The random counting variable. Barbara Russo

Transcription

1 The random counting variable Barbara Russo

2 Counting Random Variable } Until now we have seen the point process through two sets of random variables } T i } X i } We introduce a new random variable the Counting Random Variable N(t) } Source: Software Reliability, Measurement Prediction and Application (Musa, Iannino, Okumoto. Pg. 253 )

3 Preamble } Even if we have complete knowledge of the faults in the software we are not able to state with certainty when next the system will fail } Namely, we do not know } what inputs will be supplied to the system and } in what order, } So we cannot predict which fault will be triggered next and which failure will cause

4 Predict the time } We may use the local expected mean to evaluate the next time to failure E[T i ]= tf i (t)dt } Does it really answer our problem?

5 System of prediction } Theoretically yes, but the pdf and cdf are in general not known } Remember? if we consider each single state separately we can consider the system nonrepairable and the local description is given by the hazard rate } In the simplest case when hazards are constant, the {λ i } are unknown, they depend on the system!

6 Example } Assume that all the failures occur randomly. Then the model is negative exponential and f i (t i )=λ i e -λ it i } and E[T i ]=1/λ i } Inference Procedure: for each i there is an unknown parameter λ i

7 } How can we find the values of {λ i }? } We can use reliability history to estimate them!

8 Example } If we want to determine λ i we can use the average of the two previously observed values: } Since it is E[ 1 Ti] = λi } and i 1 [ ] = t + t 2 i E Ti = t i + t i λi = λi 2 t i 2 + t i 1 2 Software reliability 25/05/15

9 Open questions } Arithmetic mean of which periods? } Until when should we sum? } We do not have these answers } We need to introduce a new random variable that counts

10 Counting Random Variable } Until now we have seen the point process through two sets of random variables } Ti } Xi } We introduce a new random variable: the Counting Random Variable N(t) } Source: Software Reliability, Measurement Prediction and Apllication (Musa, Iannino, Okumoto. Pg. 253 ) Software Reliability 5/11/16

11 Counting processes } A stochastic process is counting if the X(t) is the number of items counted by time t } The random counting variable defines a counting process Software reliability 25/05/15

12 The random counting variable } N(t) = # of failures in the interval (0,t] } N((a,b])= # of failures in the interval (a,b] } N((a,b])=N(b)-N(a) Software Reliability 5/11/16

13 Counting Function N(t) } N(t) is a counting function that keeps track of the cumulative number of failures a given system is experiencing from 0 to time t } N(t) is a step function that jumps one up every time a failure occurs and stays at the new level until the next failure Software Reliability 5/11/16

14 N(t) } Note: if you have multiple failures N(t) jumps more than one N(t 4 ) N(t 3 ) N(t 2 ) N(t 1 ) 0 t 1 t 2 t 3 t 4 5/11/16 Software Reliability

15 Probabilities } N(t) number of failures in the interval (0,t] } p(k)=p{n(t)=k} probability that in the interval (0,t] there are k failures } F(k)=P{N(t) k} probability that in the interval (0,t] there are less than k failures

16 Probabilities } N(a,b]=N(b)-N(a) difference of # of failures } p(k)=p{n(a,b]=k} probability that in the interval (a,b] there are k failures } F(k)= P{N(a,b] k} probability that in the interval (a,b] there are less than k failures

17 Expected value for N(t) E[N(t)] = i ip ( i) } Problem: until when should I sum up? } Deciding this is equivalent to expecting that a software has a known number of failures in the interval (0,t]

18 } So we need to find instruments to estimate the expected number of failures this is a hard work! } With the counting random variable we can define a global process!

19 Mean function } Mean function is the expected value µ(t)=e(n(t)) } Note that for N, t is the global time! } µ(t) is the expected number of failures at time t

20 ROCOF } When µ(t) is differentiable then the Rate of Occurence of Failure is defined as } d dt λ(t) = µ(t) } It is the instantaneous rate of change of the expected number of failures

21 } Time model and Count model are equivalent

22 Time/No. Failures model equivalence } Lemma. Let fk(t) denote the density of Tk, the time to the k th failure. Then the ROCOF function is λ(t)= fk(t) Proof. P{N(t) k}= P{Tk t} and P{N(t) k+1}= P{Tk+1 t} P{N(t)=k} = P{N(t) k} - P{N(t) k+1} = P{Tk t} - P{Tk+1 t}

23 Proof µ(t) = E(N(t)) = k*p{n(t)=k} = k>=1 k*(p{t k>=1 k t} - P{T k+1 t}) = P{T k>=1 k t}= = Fk(t) k>=1 d } Then we derive µ(t) = fk(t) dt k>=1