PROBABILITY MODELS OF LIFE LENGTH

Transcription

1 Gdańsk Workshop on Stochastic Modelling in Biology, Economics, Medicine and Technology Gdańsk University of Technology Faculty of Applied Physics and Mathematics Department of Probability and Biomathematics June 11 12, 2010 PROBABILITY MODELS OF LIFE LENGTH JOACHIM DOMSTA

2 ABSTRACT The aim of this talk is to present some problems related to mathematical description of duration of life of objects of any kind - from technical to biological. In particular, the following problems will be discussed: - modelling of the life length of a single object - life probability distributions - taking into account the "reliability structure" of composed systems -- in the case of independent elements -- in the case of interacting components - influence of outer factors Accordingly, some corollaries on the duration of particular states will be formulated.

3 MODELLING OF THE LIFE LENGTH OF A SINGLE OBJECT - LIFE PROBABILITY DISTRIBUTIONS the simplest model: a positive number T statistical character of the values of T is represented by probability distributions (p.d.) of the life time; its distribution function is equal F T (t) = Pr{ T t } for all real t problems: the choice of the p.d. for correct description of the reality

4 THE GENERAL SOLUTION: Statistical decision based on given data (observations) justified by the aim of the construction of the model... supported by reasonable model with a few unknown parameters or by statistical experiment with the use of simulation etc.

5 DYNAMICAL MODELLING All said above can be proceeded by additional observations performed after the first steps of the analysis and then followed by an enhanced or more sophisticated model Here we are presenting only the initial steps of such procedures more attention is paid to variety of possible structures with respect to the time evolution and/or to the space distribution

6 EXAMPLES OF USEFUL PROBABILITY DISTRIBUTIONS - CONTINUOUS AND DISCRETE two-parameter gamma (the exponential and Erlang s p.d. included) two-parameter normal (Gaussian) three-parameter Weibull two-parameter Cauchy one-parameter Poisson two-parameter binomial two-parameter negative binomial... and many possible combinations of the above

7 ,,COLD SPAR PARTS' MODEL - WITH A SWITCH THE LIFE TIME OF SYSTEM WITH 1 PLUS n-1,cold' SPAR PARTS T U = T 1 + T 2 + T T n-1 + T n

8 ,,HOT SPAR PARTS' MODEL PARALLEL SYSTEM THE LIFE TIME OF SYSTEM WITH 1 PLUS n-1,hot' SPAR PARTS T U = max { T 1,T 2, T 3,..., T n-1, T n }

9 ,,HOT SPAR PARTS' MODEL SERIES SYSTEM THE LIFE TIME OF SYSTEM WITH 1 PLUS n-1,hot' SPAR PARTS T U = min { T 1,T 2, T 3,..., T n-1, T n }

10 FOR LARGE NUMBER OF PARTS (COMPONENTS, ELEMENTS) -- LIMIT THEOREMS ARE VERY HELPFUL IN THE CHOICE OF THE,APPROPRIATE' P.D. Central Limit Theorem Extreme Values Limit Theorem

11 CENTRAL LIMIT THEOREM THEOREM Let T 1, T 2, T 3,... be i.i.d. with finite moments m = E T, v = D 2 T. Then the normalized sums S n := Σ n (T j -m)/ n v converge in distribution to the standard normal distribution. The distribution is denoted by N(0,1) and has density proportional to exp {-t 2 /2}, for all real t.

12 THEOREM. Let T 1, T 2, T 3,... be i.i.d. such that for some normalization of the maximal value T U,n =max { T 1,T 2, T 3,..., T n-1, T n } the limit distribution function G exists. Then it must be of one of the following types GUMBEL, WEIBULL, FRECHET DISTRIBUTIONS

13 COMMENTS There are many deep results extending the above thorems to the case of dependent r.v. s, especially when the relation between the far elements is suitably close to independence In another direction - the claims remain valid also for structures perturbed if compared with the three above cases The counterpart of the second theorem related to minimal values is a simple corollary

14 DAMAGES EXAMPLARY PROCESSES,First failure instant or, Length of life time = Instant of the first exit from GOOD states (or, entrance into BAD states Clases of GOOD states and inner processes * D&D / D&R / R&D / R&R * temporary / permanent entrances into BAD * continuous / non-continuous inner process

15 exemplary processes

16 EXAMPLARY PROCESSES 1

17 EXAMPLARY PROCESSES 2 T A t iff A X t

18 EXAMPLARY PROCESSES 3 T A t iff A X t

19 EXAMPLARY PROCESSES 4 T A t iff A X t

20 EXAMPLARY PROCESSES 5 T A = inf { t : X t is in (set) A }

21 EXAMPLARY PROCESSES 6 T A = inf { t : X t is in (set) A }

22 SEMI-MARKOV EXAMPLE OF CALCULATIONS

23 SEMI-MARKOV EXAMPLE OF CALCULATIONS, CNTD.

24 STRUCTURED SYSTEMS 1

25 STRUCTURED SYSTEMS 2

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