Sample extraction ） Statistics
Figure 4．1 Monte Carlo methods
4．3 Random and Pseudorandom Numbers
The essential feature common to all Monte Carlo computations is that we have to substitute for a random variable a corresponding set of actual values， having the st atistical properties of the random variable． The values are called random numbers， on the ground that they couユd well have been produced by chance by a suitable random process ［35］． lt is convenient to obtain needed random values directly from already developed program． Here， random values are not in most
sense 狽窒浮撃?random ， although they should be random enough for practical purposes． Such values are called pseudorandom because they are determined by using deterministic equations．
The random number generator usually is subprogram that returns value from a uniform distribution of the intervals O．O to 1．0． Algorithms are available to generate random variables from distributions by making transformations on one or several values generated from the uniform random number generator． For example， based on the inverse of the cumulative distribution function， random variables for the exponential distribution can be easily obtained from the uniform pseudorandom numbers distributed between O．O and 1．0．
4．4 lnput Modeling
For the time−sequential failure logic shown in Figure 2．5， followings are assumed for failures and repairs of inputs Xi， xib ， Xn・
a） Failures of inputs are mutually statistically independent．
b） lnput xi j s characterized by the failure rate A．i．At time zero， xi is not in a
53
failed state．
c）The probability of two or more failures and repairs during the period［ちt＋dの is assumed to be o（dりf（）r any input．
d）Repait of input．￥i follows the exponential distribution with the constant repair
rate μx∫・
e）The ti狙e required to repair input苓・is a constant value，1／μxヂ
ユ
Ty：p6−1 system is defined as the system that satisfies assumptions a）， b）， c）and d）whereas type−2 system meets with assumptions a）， b）， c）and e）．
Since input。sequence−vectors are mutually exclusive，
ω（の惑9・（の・ （4．1）
V17（t） ＝ ．1111711「・（の・ （4．2）
If generic algorithm for estimation of ca．（t） or PV．（t） is found out， to（t） and J＞IZ（t）
are easily obtained using equations （4．1） and （4．2）， given that X， is explicitly
known．
4．5 Monte Carlo Simulation for Time−Seque ntial Failure Logic
In order to simulate the relationship s between inputs and the output
characterized by the time−sequential failure logic as shown in Figure 2．5 using the Monte Carlo method， attentions should be paid to the．following two major issues：
how to simulate the behavior of inputs and to evaluate the relationships between inputs and the output．
4．5．1S血Ulation of lnputs
From assu皿ptions mentioned above， the whole behavior of input x｝＝κis composed of repair−to−failure and failure−to−repair processes． Here， for the type−1 system， the random variables， 7． and y ．， denote durations of a repair−to−failure and of a failure−to−repair processes， respectively． Therefore， they are given by equations （4．3） and （4．4） based on the inverse of the cumulative distribution
function ［9， 35］：
1 10gg ，
YK ＝＝ 一 A，．
（4．3）
冒 1
10gg ， γκ冨一 ILK
（4．4）
where g and g are the uniform pseudorandom numbers distributed between n．O
and 1．0．
For the type−2 system， the duration of a rep air−to−failure process， y．， can be also given by equation （4．3）． On the other hand， the duration of a failure−to−repair process， y D， is expressed as：
