Mathematical support for preventive
maintenance periodicity optimization of
radio communication facilities
The Ministry of Transport of the Russian Federation
The Federal Agency of Railway Transport
Omsk State Transport University (OSTU)
Granada, 2016
Alexander Lyubchenko – Associate Prof. of
the Dept. Information and communication
systems and data security

Variety of radio application areas
Page 1
Building industry
Gas-and-oil producing
industry
Electric-power industry
Minerals industry

Page 2Technological radio communication devices
of industrial enterprises
Fig. 2.1 Technological radio communication network
of industrial enterprises
Base
station
PSTN
Dispatcher Control Board
Industrial transport Portable crane
Engineering and
industrial personnel
Mobile-radio
station
Mobile-radio
station Portable radio
stations
Fixed radio
station Fig. 2.2 Railway fixed radio
station RS-46MC
Russian railways have:
- 31.000 of FRS;
- 60.000 of MRS;
- 79.000 of PRS.

Page 3
Actuality of the research
Reasons:
- Idealistic conditions;
- Long time exploitation;
- No recommendations about
length of PM procedures.
Challenging problem:
Solution approaches:
Natural experiments +
expert method
- Long time tests;
- Real system availability;
- Error probability of
decision making.
Mathematical
support
Scientifically
substantiated results:
periodicity Tint and
duration tt
Reasons:
- Idealistic conditions;
- Long time exploitation;
- No recommendations about
length of PM procedures.
!!!Necessity to develop
own local regulations!!!
0
( ) ( ).i ij ij ij ij
j
T p dF

    
о д о д
1 12 13
0 0
( ) ( ) ( )
t t t t
T F F d
 
       
…...

Research objective – development of mathematical support algorithms of a
CAD system for optimization of preventive maintenance intervals of radio com-
munication devices based on a simulation model of the operational process.
Page 4
Research objective and tasks
Tasks:
1) Select and justify the optimality criterion of the operational process of the
radio communication devices;
2) Develop a conceptual and simulation model of the process taking into
account the impact of the following factors: appearance of sudden, gradual,
latent and fictitious failures, human factor of service staff and time
parameters;
3) Implement the experiments to verify the conceptual model, confirm the
adequacy of the simulation and test its stationary properties;
4) Develop algorithms for computer-aided design system allowing
optimization of the preventive maintenance periodicity.

Optimality criterion
Page 5
Fig. 5.1 Typical graphs of the dependences
KOE(Tint) and KA(Tint)
where – allowable value of availability
Objective function KOE(Tint):
where TOS(Tint), TRS(Tint) and TMS(Tint) -
mean time of operable state, repair and
maintenance, accordingly.
Availability function:
Advisable value Trat:
(1)
(3)
(4)
(5)
(6)
 


int
A.A.
(T ) maxOE
A
K f
K K
A.A.K
int
int
int int int
(T )
(T )
(T ) (T ) (T )
OS
OE
OS RS MS
T
K
T T T

 
(2)
int
int
int int
(T )
(T )
(T ) (T )
OS
A
OS RS
T
K
T T


intargmax ( )opt OET K T
1
. .( )all A A AT K K
  min ,rat opt allT T T

Fig. 6.1 State diagram of the operational process
of repairable devices
Conceptual model
Model’s parameters:
1) vector of initial state of embedded
Markov chain:
 0
0 , 1,iP P i n 
where n is a possible quantity of states.
2) matrix of transition probabilities from
state Si to Sj:
, ( 1, ; 1, )ijP i n j n 
3) vector of density functions of
time intervals for each state:
 ( ) , 1,iF F t i n 
States of the process:
- Operable state (S1);
- Misalignment state (S2);
- Nonoperable state (S3);
- Preventive maintenance of operative system (S4);
- Maintenance of system with misalignment (S5);
- Latent failure (S6);
- Maintenance of system being in latent failure (S7);
- Fictitious failure (S8).
(1)
(2)
(3)
Page 6
The model takes into account:
1) sudden and gradual failures;
2) fictitious and latent failures;
3) human factor of service staff;
4) time parameters of maintenance
operations.

