Transcript x - Operational Research Society of India

Solving Renewal Equations –
the beginning to my reliability research…
M Xie, PhD, Fellow of IEEE
Chair Professor; Dept of Systems
Engineering and Engineering Management
City University of Hong Kong
What is my research about?
T, a random variable
Lifetime, Reliability,
Distribution, estimation
Prediction of next T
Monitoring of T, SPC
Application to components
system, software, service etc.
Optimization
Repair/replacement
Cost models
Improvement
Development process
Quality and Management
etc
Which one is my “best research” done?
E.
Software Reliability (first book)
Statistical Process Control (second book)
Quality Function Deployment (third book)
System Reliability (edited volume and latest book)
Weibull distribution (another book)
F.
NONE OF THE ABOVE
A.
B.
C.
D.
Predicting the Number of Failures
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Back ground to the study
The actual problem
The existing methods/results
The proposed solution
Comparison and assessment
The learning experience
My first problem as a student
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A device is immediately replaced by a new one
after failure;
The lifetimes of each device can be assumed to
be independent and identically distributed;
We want to have an estimate of the expected
number of replacement within time (0,t).
Problem from automobile company (test of new
car) and estimation of warranty cost (free
replacement)
Modelling - Renewal Process
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Let {N(t),t>0} be a counting process and
let Tn denote the time between the (n-1)st
and the nth event of this process.
If the sequence of nonnegative random
variables {T1 , T2 , ...} is i.i.d., then the
counting process {N(t),t>0}is said to be a
renewal process.
Renewal vs Poisson Process
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For a Poisson process, times between events
should be exponentially distributed while for a
renewal process, this is not the case.
X
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Poisson process
X
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Renewal process
X X
XX X
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The Renewal Function
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The expected number of events in a renewal
process is called the renewal function as it is a
function of time t, M(t)=E[N(t)].
The Probability for m(t) can be determined by
t
M (t )  F (t )   M (t  x) f ( x)dx
0
A New Problem
t
M (t )  F (t )   M (t  x) f ( x)dx
0
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Slow convergence when numerical
algorithms or subroutines are used;
Possibly because of the singularity
when f(x) is infinity for some x;
This is common for example when
Weibull distribution is used, or when
only the empirical distribution is
available.
Analytical Results available!!!
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The elementary renewal theorem
M (t )
1
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t
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as t  .
=10, =1 and t=1?
M(t) = -0.4!
In general, we have (asymptotic result)
 
M (t )  
2

2
t
2
2
when t  .
THEY ARE NOT USEFUL
t is NOT large here!
A New (numerical) Method
t
M (t )  F (t )   M (t  x) f ( x)dx
t
0
 F (t )   M (t  x)dF ( x)
0
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Rewrite the equation;
Discretize the integral;
Obtain a set of linear equations;
Solve the linear equations.
A New (numerical) Method
t
M (t )  F (t )   M (t  x)dF ( x)
0
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b
a
The discretization is based on the
definition of the Riemann-Stieltjes Integral:
f ( x)dg ( x)  i 1 f ( xi 1/ 2 )[ g ( xi )  g ( xi 1 )]
n
The RS-Method
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M(t) can be determined by solving the
following equations:
M (ti )  F (ti )   j 1 F (ti  ti 1/ 2 )[ M (ti )  M (ti 1 )]
i
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In fact, M(t) can be calculated recursively.
Comparative Studies
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The method is surprisingly
 Accurate (no need for small step-length)
 Fast (10-fold time saving)
 Simple (20-line BASIC programme)
Reference:
 M. Xie On the solution of renewal-type integral
equations. Communications in Statistics Simulation and Computation 18(1),281-293, 1989.
