Modelling and Assessment of Survival Probability of Shock Model with Two Kinds of Shocks

The study aims at modelling and assessment of survival probability of a component experiencing two kinds of shocks namely, damage shock and fatal shock. Shocks are occurring randomly in time as events of a Poisson Process. The two cases of fixed/random threshold of components are studied. Survival probabilities of proposed models are derived. Maximum likelihood estimators (MLEs) of survival probabilities are obtained using the data from life testing experiments. Fisher information and asymptotic distribution of MLEs of parameters are obtained when a constant threshold is considered. Computation and comparison of estimators of two cases (constant threshold and random threshold) are made through simulation studies. The study recommends the consideration of threshold as a random variable.


Introduction
Study of reliability can be broadly classified into two major aspects i.e. reliability modelling and reliability assessments. Modelling reliability aims at development of reliability model under certain assumptions. Here one discusses about configuration of associated components, methods of reliability enhancement, the conditions under which the system functions at its best, etc. Here the point of interest is "system survival" and compensatory measures to increase system survival.
On the counter part, reliability assessment involves processes such as identifi-is outlined in Section 6. Results and findings are also discussed in the same Section.

Modelling Survival Probability
A component or system is experiencing shocks occurring randomly in time as events of a Poisson process with intensity , 0 λ λ > . Shocks are of two types. One is fatal shock, which causes failure of the system or component. Another is damage shock, which causes some amount of damage to the component. Damages are non-accumulating. Damages (X) are assumed to be independent and identical exponential random variables with parameter , 0 θ θ > . The system or component fails either due to experiencing a fatal shock or whenever the amount of damage due to a damage shock exceeds its threshold u. If The first part of above expression under summation represents the probability that the component experiences k number of damage shocks during ( ) 0,t , second part is the probability that the damages due to all k shocks are less than its threshold u. The third component is the probability that during ( ) 0,t the component do not experience a fatal shock. During ( ) 0,t , the component may experience 0 shock or 1 shock or 2 shocks and so on … and hence the summation with 0,1, 2, . k =  Taking the terms independent of k outside the summation and further simpli- Example 2: Credit-scoring systems aid the decision of whether to grant credit to an applicant or not. Traditionally, this is done by estimating the probability that an applicant will default. In recent years the aim has been changing towards choosing the customers of higher profit. In this case it is important to know that when a customer will default (fatal shock). It is possible that if the time to default is long, the acquired interest will compensate or even exceed losses resulting from default. Another factor that affects profitability is the case in which customers close their account early or pay off the loan early by switching to another lender. Depending on when the actual repayment occurred, the lender will lose a proportion of the interest on the loan [15]. One can consider the number of times partial repayments made as damage shocks and the amount repaid is as damage caused due to damage shock. On the other hand, if a customer pays off the loan early then it will be considered as amount of damage due to a damage shock exceeding the threshold (outstanding loan amount). In this case, one will be interested to investigate the probability that a customer will default or the probability that a customer pays off the loan early.

Assessment of the Survival Probability
Suppose r components each with threshold u having life distribution are subjected to life testing experiment and the experiment is conducted until all of them fail. Out of r components, let 1 r components fail due to damage shock i.e. damage exceeding the threshold u and 2 r ( ) 1 r r = − components fail due to occurrence of fatal shock. It is assumed that damage due to fatal shock is not observable. Let the i th component fail at th representing the amount of damage due to j th damage shock to i th component. ij X 's are assumed to be independent exponential random variables with pa- Whenever a component fails due to a damage shock, it is assumed that the damage due to a shock at which the component fails is not observable but is , , , , , , , , , , , , , , Combining the two cases, the joint distribution 1 L of all the involved r.v.'s is given by where, .. 1 1 1   , , , , , ,

Simulation Study
are generated as follows: Step 1: A random number i V is generated from  30,50 r = (large samples). In the example of heart disease, the number of patients will be moderately large and among them the cases of cardiac arrests (death due to fatal shock) will be small. To get the non-zero number of cases of failure due to fatal shocks, r needs to be moderately large.

Modelling Survival Probability When Threshold (Ui) Is Random Variable
The threshold of the component or system is considered as random variable by many researchers. It is trivial assumption as the damage sustaining capacity depends on threshold and threshold of the component may vary due to raw materials used in its manufacturing, technology with which it is manufactured, inbuilt capacity, physical properties, random factors that influence its shock sustaining capacity, etc. It is assumed that threshold i U of the component is exponential random variable with parameter , 0 σ σ > .
The survival probability of the component at time t is given by:

Assessment of the Survival Probability
Suppose r components each with threshold , 1, 2, , are subjected to life testing experiment and the experiment is conducted until all of them fail. Continuing the life testing experiment as in Section 3.
The joint distribution 2 L of all the involved r.v.'s is given by ( ) where, ( )

Simulation Study
The estimators are computed using Monte-Carlo simulation for random thre-    [17].   S t , so the study advocates consideration of threshold as a random variable and is a realistic assumption also.