Estimation in delay time modelling


Delay time models for warranty case are proposed in this paper. The models are distinguished into Homogenous Poisson Process (HPP) and Non-Homogenous Poisson Process (NHPP) and apply in two types of delay time distribution which are Exponential and Mixed Exponential. From six models proposed, it will be chosen the best model by using Akaike Information Criteria (AIC).

Keywords: Delay Time, Parameter Estimation, Maintenance, Warranty


Warranty is one of the important things in automobile industry. Sellers would use warranties as a marketing tool to increase sales of a product, whereas customers usually use warranties as a protection, i.e. longer warranty period would be interpreted as a better sign of quality and reliability. This is especially true for new products in which customers have little or no information about quality and reliability. Sellers also offer warranties as a means of protection against exceptional claims by requiring certain responsibilities from customers, for instance warranties do not cover any item failures due to misuse of the product, (Blischke and Murthy 1992).

This paper concern on modelling and estimating the delay time parameters based on operational data in context of warranty case study. This model applies two events as inputs to the likelihood function for the observed data namely failure cycles and inspection cycles. This delay time model shows allowed defect repair at failure point. All the data and information has been used in the model Homogenous Poisson Process (HPP) and Non-Homogenous Poisson Process (NHPP) and will apply in two types of delay time distribution which are (i) Exponential and (ii) Mixed exponential. All the delay time models has been tested with three different form of defect arrival which are (i) Simple model is constant, (ii) Asymptotically constant defect arrival and (iii) Power law.

Delay Time Model


Before the models are presented, some general assumptions for this model are:-

  1. Defects are independent of each other.
  2. Defect arises as a non-homogeneous Poisson process, and the instantaneous rate of occurrence of defects (ROCOD), at time is.
  3. The delay time of a defect is independent of the time of origin and all defects share a common delay time pdf and cdf.
  4. Inspections are both perfect and fault less in that any defects present will be recognized and the process of inspection will not cause a failure to occur. Inspection is assumed to be instantaneous.
  5. Defects are repaired on timescale that may be regarded as instantaneous.
  6. Planned inspection takes place at time from new, where denotes the inspection policy over the systems lifetime.
  7. Opportunity inspection takes place upon failure.
  8. The system is replaced at time i.e. is the systems lifetime.

At the moment, first we are looking HPP model as initial stage and assume is a constant. Let and two consecutive inspection epochs where and the cycle can be anyone of these two types, which are: (i) a failure cycle with a system failure at, when additional defect are identified. And (ii) an inspection cycle where it is a planned inspection takes place and otherwise working system, with time when defect are identified.

Classic Delay Time Model

The classic delay time does not really apply fleet vehicle because the vehicle have multi component system. What we can do is repair at failure for unrelated system. But Baker and Wang (1991) highlight the objective data method has been developed in the context either a single component system or a system with only a few key components, so that the individual behaviour for each component can be modelled separately and a simple system model constructed.

Delay Time Model for HPP,

This is a different delay time model because defect are corrected at the same time repaired the other components. Regarding the failure, they also do the inspection which is unscheduled inspection (unscheduled maintenance) and at the same time defect detected at failure point. Start from the last inspection until the failures occur this is failure cycle. Then, from the failure occur until next inspection we call it as inspection cycle. In HPP case, at failure point we assume system is renewal, this is because of failure and defect will replace and repaired at this point. The system renews and it will start a new cycle in the next operation time.

In figure 2, we found and mean two defects and one failure identified at failure cycle. Then, only one defect identified at inspection cycle. The point for this historical vehicle is they have some failure cycle and some inspection cycle. The system is renewal at each inspection cycle and all are independent. We can look at the contribution at failure and inspection cycle and multiply them altogether or add them in likelihood function. 2.4 Delay time model with Non Homogeneous Poisson Process (NHPP)

Inspection cycle

Failure cycle

Based on figure 3 above, we found there are 5 inspection terms and only 2 failure cycles happen in the system. It is found 2, 1 and 1 defect in the first, second and the last inspection cycle respectively. In the first failure cycle, we found two defects and one failure in that cycle. And just only one defect found in the second failure cycle.

Each cycle have starting and ending point such as or for inspection cycle and failure cycle, respectively. For inspection cycle, as we mention above it have time cycle represent as. The ending time represent as an in existing vehicle historical data and state as. But it is different approach found in paper Christer and Wang (1995), every ending point in failure cycle have respectively. In other word, if any failure occur in the system refer figure 3, we have to give as well. The failure time that we defined is only place appearance in likelihood function.