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programme on the MATLAB Central File Exchange at
http://www.mathworks.com/matlabcentral/fileexchange/2265
-an-introduction-to-stochastic-processes
MATLAB Central > File Exchange > Companion Software
For Books > Statistics and Probability > An Introduction to
Stochastic Processes An Introduction to Stochastic Processes
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function [X]=c3_mt_f(F,g,t)
%
% Find the renewal function given cdf F
% Ref: Xie, M. "On the Solution of Renewal-Type Integral Equations"
% Commun. Statist. -Simula., 18(1), 281-293 (1989)
%
[n,m]=size(F); g0=g(1); g(1)=[];M=F;dno=1-g0;
M(1)=F(1)/dno;
for i=2:m
sum=F(i)-g0*M(i-1);
for j=1:i-1
if j==1
sum=sum+g(i-j)*M(j);
else
sum=sum+g(i-j)*(M(j)-M(j-1));
end
end
M(i)=sum/dno;
end
%
% Output the results
%
d=20;
x=[0]; Mt=[0];
Some further research
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Error bounds
Approximation on renewal function
Bounds of renewal function
Bounds/approximations of renewal-type functions
…
Bounds and approximations
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Using any M1, we can get M2 (analytically)
t
M 2 (t )  F (t )   M1 (t  x)dF ( x)
0
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If M1 is a bound, M2 is also a bound.
Error bounds and convergence.
Some of our papers
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Title: Some analytical and numerical bounds on the renewal function
Author(s): Ran L, Cui LR, Xie M
Source: COMMUNICATIONS IN STATISTICS-THEORY AND
METHODS Volume: 35 Issue: 10 Pages: 1815-1827 Published: 2006
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Title: Some normal approximations for renewal function of large Weibull shape
parameter
Author(s): Cui LR, Xie M
Source: COMMUNICATIONS IN STATISTICS-SIMULATION AND
COMPUTATION Volume: 32 Issue: 1 Pages: 1-16 Published: 2003
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Title: Error analysis of some integration procedures for renewal equation and
convolution integrals
Author(s): Xie M, Preuss W, Cui LR
Source: JOURNAL OF STATISTICAL COMPUTATION AND
SIMULATION Volume: 73 Issue: 1 Pages: 59-70 Published: JAN 2003
“Cost” Analysis
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The company paid US$20,000.00 for solving this
problem (not to me, but to the university);
We gave a 20-line BASIC programme;
For each model of car, they sell 100,000
worldwide, say at $10,000.00 each;
1 percent will be 10 million;
AND this is very much related to “profit”.
Learning Experience
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Typical application of probability models
Unexpected problems need to be solved
(different types of knowledge might be needed)
Software packages, subroutines, and algorithms
are not reliable
A “pity” that I did not move into software
development;
Got interested in “software reliability”…
Questions?
Recent papers on this topic…
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Title: A Gamma-normal series truncation approximation for computing the Weibull renewal function
Author(s): Jiang R
Source: RELIABILITY ENGINEERING & SYSTEM SAFETY Volume: 93 Issue: 4 : 616-626 2008
Title: Estimating the renewal function when the second moment is infinite
Author(s): Bebbington M, Davydov Y, Zitikis R
Source: STOCHASTIC MODELS Volume: 23 Issue: 1 Pages: 27-48 Published: 2007
Title: Nonparametric estimation of the renewal function by empirical data
Author(s): Markovich NM, Krieger UR
Source: STOCHASTIC MODELS Volume: 22 Issue: 2 Pages: 175-199 Published: 2006
Title: Some new bounds for the renewal function
Author(s): Politis K, Koutras MV
Source: PROBABILITY IN THE ENGINEERING AND INFORMATIONAL SCIENCES Volume: 20 : 231-250
2006
Title: Approximation of partial distribution in renewal function calculation
Author(s): Hu XM
Source: COMPUTATIONAL STATISTICS & DATA ANALYSIS Volume: 50 Issue: 6 Pages: 1615-1624 2006
Title: Parametric confidence intervals for the renewal function using coupled integral equations
Author(s): From SG, Tortorella M
Source: COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION Volume: 34 Issue: 3 Pages:
663-672 Published: 2005
Recent papers on this topic…
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Title: An Approximate Solution to the G-Renewal Equation With an Underlying Weibull Distribution
Author(s): Yevkin, Olexandr; Krivtsov, Vasiliy
Source: IEEE TRANSACTIONS ON RELIABILITY Volume: 61 Issue: 1 Pages: 6873 DOI: 10.1109/TR.2011.2182399 Published:MAR 2012
Title: Moments-Based Approximation to the Renewal Function
Author(s): Kambo, Nirmal S.; Rangan, Alagar; Hadji, Ehsan Moghimi
Source: COMMUNICATIONS IN STATISTICS-THEORY AND METHODS Volume: 41 Issue: 5 Pages: 851868 DOI:10.1080/03610926.2010.533231 Published: 2012
Title: CONCAVE RENEWAL FUNCTIONS DO NOT IMPLY DFR INTERRENEWAL TIMES
Author(s): Yu, Yaming
Source: JOURNAL OF APPLIED PROBABILITY Volume: 48 Issue: 2 Pages: 583-588 Published: JUN 2011
Title: Refinements of two-sided bounds for renewal equations
Author(s): Woo, Jae-Kyung
Source: INSURANCE MATHEMATICS & ECONOMICS Volume: 48 Issue: 2 Pages: 189196 DOI:10.1016/j.insmatheco.2010.10.013 Published: MAR 2011
Title: Bayesian estimation of renewal function for inverse Gaussian renewal process
Author(s): Aminzadeh, M. S.
