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 2 Technological 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

intint

int int int

(T )(T )

(T ) (T ) (T )OS

OE

OS RS MS

TK

T T T

(2)

intint

int int

(T )(T )

(T ) (T )OS

A

OS RS

TK

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:

1ξ 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 ijj 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 xt

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 × £ £íïïï £ 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 S2y/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 140000.996

0.9965

0.997

0.9975

0.998

0.9985

0.999

0 2000 4000 6000 8000 10000 12000 140000.996

0.9965

0.997

0.9975

0.998

0.9985

0.999

0.9995

1

intT ,hours

intT ,hours

int

(T)

OE

K

int

(T)

AK

0 2000 4000 6000 8000 10000 12000 14000 160000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

intT ,hours

int

()

pF

T

#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 100000.9988

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)

OE

K

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

synthesisCalculation

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