， 1 Y K ＝一 ILtK
（4．5）
For the type−1 system， the relationships between the time histories of i−th repair−to−failure and i−th failure−to−repair processes can be expressed as follows
（see Notation given in 4．1）：
55
t．，i ＝ t ．，i−1一一 ￡1一 1ogg．，i （t ．，o g O），
（4．6）
1
t
j，i ・一一 t．，i 一一 ÷． logg ．，i．iLtK
（4．7）
Here， g．，i and g ．，i are the uniform pseudorandom numbers distributed between O．O and 1．0 obtaine d in the context of the i−th repair−to−failure and i−th failure−to−
repair processes for input K， respectively．
Si皿ilarly；fbr the type・2 system， the time histories can be o胱ained by the fbUowing equations：
t．，i＝t ．，i一1一；1）一 logg．，i （t ．，o Ei o）， （4．s）
1
t
j，i ＝ tK，i ＋ 一＝一一 ibeK（4．9）
4．5．2 Simulation of the Output
The preceding section shows that the behavior of any input can be simulated
using the variable g．，i and g ．，i by putting K＝xi， xth ， or x． and i一一一1， 2， ．．．for
the type−1 system． Here， it is supposed that the failures of the output during ［O， e are caused through a p articular sequential vector x ＝ （xi， x2， …， x．］EXi ． Then， the relationship between inputs x， and x． is analyzed at first using the following three
steps． 一
Step 1：Find out the earliest time tx、，ノ（ノ≠iandノ＝10r 20r＿）， under the condition that the 7 一th failure of inp ut x2 occurs after the i：th failure of input x，．
When the time is found out， then go to step 2．
Step 2： Judge if the i−th repair of input x， is comp．leted aft er the failure of input
X2 i・e・， t ．，，i ＞t．，，i． Once it is true， go to step 3．
Step 3：Examine， which repair of input xl or xz i．e．， t x、，ゴor tlx2，ノ， appears earlier． If the repair of xi precedes that of砺i．e．， t曾x、，f＜t x、，ノ，record the times
Th ＝t．，，f（h＝1， 2， ．．．） and T ．b＝t ．，，ias the time histories of compound input x，x．
respectively．且ere， the compound input xix2…x． is defined as the virtual outp ut of the time−sequential failure logic generated through an input−sequence−vector
［xi，x2，…，x．］ （msn）． Then go to step 1 and iterate the step s for next （i ＋1）一th failure of xl． On the other hand， if t x、，i＞t馨x、，ノ， similarly；record the times
Th冒tx、，ノand T「h。t雪x、，ノas the time histories of the compound inputろ梅
respectively． Then iterate steps 2 and 3 for next U＋1）一th failure of x2
When the simulation time for inputs xi and x2 is equal to or greater than the given time limit， T， the simulation for inputs x， and x2 is completed and the overall ti皿e histories of the compound input xix2 are obtained．
Then， the time histories of the compound input x，x2 are analyzed with the input x3 using the same procedures as those of inputs x， and x2 described above in order to generate the overall time histories of the compound input xi xptr3．
Thus， the same procedures are repeated until the overall time histories of the
compound inp ut xixi x．， Th and Tb （h ＝1， 2， ．．．）， i．e．， the overall time historie s of the outp ut generate d through a p articular input−sequence x＝［xi， x2，…，x．1， are finally obtained． Therefore， the number h signifies PI7（7）i， i．e．， the number of failures of the outp ut during ［O， T） for the i−th simulation trial． The fiow chart of the Monte Carlo simulation is shown in Figure 4．2．
For the type−2 system， the time history of the output can be obtained through the same simulation procedures as those of the type−1 system using equations
（4．3）， （4．5）， （4．8） and （4．9）．