Page 7
Simulation model’s algorithms
Start
T < Tk
|Pij|, P0, Tоб,
Т = 0
-
+
N(I) = N(I) + 1;
Determination
of initial state
Calculation of
the vector of
density functions
F(I) < Tоб
-+
T+F(I) >= Tk
- +
F(I) = Tk – T;
T = Tk;
T(I) = T(I)+F(I);
T(I) = T(I)+F(I);
T = T + F(I);
F(I) = Tоб;
T+F(I) >= Tk
- +
F(I) = Tk – T;
T = Tk;
T(I) = T(I)+F(I);
T(I) = T(I)+F(I);
T = T + F(I);
Determination
of the next state
I=J
N = N + 1
End
Fig. 7.1 Flow chart of the simulation algorithm
Steps of simulation:
1) Determination of the first state of the
process according to the vector of initial state;
2) Calculation system’s duration of stay in
current state:
1
ξ ln(1 ),
λ
u  
where u is a uniformly distributed number in the
range [0,1].
3) Definition of the next state Sj according to the
matrix of transition probabilities when the following
inequality is correct (j=f):
1
1 1
, , 1,8.
f f
ij ij
j j
P u P i const j

 
    
(1)
(2)

Page 8
Fig. 8.1 Algorithm of estimation
of output parameters
-
+
Process simulation
Outputs’ estimation
+
+
-
-
-
+
End
Start
7
int intT Т 
int.max RT ,
RN 0,N 0  
int int.maxT T
R RN N 1 
A OEK K, Estimation of
RN 120
A OEK K R,  
N N 1  
N 10 
int int intT T Т  
Accuracy of the estimations:
/2, 1
[ ]
,pN
P
S x
t
N
   (1)
where S[x] is a sample standard deviation of random value x;
Np is a sample size; ta/2,Np-1 is a fractile of t-distribution.

Page 9
Smoothing and interpolation of simulation results
ES – exponential smoothing;
MA – moving-average method;
SGF – Savitzky-Golay filter;
LOWESS – locally weighted scatterplot smoothing.
FT – polynomial fit of the 4th order ;
LI – linear interpolation;
CI – cubic interpolation;
SSI – smoothing spline interpolation.
Table I. Results of average squared error (εASE) estimation
Table II. Results of estimation of εASE and determination
coefficient (R2)
Fig. 9.1 Dispersion before and after the
use of the method of significant sample
Fig. 9.2 Results before (red line) and
after (dotted line) the use of the method

Page 10
Estimation of parameters and adequacy testing
2
int
5 7
int
( , ) min;
5 10 10 ;
100 2000 .
o k
k
S J T T
T hours
T hours
ìï ¢= D ®ïïïï × £ £í
ïïï £ D £ïïî
 
 
6
int
10 10
600
kT hours
T hours
Fig. 10.1 Graph of the residual dispersion Fig. 10.2 Analytical and simulation estimations
Adequacy estimation criterion:
  
2
/
2
,if 2Y X
o
S
S
where S2
y/x is a dispersion cased by the model; S2
o is a residual dispersion.
the simulation model is adequate to analytical.

Page 11
Numerical experiments
0 2000 4000 6000 8000 10000 12000 14000
0.996
0.9965
0.997
0.9975
0.998
0.9985
0.999
0 2000 4000 6000 8000 10000 12000 14000
0.996
0.9965
0.997
0.9975
0.998
0.9985
0.999
0.9995
1
intT ,hours intT ,hours
int(T)OEK
int(T)AK
0 2000 4000 6000 8000 10000 12000 14000 16000
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
intT ,hours
int()pFT
#1lin
#1non
#2non
#3non
#4non
Fig. 11.1 Results for different distribution functions of service staff errors probability

Page 12
Numerical experiments
Fig. 12.1 Results for different values maintenance time parameters
0 2000 4000 6000 8000 10000
0.997
0.9975
0.998
0.9985
0.999
0 2000 4000 6000 8000 10000
0.9988
0.9989
0.999
0.9991
0.9992
0.9993
0.9994
0.9995
0.9996
0.9997
0.9998
intT ,hours intT ,hours
int(T)OEK
int(T)AK
Default values tt=2, ta=1, ts=3 and tr=3 hours

Module of
multivariant
analysis
No
Yes
Increasing of
Тоб.max value
Start
Selecting a device
from DB
Simulation is
finished ?
Module of
parametric
synthesis
Calculation
results
Output of
results
End
Input
parameters
Algorithms of CAD system
Fig. 14.2 Block chart of CAD system
functioning
Core
GUI
CAD Simulation model
prepared in VC++
as external mex-
function of MATLAB
GUI developed
by GUIDE of Matlab
software
Page 13
Fig. 13.1 Structure of the software
The core contains:
1. Module of multivariate analysis including:
a) main simulation algorithm;
b) algorithm of calculation of availability and
operating efficiency mean values;
c) smoothing and interpolation of results.
2. Module of parametric synthesis allowing
to calculate recommended value Trat with
accordance to the optimality criterion.

Fig. 14.1 GUI of CAD system
Page 14