Intelligent Information Management, 2011, 3, 49-55
doi:10.4236/iim.2011.32006 Published Online March 2011 (http://www.SciRP.org/journal/iim)
Copyright © 2011 SciRes. IIM
Semi-Markovian Model of Unreliable Control of
Restorable System with Latent Failures
Yuriy E. Obzherin, Aleksei I. Peschansky, Yelena G. Boyko
Sevastopol National Technical University, Sevastopol, Ukraine
E-mail: vmsevntu@mail.ru
Received December 27, 2010; revised January 3, 2011; accepted January 5, 2011
Abstract
Semi-Markovian model of control of restorable system with latent failures has been built with regard to con-
trol errors. Stationary reliability and efficiency characteristics of its operation have been found. The problem
of control execution periodicity optimization has been solved.
Keywords: Control, Control Errors, Latent Failure, Semi-Markovian Process, System Stationary
Characteristics
1. Introduction
An important factor providing reliability, high quality,
and efficiency of technological complexes is the presence
of control systems in them. The reviews of the results
concerning control model building are contained in [1,2].
In the present article the model of control execution
and restoration of a single-unit system with latent fail-
ures has been investigated under the condition of control
errors occurrence. The latent failure is the one that does
not show up till the control is executed. While control is
executed errors of first and second kinds are possible [3].
It reduces technological complex operation efficiency.
In the work [4] the model of control of the system with
possible failures was studied, but there reliability charac-
teristics were defined under the assumption of exponential
distributions of random values characterizing the system.
The problems of technological complexes’ control are
closely connected with their maintenance. In the work [5]
maintenance models were built by means of semi-
Markovian processes with a common phase field of
states [6]. In the present article this apparatus is used to
build the model of control under the condition of latent
failures occurrence. And herewith, random values char-
acterizing the system are supposed to have distributions
of general kind.
2. The Problem Definition and Mathematical
Model Building
Let us investigate the system operating in the following
way. At the time zero the system begins operating, and
the control is on. System failure-free operation time is a
random value (RV)
with distribution function (DF)
F
tP t
and distribution density (DD)
f
t.
The control is executed in random time
with DF
Rt Pt
and DD
rt. The failure is detected
only when control is carried out. Control duration is RV
with DF
tVt P
and DD
vt. When the
control is on, the system does not operate. After failure
detection system restoration begins immediately and the
control is deactivated. System restoration time is RV
with DF
Gt Pt
and DD

g
t. After the
system restoration all its properties are completely re-
stored. All the RV are supposed to be independent, have
finite assembly averages and variances.
While control execution errors of first and second
kinds can take place. Control system error of first kind
lies in regarding system in up state as a failed one (false
failure). The probability of such an errror is equal to р1.
The error of second kind is taking system’s good state
for bad one (failure omission). The probability of such an
error is equal to р0.
The purpose of the present article is to find stationary
reliability and economical characteristics of the sin-
gle-unit restorable system under the condition of latent
failures occurrence, with regard to control errors.
To describe the system operation let us use semi-
Markovian process
t
with the following field of
states:
111, 212, 211,101, 202, 201, 200, 210Еххх,
Y. E. OBZHERIN ET AL.
50
;
;
where
111 means the system has been restored, the control is
activated;
212х - control has begun, the system is in up state, it
does not operate during control execution, time х is left
till the latent failure (regardless of deactivation time);
211х - control has ended, the system in up state has
been treated correctly and continues to operate, time х is
left till the latent failure;
101 х - latent failure has occurred, time х is left till
control execution;
202 - control has begun, the system is in a state of la-
tent failure, it does not operate;
201 - control has ended, failed system is regarded as
the one in up state (error of second kind), its operation
has begun;
200 - control has ended, failed system is regarded as a
failed one, system restoration has begun, control is deac-
tivated;
210 - control has ended, system in up state is taken for
failed one (error of first kind), system restoration has
begun, control is deactivated;
Time diagram of the system operation is repre-
sented in the Figure 1. And in the Figure 2 the system
transition graph is given, where a set of up states is,
is a set of down states.
E
E
Time diagram of the system operation and the system
transition graph are shown in Figure 1 an d Figure 2
respectively.
Let us define the probabilities of the embedded Mark-
ovian chain (EMC) transitions:
,0
nn




212
111
0
101
111
0
101
211
212
211
,0
,0
,0;
,0 ;
x
x
y
x
y
x
pfxtrtdtx
prxtftdtx
prxyy
prxyyx
 