57
Start lteration
4，i 一一 t3，i−i 一（ 11 ADIn g
t P，ノ＝@ち，ゴー（1！μi）1nξ
No
ち，ゴ≧10000Yes
Record 1＝ i
No
K〈n
Yes
trtノニ・t ﾈ，ノ＿1一（1／λx）lnξ
t rc，ノ＝tK，ノ
No ノ＜I Yes
tl，ゴく瓶ノ
Record Frequency W（のニb110000
No
No
Yes
Stop lteration
t P，ゴ≧trc，ノ
Yes
t 1，i g一 t
j，J彦1，i＝Th
t電P，ゴ＝T h
No
Yes
Th＝t、，，ノ T h＝ t 1．i
戸f←1
Th＝tr，ノ T h＝ t rc，i
メ＝ノ←1
戸吾1
Figure 4．2 The fiow chart for Monte Carlo simulation
4．6 Comparison between Two Approaehes
The statistically expected number of failures of the output during ［O， e for the type−1 and 一2 systems with three inputs are estimated using the multiple inte gration method as well as the Monte Carlo simulation． Re sults from the Monte Carlo simulation are compared with the standard values obtained through multiple integration method in order to demonstrate the validity of the Monte
Carlo simulation． ．
4．6．1 Multiple lntegration Method
For the type−1 system， the statistically expected number of failures of the output （generated through an input−sequence−vector x E X，） per unit time at time
t， tu．（t）， is given by the following equation ［23］：
ω．（の一ω。。（のS：sf…sl．，4、（τ1）4，（τ・）…4。一、（・。．・）drn一・dTn−2…drl， （4…）
where a）．．（t） and 4，（Ti） can be given using equations （3．．10） and （3．11）
From equations （4．2） and （4．10） and by integrating ca．（t）， J7J7．7（e can be easily obtained using Mathematica ［33］．
4．6．2Nu皿eric Analysis
In systems of type−1 and 一2 with three inputs， suppose that the only input−
sequence−vector x＝［1， 2， 3］ can generate the output． Statistics of the inputs are characterized as shown in Table 4．1 and Table 4．2， respectively．
59
Table 4．1 Statistics assigned to inputs of case 1 and case2
：Failuエe rate
@ llh
Repair rate
@ 1！h
Time to repair
Case
λ1 λ2 λ3 μ1 μ2 μ3 11μ1 11μ2 1／μ3
1 0．01 0．04 0．03 0．1 0．15 0．02 10 6．67 50
2
0．1 0．04 0．03 0．1 0．15 0．02 10 6．67 50Table 4．2 Statistics assigned to inp uts of case 3 and case 4
：Failure rate
@ llh
Repahl rate
@ llh
Time to repahl
Case
λ1 λ2 旭
μ1 μ2 μ3 1！μ11／μ．1／μ33
0．1 0．04 0．03 0．01 0．15 0．02 100 6．67 504
0．1 0．04 0．03 0．1 0．15 0．02 10 6．67 50For type−1 system， M7．T（n can be estimated using the multiple integration or Monte Carlo simulation approaches described above whereas for type−2 system，
only the Monte Carlo simulation can evaluate W．（n． Assume that MZimlf（n and WMcsl n stand for W．（ 7） obtained by multiple integration method and Monte Carlo simulation， respectively． MZMcsl T） is computed from：
WMcs（T）＝
黒照）・
れM
（4．13）where Mis the number of simulation trials and M7（ni i s the i−th simulation result．
In Table 4．3 and Table 4．4， the values of PIZMcsl 7） are shown when the simulation trails are 35， 70 and 100 times， respectively．
Deviation rate R（7）is de丘ned as：
R（T）＝．mptMi？i1 （．T．）一ljlmll￥cs（T）1．
MZ．，，．（T）
（4．．14）
4．6．3 Discussions
Exponential repair distributions are assumed fbr the i叩uts of the type・1 system． According to the definitions wide1y accepted， the mean time to repair of input K of this type is 1／pt． ［hours］． So the mean time to repair is theoretically equivalent to the time to repair of input K involved in the type−2 system． For some cases， items would be better modeled on constant times to repair rather than exponential repair distributions． lt is interesting to cpmpare the results obtained from the type−1 and 一2 systems．
61
Table 4．3 Results of case 1 and case 2 obtained through multiple inte gration method and Monte Carlo simulation approaches