Source: JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION Volume: 81 Issue: 3 Pages: 331-341 Article Number: PII
921405508 DOI: 10.1080/00949650903325153 Published: 2011
Min Xie, Chair Professor of Industrial Engineering
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Research in Quality and Reliability Engineering
William Mong Visiting Fellow to Univ of Hong Kong (1996)
Invited professor at UNPG, Grenoble, France (2000)
Recipient of Lee Kuan Yew research fellowship in
Singapore (1991)
Fellow of IEEE (2006)
Supervisor of over 30 PhD students
Published about 200 journal papers and 8 books,
Editorial services in over 20 international journals.
Organizer, chairman and keynote speaker at numerous
conferences.
Contact me:
[email protected]
Professor M Xie
Dept of Systems Engineering and
Engineering Management
City University of Hong Kong
Department of Systems Engineering
Engineering Management (SEEM)
4/2/2016
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SEEM
vision & mission
Vision
SEEM aspires to be a centre of excellence in education and research in
industrial and systems engineering, engineering management.
Mission
to provide high quality educational and research experience in
disciplines related to industrial and systems engineering, engineering
management and to prepare our graduates for professional and leadership
roles in industry and academia;
to conduct and disseminate research to advance knowledge and knowhow
in industrial and systems engineering, engineering management;
to provide expert services to professional institutions and learned societies,
and consultancies to industrial and governmental organizations, in disciplines
related to industrial and systems engineering, engineering management.
4/2/2016
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SEEM
Disciplines
Systems Engineering
Industrial Engineering
Engineering Management
Healthcare Informatics
Ergonomics and Human Factors
Quality Management
Information and Knowledge
Management Systems
Occupational Safety and Health
Engineering
Product Development Strategies and
Systems
Intelligent Decision Making Systems
Performance Analysis and Quality
Control
Quality and Reliability Engineering
Conflict Management
Supply Chain and Logistics
Management
Quality Engineering and Robust Design
Energy Management and Auditing
ERP and IT Systems Management
Statistical Process Control and
Monitoring
Project Management
Systems Reliability Modelling
Process Design and Optimization
System and Experimental Designs
Computational Optimization and
Applications
Technological Innovation and
Entrepreneurship Management
Lean Manufacturing and Value
Stream
System integration
Failure analyses and quality inspection
Learning Organisation and
Organisational Learning
Modeling and Analysis of Complex
Systems
Green engineering
Manufacturing Strategy
Prognostics and System Health
Management
System Informatics and Data Mining
Computer Model Validation and
Calibration
4/2/2016
Intelligent Maintenance and
Management
Engineering Asset Management
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SEEM
people
SEEM
 17 academic staff (12 faculties, 2
lecturers & 3 Instructors) – more are coming
 20+
4/2/2016
research staff
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SEEM
UG Programmes / Majors
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4/2/2016
BEng in Total Quality Engineering
BEng in Mechatronic Engineering
BEng in e-Logistics and Technology
Management
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SEEM
PG Programmes
PhD and MPhil
 Engineering Doctorate (EngD) in
Engineering Management
 MSc in Engineering Management
(MScEM)
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4/2/2016
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SEEM
Focused research areas:
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4/2/2016
Quality and Reliability Engineering
Prognostics and Health Management
Product Design and Development
Process and Equipment Fault Diagnosis and Evaluation
Data and Knowledge Mining
Decision Making Systems and Methodologies
Logistics and Supply Chain Systems / Management
Optimization and Operation Research
4/2/2016
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