 
 
 
(1)
211 210
2121 2121
202202111 111
101201200210
201 200
2020 2020
,;
1;
,1,
x
xx
x
ii
PpPp
PPPP
PpPpp pi



0,1.
Let us indicate ,

111
202
, ,

201
210
,
the values of EMC
stationary
distribution for the states 111, 202, 201, 210, 200 respec-
tively and assume the existence of stationary densities
200
,0n
n

212
х
,

211
х
,
101
212x
111 211x202 201 202 200210
111 212x
t
t
t
1
K
s
х
х
111
101х
Figure 1. Time diagram of the system operation.
111
211x
212x
210
101x 202
200
201
Е
+
Е
-
Figure 2. System transition graph.



 


 
 
 
 
 

0
0
x
1
0
x
0
0
0
1
0
0
0
0
111200210 ,
212 111
211 ,
211212 ,
101 111
211 ,
202101201 ,
201202 ,
200202 ,
210212 ,
2202 201212
211
x
fxtrtdt
yryx dy
xp x
xftrxt
yryx dy
xdx
p
p
px
dt
x
dx
x
 



 



 








0
101 1.dxx dx

(2)
The last equation of the system (2) is a normalization
requirement.
To solve the system (2) let us exclude the function
211
x
from the second and third equations:
 
01
0
212 212
x
х
f
xtrtdt pyryxdy
 
 

(3)
х
for the states 212
х
,
211
х
, 101
х
respectively. Now we can make the system
of integral equations for them:
Copyright © 2011 SciRes. IIM
Y. E. OBZHERIN ET AL.
Copyright © 2011 SciRes. IIM
51
Let us indicate

1
212х
х

, then the Equation
(3) will take the form:
Let us introduce the function
1
rxp rx
and
the integral operator

 
r
x
A
xryxyd


y
.
 

10
11
х
x
хftrt хdt
p
yr yxdy



(4) Then the Equation (4) can be rewritten in the following
way:
0
11
1
rr
A
Af
p



.
The solution of this equation will be defined with the
help of method of successive approximations. The solution of this equation is defined by the formula
   
0
10
110
1
212 ,
rr
x
х
x
hyxfydyhyfxydy
pp




(5)
where is the density of renewal function

r
ht
r
H
t
generated by the RV with improper DF

1
pRtRt
;



*
1
n
rn
htr t
, where

*n
rt
is n-fold convolu-
tion of the function

.
1
rtp rt
Using Formula (5), one can define the rest of station-
ary densities
  
0
0
1
00
211, 101,,
rr
x
h yfx ydyxfzzxdz
p
 

 

(6)
where
 
0
,
z
rr
zxr xzr xyhzydy
 
 is the ated by RV with improper distribution density
rt
.
The values of stationary distribution for the states 210,
202, 201, 200 are defined from the system (2):
density of direct residual time of renewal process gener-
   
   
11
00
10100
011
00
01 1
00
1
210, 2021,
2011, 2001.
rr
rr
pp
H zfzdzH zfzdz
ppp
ppp
H
zf zdzHzf zdz
pp p
 
 






 






(7)
Here



*
0
n
rn
H
tR
t
, where

*n
Rt
is n-fold tionary operation time T
, mean stationary restoration
time T
, stationary availability function Kг.
convolution of the function . The constant

Rt
0
is
found with the help of normalization requirement. The sets of up states and down states
EE
are
the following ones:
111,211,212, 101,202,201,200,210.ExExx


3. Definition of System Stationary
Characteristics Mean stationary operation timeT and mean sta-
tionary restoration tim T
e
can be found with the help
of Formula [6
Let us define system stationary characteristics: mean sta- ]:


,
,,
ЕE
EE
me deme de
ТT.
P
eE dePeE de









(8)
Here is the EMC

de
,0
nn
stationary dis-
tribution; are mean values of system dwelling
times;

me
,
P
eE
are the probabilities of EMC
,0
nn
transitions from up into down states.
Mean values of system dwelling times in the states
are:
   
00
111; 211; 212202;
201; 101;210200
x
mFtRtdtmxRtdtmxm
М
mМmxхmmМ




(9)
Y. E. OBZHERIN ET AL.
Copyright © 2011 SciRes. IIM
52
With regard to Formulas (5-7) and (9) one can define the functionals contained in (8):
  
    
 