Method System
Case
12
No． of
srialS
Time
104［hr］
1．0 2．00 5．00 1．00 2．00 5．00
MIM
丁夕pe−1 脇田（の 1．38 2．76 6．90 7．58 15．16 37．9035
脇α1〈の
N（n（o／，）
1．27 2．56 6．31 W．69 7．25 8．55
6．91 14．20 34．95 W．83 6．33 7．81
Type−1
70
脇㏄（の
d（の（％）
1．29 2．69 6．51 U．52 2．54 5．62
7．13 14．31 35．48 T．94 5．60 6138
100
脇㏄（乃
N（n（o／，）
1．32 2．73 6．71
S．35 1．09 2．75
7．33 14．59 35．64 R．30 3．76 5．96
MCS
35
職㏄（の
d（の（％）
1．51 2．97 7．48 X．42 7．61 8．41
8．03 16．62 39．97 T．94 9．63 5．46
Type−2
70
職。β⑦
n（n（e／，）
1．47 2．91 7．41 U．52 5．43 7．39
7．81 16．06 39．06 R．03 5．94 3．06
100
脇α9（の n（の（％）
1．46 2．85 7．30 T．80 3．26 5．80
7．77 15．54 38．46 Q．51 2．51 1．47
Table 4．4 Results of case 3 and case 4 obtained through multiple integration method and Monte Carlo simulation approaches
Method System
Case
34
No． of
srials
Time
104［hr］
1．00 2．00 5．00 1．00 2．00 5．00
MIM
Type−1陥（の
2．15 4．30 10．75 7．58 15．16 37．9035
職㏄（の
d（n（o／，）
1．95 4．09 9．87 X．30 4．88 8．19
6．92 14．20 34．95 W．83 6．33 7．81
Type・1
70
職α；（7）
N（n（o／，）
2．02 4．17 10．02 U．05 3．02 6．80
7．13 14．31 35．48 T．94 5．60 6β8
100
脇α3（の
n（n（o／，）
2．08 4．24 10．27 C3．26 1．40 4．47
7．34 14．59 35．64 R．30 3．76 5．96
MCS
35
脇㏄（の
n（n（o／，）
2．27 4．45 11．42 T．58 3．49 6．23
8．04 16．62 39．97 T．94 9．63 5．46
Type−2
70
脇α3（の
d（n（o／，）
2．24 4．41 11．31 S．19 2．56 4．95
7乙82 16．06 39．06 R．03 5．94 3．06
100
脇㏄（7）
n（の（％）
2．23 4．40 11．19 R．72 2．33 4．09
7．78 15．54 38．46 Q．51 2．51 1．47
63
For the examples with three inputs， the followings are found from Tab1e 4．3 and Table 4．4：
a） For the type−1 system， P12Mr cs（7） is always estimated less than WMiM（7）， but the deviation rates are less than 100／o．
b） For the typ e−2 system， most of WMr cs（7） are counted in excess of MZMiM（7＞ with the deviation rates less than 100／o．
c） With the increase of the trial runs of simulation from 35 to 100 times， the deviation rate decreases quite a bit．
The examples in（licate that when trial runs of simulation are repeated more than 100 ti皿es， the Monte Carlo simUlation gives very similar values with those through the multiple integration method for both the type−1 and 一2 systems． ln erder to obtain more generic conclusions on the relationships between the type−1 and 一2 systems irrespective of the examples， much more simulation data will be required． There are stil l various problems to be studied hereafter．
4．7 Conclusions
The chapter discusses how to analyze the time−sequential failure logic using Monte Carlo simulation． TWo types of systems are set up according to inp ut modeling． ln systems of type−1， the inputs are mode1ed on the exponential distributions with constant failure and repair rates． For the type−2， it is assumed that inputs are characterized by constant failure rates and constant times to repair． Next， how to simulate the behavior of inputs and to evaluate the relationship s between inputs and the output are described． A fiow chart for computer simulation is introduced． Finally， the estimation of statistically expected
number of failures of the output for both the type−1 and 一2 systems with three
inputs is demonstrated using the multiple integration and Monte Carlo
simulation approaches． The results obtained indicate that deviation rates are within 100／o for ail cases examined．
65
Chapter 5
Systems Modeling
5．1 Introduction
As well known， the statistically expected number of failures and the probability that the system is in a failed state are very usefu1 for predicting repairable， non−
