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Solving Renewal Equations – the beginning to my reliability research …

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?. Application to components system, software, service etc.

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Solving Renewal Equations – the beginning to my reliability research …

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  1. 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

  2. What is my research about? Application to components system, software, service etc. T, a random variable Lifetime, Reliability, Distribution, estimation Optimization Repair/replacement Cost models Improvement Development process Quality and Management etc Prediction of next T Monitoring of T, SPC

  3. Which one is my “best research” done? • 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) • NONE OF THE ABOVE

  4. 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

  5. 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)

  6. 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.

  7. X X X X X X X Poisson process X X X X X X Renewal process X X X X X X X 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.

  8. 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

  9. 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.

  10. Analytical Results available!!! • The elementary renewal theorem • In general, we have (asymptotic result) m=10, s=1 and t=1? M(t) = -0.4! THEY ARE NOT USEFUL t is NOT large here!

  11. A New (numerical) Method • Rewrite the equation; • Discretize the integral; • Obtain a set of linear equations; • Solve the linear equations.

  12. A New (numerical) Method • The discretization is based on the definition of the Riemann-Stieltjes Integral:

  13. The RS-Method • M(t) can be determined by solving the following equations: • In fact, M(t) can be calculated recursively.

  14. 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

  15. 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); elsesum=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:my=i*t/m; x=[x y]; Mt=[Mt M(i)];end m=length(x); X=zeros(m,2); X(:,1)=x'; X(:,2)=Mt';

  16. Some further research • Error bounds • Approximation on renewal function • Bounds of renewal function • Bounds/approximations of renewal-type functions • …

  17. Bounds and approximations • Using any M1, we can get M2 (analytically) • If M1 is a bound, M2 is also a bound. • Error bounds and convergence.

  18. 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

  19. “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”.

  20. 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”…

  21. Questions?

  22. 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

  23. 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

  24. 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.

  25. Contact me: minxie@cityu.edu.hk Professor M Xie Dept of Systems Engineering and Engineering Management City University of Hong Kong

  26. Department of Systems Engineering Engineering Management (SEEM)

  27. 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.

  28. SEEM Disciplines

  29. SEEM people SEEM • 17 academic staff (12 faculties, 2 lecturers & 3 Instructors) – more are coming • 20+ research staff

  30. SEEM UG Programmes / Majors BEng in Total Quality Engineering BEng in Mechatronic Engineering BEng in e-Logistics and Technology Management

  31. SEEMPG Programmes PhD and MPhil Engineering Doctorate (EngD) in Engineering Management MSc in Engineering Management (MScEM)

  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 9/20/2014 32

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