  
00
0000
1
00 0
1000 0
00
00
00
1100
,
,2111,
11
x
r
E
y
rr
r
E
r
E
medeF t Rt dtdxRt dthyfxy dy
p
Ft R t dtFy dyhyt R t dtMHy Fy dy
p
PeEdexdxHzf zdz
m edeMdxhyfxydyx
pp
 
 
 
 

 


 

 



 




 




  
  
 
00
0
111
000
01 011000
1
000
111
0000
1
0
010
,
1111
11
,
11
rrr
rr
r
dxzxfzdz
p
ppp
MHzfzdzMHzfzdzMHzfzdz
pp ppp
p
MHzfz dzMHyfy dyfz dzxzxdx
ppp
p
MHzfzdz
pp
  
 





  


 







  
  
00
1
000
01 100
01 1
000000
100 1
0 0
1
11 .
rr
r r
pp
p1
p
M
Hzfzdz MMHzfzdz
pp p
pp p
MHzfzdzMMMMHxFxdx
pppp
 


 

 
 
 




 
Thus, mean stationary operation time is defined by the ratio
T

 
1
10
0
,
1
r
r
p
M
FyH ydy
p
T
FzdHz
(10)
and mean stationary restoration time is determined by the formula:
T
    
 
01 1
01 01
0 0
0
1
1
rr
r
pp p
M
MFzdHzMMMMHtF
pp pp
T
FzdHz
 
 



tdt
(11)
Stationary availability function is found from the ratio

г
КТТТ

.
We get
 
1
10
01
01 0
0
() ()
1
() ()
г
r
r
К
p
MFyHydy
p
pp
M
MFzdHzMM
pp p
M




(12)
Important characteristics for system operation quality
testing are economical criteria, such as mean income S
per unit of calendar time and mean expenses C per time
unit of system’s up state. To define them let us use the
Formula [7]:

 
 
 
s
E
E
c
E
E
mеfеdе
Smеdе
mеfеdе
Cmеdе
, (13)
Y. E. OBZHERIN ET AL.53
where
 
,
sc
fе
f
e are the functions defining income
and expenses in each state respectively.
Let c1 be the income received per time unit of sys-
tem’s up state; c2 - expenses per time unit of restoration;
c3 – expenses per time unit of control; c4 are wastes
caused by defective goods per time unit of latent failure.
For the given system the functions

,
sc
fе
f
e are the
following:







1
22
33
44
,111,211 ,0,111,211 ,
,200,210,,220, 210,
, 212,202,, 212,202,
,101 ,201,,101 ,201.
sc
се хeх
се сe
fe fe
се хсeх
cех ce х



 



 


 

(14)
With regard to (13) and (14) mean income is defined by the ratio

01 1
14 23414
001 1
00
01
001
0
1
()()()()() ()()
1() ()()
rr
r
pp p
M
cccMcMcMFzdHzccFtHtdt
ppp p
Spp
MM MMFzdHzM
ppp

 


 


 


and mean expenses are determined by the ratio

  

 
01 1
24 431434
011 0
00
1
10
1
rr
r
pp p
cMcMcMcMFzdHzccF tHt dtcMcM
ppp p
Cp
MFtHtdt
p
 

 


Let us consider the case of non-random control execu-
tion periodicity 0
. Taking into account that in this
case
1Rtt
, where const
, stationary avail-
ability function is defined by the formula


 
(1)
1
1
01
1
1
001
1
1
n
n
nn
г
n
n
MpFtdt
Кpp
M
MM pF
ppp
n
 


(15)
mean income can be defined in the following way
 







 
1
01
1423 413 4141
11
01 0
01
1
1
001
11
1
n
nn
nn
n
n
n
pp
M
c ccMcMcpFncMcccpFtdt
pp p
Spp
MM pFnM
ppp
 
 


 
 

(16)
mean expenses are found with the help of the ratio


 







1
01
24 43114134
11
01 0
1
1
1
1
1
1
n
nn
nn
n
n
n
nn
pp
cMcMccMpFnccpF t dtcMc
pp p
C
MpFtdt
 


 


(17)
Copyright © 2011 SciRes. IIM
Y. E. OBZHERIN ET AL.
54
Let us investigate some special cases of the system op-
eration. If errors of one kind only occur, two cases are
possible.
Let p= 0, p 1, then Formulas (10-12) take the form
0 0
 
1
1
r
p
MFyHydy
p
 
   