catastrophic or partial system failures ［9］． This kind of system failure would not result in complete destruction of the system and would repeat a number of times during a system s lifetime． For instance， although many fatal trafific accidents occur every year， the whole tra茄．c system is never complete1y destructed．
On the other hand， the system reliability or system unreliability is more useful for describing catastrophic， non−repairable or complete system destruction such as worldwide nuclear war． The system unreliability is the probability that the system suffers the缶st failure durillg［0， e， and is equivalent to the statisticaHy expected number of system failures during ［O， e since the system can not be repaired．
The preceding chapters deal with the quanti丘catioll of the time−sequential failure logic using analytical and numerical approaches for repairable systems．
This chapter discusses the feasibility of． the multiple integration method for non−
rep airable systems， by comparing repairable and non−repairable systems using the Markov model．
5．2 Repa irable systems
5．2．1 Three−lnput Systems
Consider a system of three inputs which have the constant failure rates Ai （i ＝1，
2， 3） and repair rates pah respectively． The Markov transition diagram of the repairable system is shown in Figure 3．2 ［9， 48］．
As discussed in Chapter 3， when a single sequence of input failure is taken into account， the Markov model is applicable only to the special case where inputs are characterize d by common failure and rep air rates， A and pa Then， the transition diagram of Figure 3．2 is simplified as shown in Figure 5．1．
None of three inputs is true．
@ State prob． P。（の
μ 3λ
One input is true．
rtate prob． P 1（の
2μ 2λ，
Two inputs are true．
@State prob． P2（の
3晒 λ
Three inputs are true．
@ State prob． P3（の
Figure 5．1 The s血phfie d transition diagram for three inp uts
67
The corresponding d瞳fferential equations of Figure 5．1 are
り
P・（の一 一3λLP。（の＋μ・P， （t）
の
1）1（の雷3λif？o（の一（2λ＋μ）P，（t）＋2PtP，（の ， （5．1）
1）2（t）富22〕P，（の一（λ＋2μ）P，（の＋3μ2略（の
P・（の旨λ弓（の一3μaP，（の
where the initial condition is
，
｛Po（0），君（0）， P，（0）， P，（0）｝＝［1，0，0，0］． （5．2）
Taking Laplace transfbrm to equation（5．1）and inserting the initial condition
（5．2）into it， equation（5．1）becomes
（s＋3λ）1）o（S）一μ君（5）＝1
−3λPo（s）＋（s＋2λ＋μ）君（5）一2μ2宅（5）富0 。 （5．3）
一2AP，（S）＋（5＋λ＋2μ）P，（5）一 3paP，（5）零0
一λ1：｝（s）＋（5＋3μ）P，（5）富0
Equation （5．3） can be stated in matrix form as mp ＝b ：
s＋3A
−3A o o
一一 pt o o s＋2A＋pt 一2pa O
−2A s＋A＋21tt 一31・t
O 一A s＋3i［，t
Po （s）
P， （s）
P， （s）
P， （s）
1
0 0 0
，
（5．4）
where
m＝
s＋3A
一 3A
o o
一 pt
s＋2A＋pa
−2A
o
o
一 2pt
s＋A＋2pt
−A
o
o
−3u s＋3u
，P＝
Po （s）
P， （s）
P， （s）
P， （s）
and b＝
1
0 0 0
According to Crame formula ［47］， solving the above equation by means of Mathematica， P，〈s） can be calculated． Then，
P・（・）一
iλ誓）・｛f＋5鴇一5愚麦呈μ）・。＋f，．μ）｝・
（5．5）
If the inverse of the above：Laplace transforms is taken， the probab丑hty can be
determined．
P・（ 3A2
煤j ＝@ （A ＋ ILt）・｛μ・（λ一2μ）び・神・診一（2λ一μ）び2（一・λび3（λ畔｝・ （5．6）
69
Since e ach failure sequence has the same probability of occurrence， the statistically expecte d number of failures of the outp ut per unit time at time t is given as follows：
ω（の呂鳩（t） ＝
3ズ ｛iu ＋ （A． 一 2Lt）e一（A ） 一 （2 L 一・ LL）e−2（A ）t ＋ Ae−3（A ）t｝ ．
（A ＋ pt）3
（5．7）
Therefore， the expected number of failures of the output during ［O，e ．i s obtained by：
w（t） 一 f6 tu（t）dt