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”…

Questions?

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

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Department of Systems Engineering Engineering Management (SEEM)

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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.

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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

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SEEM people

SEEM 17 academic staff (12 faculties, 2

lecturers & 3 Instructors) – more are coming

20+ research staff

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SEEM UG Programmes / Majors

BEng in Total Quality Engineering BEng in Mechatronic Engineering BEng in e-Logistics and Technology

Management

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SEEMPG Programmes

PhD and MPhil Engineering Doctorate (EngD) in

Engineering Management MSc in Engineering Management

(MScEM)

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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