solving renewal equations – the beginning to my reliability research … m xie, phd, fellow of...
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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 variableLifetime, Reliability,Distribution, estimation
Application to componentssystem, software, service etc.
Prediction of next TMonitoring of T, SPC
Optimization Repair/replacement Cost modelsImprovementDevelopment processQuality and Managementetc
Which one is my “best research” done?
A. Software Reliability (first book)
B. Statistical Process Control (second book)
C. Quality Function Deployment (third book)
D. System Reliability (edited volume and latest book)
E. Weibull distribution (another book)
F. NONE OF THE ABOVE
Predicting the Number of Failures
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
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
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 For a Poisson process, times between events
should be exponentially distributed while for a renewal process, this is not the case.
Poisson process
X X XXX XX
Renewal process
X X XX XX
X X XXX XX
The Renewal Function 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
dxxfxtMtFtM0
)()()()(
A New Problem
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.
t
dxxfxtMtFtM0
)()()()(
Analytical Results available!!! The elementary renewal theorem
In general, we have (asymptotic result)
.1)(
tast
tM
.2
)(2
22
twhent
tM
THEY ARE NOT USEFUL t is NOT large here!
=10, =1 and t=1?M(t) = -0.4!
A New (numerical) Method
Rewrite the equation; Discretize the integral; Obtain a set of linear equations; Solve the linear equations.
t
t
xdFxtMtF
dxxfxtMtFtM
0
0
)()()(
)()()()(
A New (numerical) Method
The discretization is based on the definition of the Riemann-Stieltjes Integral:
t
xdFxtMtFtM0
)()()()(
n
i iii
b
axgxgxfxdgxf
1 12/1 )]()()[()()(
The RS-Method
M(t) can be determined by solving the following equations:
In fact, M(t) can be calculated recursively.
])()()[()()(1 12/1
i
j iiiiii tMtMttFtFtM
Comparative Studies 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.
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 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];for i=d:d:m y=i*t/m; x=[x y]; Mt=[Mt M(i)];end m=length(x); X=zeros(m,2); X(:,1)=x'; X(:,2)=Mt';
Some further research
Error bounds Approximation on renewal function Bounds of renewal function Bounds/approximations of renewal-type functions …
Bounds and approximations
Using any M1, we can get M2 (analytically)
t
xdFxtMtFtM0 12 )()()()(
If M1 is a bound, M2 is also a bound. Error bounds and convergence.
Some of our papers Title: Some analytical and numerical bounds on the renewal function
Author(s): Ran L, Cui LR, Xie MSource: COMMUNICATIONS IN STATISTICS-THEORY AND METHODS Volume: 35 Issue: 10 Pages: 1815-1827 Published: 2006
Title: Some normal approximations for renewal function of large Weibull shape parameter Author(s): Cui LR, Xie MSource: COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION Volume: 32 Issue: 1 Pages: 1-16 Published: 2003
Title: Error analysis of some integration procedures for renewal equation and convolution integrals Author(s): Xie M, Preuss W, Cui LRSource: JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION Volume: 73 Issue: 1 Pages: 59-70 Published: JAN 2003
“Cost” Analysis
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
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”…
Recent papers on this topic… 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… 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: 68-73 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: 851-
868 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: 189-196 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
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: minxie@cityu.edu.hk
Professor M Xie
Dept of Systems Engineering and Engineering Management
City University of Hong Kong
04/19/23 27
SEEM vision & mission
VisionSEEM aspires to be a centre of excellence in education and research in industrial and systems engineering, engineering management.
Missionto 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.
04/19/23 28
SEEM Disciplines
Systems Engineering Industrial Engineering Engineering ManagementHealthcare 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
Conflict Management
Prognostics and System Health Management
Quality and Reliability Engineering Intelligent Maintenance and Management
System Informatics and Data Mining Supply Chain and Logistics Management
Energy Management and Auditing
Computer Model Validation and Calibration
Quality Engineering and Robust Design
Engineering Asset Management
ERP and IT Systems Management Statistical Process Control and Monitoring
Project Management
Systems Reliability Modelling Process Design and Optimization Technological Innovation and Entrepreneurship Management
System and Experimental Designs Computational Optimization and Applications
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
04/19/23 29
SEEM people
SEEM 17 academic staff (12 faculties, 2
lecturers & 3 Instructors) – more are coming
20+ research staff
04/19/23 30
SEEM UG Programmes / Majors
BEng in Total Quality Engineering BEng in Mechatronic Engineering BEng in e-Logistics and Technology
Management
04/19/23 31
SEEMPG Programmes
PhD and MPhil Engineering Doctorate (EngD) in
Engineering Management MSc in Engineering Management
(MScEM)
04/19/23 3204/19/23 32
SEEMFocused research areas:
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
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