 
0
0
11
1
00 10
00
,
1
1
,
11
r
rr r
г
rr
T
FzdHz
pp
MMFzdHzMMHtFtdtMFyHydy
pp
TК
FzdHz
M
MFzdHzM
 
 


 









If , then
10
0, 1pp
rt rt
,

rr
H
tHt
with DF is renegeneratwal function ed by RV
Rt ,



*
0
n
rn
H
tR
transformed into
t. In this case Formulas (10-12) are
 
 
 
 
 
0
1
0010
000
,
1
1
,.
1
1
r
rr
г
rr
M
T
FzdHz
p
MMFzdHz MMMFyHydy
pp
TК
FzdHzMMFzdHz M
p

 

 




 


In case of reliable control
01
0pp system sta-tionary characteristics are defined by the ratios
 
 
 
 

 
0
1
010
00
,
1
1
,
11
r
rr
г
rr
M
T
FzdHz
p
MMFzdHzMMMFyHydy
p
TК
FzdHz
M
MFzdHzM

 

 




 


which coincide with ones found for a single-component
stem with reliable control.
rol Execution
ontrol execution periodicity optimiza-
on is reduced to analysis of extremums of the system
us suppose RV
,
and
of
to have Erlangian distri-
bution. For the calculation optimal value
s
opt
sy pro-
nd
viding maximal mean income
S per calendar
time unit aof optimal value opt
s
opt
c
providing minimal
mean expenses
opt
C
per time unit of system’s good
state, the following initial data have been take1 = 2
c.u./h; с2 = 2 c.u./h; с3 = 1 c.u./ c.u./h. The results
of these calculations are represented in the Table 1. The
graphs of s
c
n: с
h
4. Optimization of Cont
Periodicity
e problem of c;
funct
с4 = 1
ion
г
K
,

,S

C
for the case
of Erlangian distribution of the 2nd order and 10,3,p
00, 25p
are shown in Figures 3, 4 and 5.
5. Conclusions
pparatus of semi-Markovian proce
Th
ti
characteristics Кг, S, C as functions of a single variable
.
Using Formulas (15-17), one can define an optimal
ripeod of control of the system investigated for different
distribution laws of random values. The initial data for
calculations of optimal values of control periodicity are:
mean time of failure-free operation
M
, mean restora-
tion time
M
, control duration
M
. For example, let
U
co
sing an a
mm
s
ne
ses with a
reliability on phase field it is possible to defi
Copyright © 2011 SciRes. IIM
Y. E. OBZHERIN ET AL.55
Table 1. Optimal control ex
Initial data
ecution period definition.
Results
, h
s
opt
S
,
c.u./h
c
opt
, h
c
opt
C
Distribution laws of ran-
dom values β, h Mγ, h p1P0 , h
k
гopt
K
opt
Mα, hMk
opt
, c.u./h
Exponential 60 0,5 0,2 0 0 4,8330,916 4.833 1,739 4,833 0,101
Exponential 60 0,5 0,2 0,30,255,0940,867 5,522 48 1,292
Erlangian of the 2nd order 60 0,5 0,2 0 0 7,7440,876 7,744 1,621 7,744 0,15
Erlangian of the 2nd order 60 0,5 0,2 0,30,255,9110,904 6,302 1,684 45,535 0,978
Erlangian of the 2nd order 60 0,5 0,2 0,20,255,0070,903 5,259 1,688 36,112 0,817
1,568
)(
г
020 50
1
0,8
0,6
Figure 3. Graph of stationary availability function
г
К
against control periodicity
.
)(
S
3
2
1
0210
Figure 4. Graph of mean income
0
30
S
against control
periodicity
.
2
1
0620 40
3
0
)(
C
Figure 5. Graph of mean expenses
C
against contro
periodicity
l
.
proble coexe pey option
andeconomical stationary performance indexes of re-
ematical
m
d M. Parlar, “A Survey of Maintenance
ti-Unit Systems,” European Journal of
ms ofntrol cutionriodicittimiza
storable system with latent failures under the assumption
of possibility of control errors. It allows solving the for
gaining best system economical characteristics.
Later on it is planned to use the method suggested in
the present article to build and investigate math
odels of multicomponent systems and of different
kinds of control.
6. References
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Models for Mul
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lti-
ith Economic De-
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d
[4] A. M. Polovko and S. V. Gurov, “Reliability Theory Fun-
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[5] Yu. E. Obzherin and A. I. Peschansky, “Semi-Markovian
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dumka, Kiev, 1982.
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