pump 3 ReliaSoft BlockSim 7 - www.ReliaSoft.com
Block Reliability vs Time
Time, (t)
Reliability, R(t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
0.000 1.000
0.200 0.400 0.600 0.800
Reliability Diagram1 System Pump 1 Pump 2 Pump 3
Fig. 6.36 Parallel system, exponential distributors.F .t /,R.t /,f .t /, and(t /. ReliaSoft®software
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Reliability Importance vs Time
Time, (t)
Reliability Importance Value
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
0.000 0.999
0.200 0.400 0.599 0.799
Importance Diagram1
Starting Block Ending Block Pump 3 Pump 2 Pump 1
Fig. 6.37 Parallel system. Reliability importance of components within the system. ReliaSoft®software
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Static Reliability Importance
Time = 10000
Reliability Importance Value
Pump 1 Pump 3 Pump 2
0.000 0.617
0.123 0.247 0.370 0.493
Reliability
3 Item(s) 100%
50%
0%
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Static Reliability Importance
Time = 4000
Reliability Importance Value
Pump 1 Pump 3 Pump 2
0.000 0.212
0.042 0.085 0.127 0.169
Reliability
3 Item(s) 100%
50%
0%
Fig. 6.38 Parallel system. Reliability importance of components within the system.t D4;000 h andt D10;000 h. ReliaSoft® software
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Block Unreliability vs Time
Time, (t)
Unreliability, F(t)=1-R(t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
0.000 1.000
0.200 0.400 0.600 0.800
Unreliability Diagram1
System Pump 1 Pump 2 Pump 3
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Block Failure Rate vs Time
Time, (t)
Failure Rate, f(t)/R(t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
0.000 1.000E+27
2.000E+26 4.000E+26 6.000E+26 8.000E+26
Failure Rate Diagram1
System Pump 1 Pump 2 Pump 3 ReliaSoft BlockSim 7 - www.ReliaSoft.com
Block Reliability vs Time
Time, (t)
Reliability, R(t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
0.000 1.000
0.200 0.400 0.600 0.800
Reliability Diagram1 System Pump 1 Pump 2 Pump 3
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Block Probability Density Function
Time, (t)
f(t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
0.000 2.000E-4
4.000E-5 8.000E-5 1.200E-4 1.600E-4
Pdf Diagram1
System Pump 1 Pump 2 Pump 3
Fig. 6.39 Parallel system, mix of distributions.F .t /,R.t /,f .t /, and.t /. ReliaSoft®software variables ttf and ttr are assumed to be exponential. In
particular the values of MTTF and MTTR are the fol- lowing:
• MTTFPump1 D10;000 h;
• MTTFPump2 D6;000 h;
• MTTFPump3 D7;000 h;
• MTTRPump1 D MTTRPump2 D MTTRPump3 D 100 h.
Figure 6.42 illustrates the state diagram, reporting the state of the components and of the system for differ- ent values of timet obtained by the application of the Monte Carlo simulation analysis. In the case of “full
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Reliability Importance vs Time
Time, (t)
Reliability Importance Value
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
0.000 1.000
0.200 0.400 0.600 0.800
Importance Diagram1
Starting Block Ending Block Pump 1 Pump 2 Pump 3
Fig. 6.40 Parallel system. Reliability importance of components within the system. ReliaSoft®software
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Static Reliability Importance
Time = 10000
Reliability Importance Value
Pump 1 Pump 3 Pump 2
0.000 0.819
0.164 0.327 0.491 0.655
Reliability
3 Item(s) 100%
50%
0%
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Static Reliability Importance
Time = 4000
Reliability Importance Value
Pump 1 Pump 3 Pump 2
0.000 0.351
0.070 0.140 0.210 0.281
Reliability
3 Item(s) 100%
50%
0%
Fig. 6.41 Parallel system. Reliability importance of components within the system.t D4;000 h andt D10;000 h. ReliaSoft® software
redundancy” the system fails if all the components fail.
In other words the number of expected failures for the system is close to 0.
In fact if the components introduced are used as parts of a redundant parallel system, the value of the system availability is very close to 1 as shown in the Fig. 6.43 reporting the simulated analysis conducted by ReliaSoft®software.
If the value of MTTR passes from 100 to 600 h (C500%), the trend of the state diagram (the so-called
up/down diagram) related to the three components and to the system changes as illustrated in Fig. 6.44. This simulated analysis is called “B” in order to distin- guish it from previous one, called “A,” which relates to MTTR equal to 100 h.
By the analysis of the simulated scenario, in config- uration B the system is always in the state of function (up state).
The system availability versus reliability diagram changes as illustrated in Fig. 6.45.
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Block Up/Down
Time, (t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
System Pump 1 Pump 2 Pump 3
State Operating Time Time Under Repair
Fig. 6.42 Parallel configuration. Repairable components, simulation analysis A. State diagram of the system. ReliaSoft®software
ReliaSoft BlockSim 7 - www.ReliaSoft.com
Availability and Reliability vs Time
Time, (t)
A(t), R(t)
0.000 10000.000 20000.000 30000.000 40000.000 50000.000
0.997 1.000
0.998 0.998 0.999 0.999
Diagram1
Point Availability Line Point Reliability Line
Fig. 6.43 Repairable components, simulation A. Availability and reliability. ReliaSoft®software