The Likelihood Function for the Simple Delay Time Model (HPP)

According Christer and Wang (1995), they assumed defect arrival in the model HPP as and delay time distribution as an exponential distribution. Pdf for the is given by, Let and.

In HPP model, all the starting point is zero it because of is constant and the ages are identical. Every time in the cycles, we will set and the age of the vehicle will represent as, refer figure 2.

This formulation respects to estimate delay time parameter and. This model describes defect found at failure point when the inspection is executed. The formula is valid for exp delay time but for the mix exponential model it has to be modified and derive the formula. As the starting, try to estimate the parameters for the HPP model.

Development For HPP And NHPP Model With Consider The Exponential And Mixed Exponential Delay Time.

DTM (Delay Time Model) With Defects Found and Repaired At Failure

The operation time starts at time and finish at time and this will happen for any inspection. The notional inspection time is. is the age at the previous inspection or might be failure point or might be inspection, defects will take at these point. And the system is not renew when the defect remove at these point. The time frame is just show for one cycle and one vehicle. Essentially, for this cycle is notional inspection time, does no failure and defect can reach this time and we observe the defect at this point.

The question is what the probability defect observe at inspection cycle or at point, mean that no failure at this point. The mean number of defect arrive over this time would be (this is mean of the Poisson Process for defect arrival which is Poisson probability for the mean ) and the rate of defect arrival is

Probability defect cause failure at time

Probability all the other defect surviving beyond

The probability defect observes the failure at failure point is slightly difference. When defect is observed at failure point and pointed out it must be defect in that cycle. One of them will cause failure and the other is still defect. We have to get the probability to correspond to the one failure. Defect arose at time and density of the delay time. And then the probability element corresponds to one defect causing a failure at time must be added to probability element all the other defect surviving beyond.

Likelihood contribution on inspection and failure cycle

These are the likelihood contribution fordefect in inspection cycle. Then, denote as inspection cycle

And another likelihood contribution for the failure cycle, , denote as failure cycles

The likelihood contribution for the inspection and failure cycle can be derived as below:

It assumes the likelihood for as failure cycles and as inspection cycles, hence the formula likelihood function for the general case is:-

General formula

The general formula for the log likelihood is:

In this research, especially in all these models it is interested to see how the systems of interest perform as a part of a fleet. For this case study the collected data needs verification to know it has same configuration, same maintenance policy and same operational environment for obtaining the consistent result for the analysis. All the delay time models has been developed

In power law case, most complex systems such as automobiles, aircraft engine, medical diagnostics systems, train locomotives, etc. they are repairable and replaced when they fail. When these systems are fielded or subjected to a customer use environment, it is often of considerable interest to determine the reliability and other performance characteristics under these conditions. Areas of interest may include assessing the expected number of failures during the warranty period, maintaining minimum mission reliability, addressing the rate of wear out, determining when to replace or overhaul a system and minimizing life cycle costs.

The Power Law model for application to data for multiple systems was first introduced by Crow (1974) and see also Ascher and Feingold (1984) for additional information on the Power Law model. This article will give some practical background on the model, discuss data collection and give an application for reducing the life cycle costs for a fleet.

The most popular process model is the Power Law model. This model is popular for several reasons.

  1. It has a very practical foundation in terms of minimal repair. This is the situation when the repair of a failed system is just enough to get the system operational again.
  2. If the time to first failure follows the Weibull distribution, then each succeeding failure is governed by the Power Law model in the case of minimal repair. From this point of view, the Power Law model is an extension of the Weibull distribution. In other words, the Weibull distribution addresses the very first failure and the Power Law model addresses each succeeding failure for a repairable system.
  3. The popularity of the Power Law model is that it generalizes the Poisson process based on the exponential distribution.

Then, we ignore the parameters because it is a constant, the final likelihood function for the case of NHPP model with power law arrivals and as in equation (11) as below. This model estimates three parameters which are and.

Normally, the power law arrival model may represent a sad system when parameter in other words the failure rate increasing (wear out) or happy system and failure rate decreasing (infant mortality) and constant useful life.

Case NHPP model, power law arrivals and mixed exponential

As same as in the previous model which is NHPP model on asymptotically constant for and delay time as mixed exponential, it still need to obtain the contribution to the likelihood for the failure at time with condition an event defect at and defect arose before for this model, so probability of failure at is:

The likelihood function in previous model implies that arose defect for the each vehicle we consider in this research is taken and the delay time is Mixed exponentially distributed with scaled parameterand. These values have been estimate from maintenance data record and imply that defects arrive on average every 2.1days and the mean delay time. And the probability zero delay time when the defect immediately failure is 18.05%. The likelihood function was maximized to give estimated values of parameters with standard error 0.6737, 0.0047 and 0.1311 respectively. The actual value for parameters and are 0.7210 and 0.01, respectively. Comparison with the actual parameter setting for all parameters shows that the likelihood function has recovered the underlying process reasonably well

To select the best model of delay time, the Akaike Information Criteria (AIC) can be used to compare the entire of alternative six models and the best model is chosen as that which minimizes the AIC. When various models were fitted using the Akaike information criterion on AIC , we choose as starting values of those parameter values from the best previously fitted model.

Before fitting the model to the real maintenance data record, we have to specify the functional form of the delay time distribution. The choice and selective the best distribution of delay time from a family plausible distribution is made by using the minimum of the Akaike Information Criteria (AIC) measurement. This is a method of evaluating the goodness of fit model. In this approach the AIC is derived under the assumption that to describe the true distribution when its parameters are suitably adjusted

In this case we assume the preventive maintenance is perfect when identifying the defect during inspection point. We have two categories of delay time distribution with various types of defect arrival form which are (i) Exponential and (ii) Mixed exponential. The Mixed exponential delay time is introduced since some defects may have zero delay time which means the defects immediately failure. The pdf is given by, where is the pdf of the delay time and is the proportion of defects that have zero delay time. The selection of the delay time distribution, the fitted values of parameters are shown in Table 6.8. The comparison for the AIC values for the choices of the delay time distribution can be seen in Table 6.7. The Non Homogeneous Poisson Process (NHPP) with mixed exponential distribution on asymptotically defect arrival form is selected as having lowest AIC value.

However, since the AIC mentioned in this model that do not tell us how good the model fits. These show to us that it is necessary to use a goodness of fit approach before the final best model selected refers to Baker and Wang (1991).

In the statistical analysis it is necessary to consider how well the model fits to the data. Normally, the chi squared test of goodness of fit is the right approach to apply in this problem. In general form we used the standard test of statistic for chi square test as below:

In this chi square test of goodness of fit, it is important that we can obtain histogram of events and values of from the fitted model.

We sorted out the data time from PM to failure for all vehicles then group it into a class or bin group based on this formula, if the data is more then hundred then the class for the chi square goodness of fit test is. Where is the number of observation for maintenance data records. At the same time, it is necessary to identify the minimum and maximum time from PM to failure and this will represent as a histogram of failure include the class of data failure.

By given the information of defect arises at time then we can calculate the probability of failure in the time of. Where is the time from PM to failure before and after class.

The delay time distribution and pattern of defect arrival has been selected in previous section. So we assume the delay time distribution as a Mixed exponential with pdf, and the defect arrival form as asymptotically constant with.

By using the fitted parameters for mixed exponential delay time then we can find the probability of failure in as below:-

Given the probability of failure in, we can calculate the expected number of failure within this particular time by using the formula as below. Hence, we have the probability of failure here. In order to calculate the expected number of failure we need to multiply the probability of failure with the number of failure at risk inspection period:

We used the above formula to calculate the probability of failure for each class and also find the expected number of failure in order to calculate the Chi square for this model. We have difference probability of failure occur for each class, for detail information regarding Chi square goodness of fit test please refers Table 3 as below.

By using the equation probability of failure in equation (1), the expected number of failure has been calculating by using the fitted parameters for Mixed exponential delay time. The expected and actual number of failure for MTB160 (Malaysian Truck) can be observed and compared as shows in Figure 6 as below.

Figure 6 shows that the model something is not fit to the data. To be clear we check this and compare the chi squared test statistic with the critical value of the chi square distribution table. The number degree of freedom is , where the number of model parameters is and the critical value of the chi square distribution is with the significance level of. So, based on calculation the chi square test statistic is. Since the actual statistic is greater than the critical value we cannot accept the null hypothesis at 5 percent of level significance. This means that this model does not fit to the observe data in this model at the significance level of.

When an analyst attempts to fit a statistical model to observed data, we notice how well the model actually reflects the data. How "close" are the observed values to those which would be expected under the fitted model? One statistical test that addresses this issue is the chi-square goodness of fit test. If the computed test statistic is large, then the observed and expected values are not close and the model is a poor fit to the data.

The chi-square goodness-of-fit test is applied to binned data (i.e., data put into classes). This is actually not a restriction since for non-binned data you can simply calculate a histogram or frequency table before generating the chi-square test. However, the values of the chi-square test statistic are dependent on how the data is binned. Another disadvantage of the chi-square test is that it requires a sufficient sample size in order for the chi-square approximation to be valid.


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