Finding reiliability on mulitple components in a system

[John King]

I am trying to find the reliability of batteries on semi tractors for a period of 2 years. I have the miles on the tractor when the battery was replaced. The problem is that there are multiple batteries on a tractor and when a battery is replaced I do not know which one is replaced, so when another battery is replaced again on the same tractor I do not know how long the battery has lasted. Is there anyway of finding the reliability on the batteries without having the exact time to failure of the individual battery, keep in mind not all batteries are replaced at the same time. Am I able to predict how long a tractor will go (miles) until a battery will need to be replaced on the tractor?

[Sergey]

Hi, John. First, you can get mean estimations:
Mean_Miles_Between_Replaced_Per_Single_Battery = Average_Miles_Between_Replaced*Amount_of_Batteries
Second, you can get non-parametric estimation about MCF (Mean Cumulative Function) for repairable failures: MCF(L[i]) = i/n, where
i - number of replacement, n - amount of batteries, L(i) - miles for i-th replacement.
Third, you can get some parametric estimations. If you assume, that replacement rate for SINGLE battery is according NHPP Power Law (i.e rate(L)=Lambda*Beta*(L^(Beta-1)), than replacement rate for ALL batteries will have same Law but instead Lambda will be n*Lambda. So, we can define parameters (n*Lambda) and Beta from Maximum Likelihood Estimations for NHPP Power Law or from Linear Regression using for Log(MCF) and Log(L) - we assume, that MCF(L) = Lambda*(L^Beta)
Regards, Sergey.

[tarik]

To facilitate the types of analysis that sergey suggested, use Weibull++ for the first suggestion, RDA (or Weibull++ 7.0) for the second, RGA for the 3rd suggestion.
http://www.reliasoft.com/products.htm

[Sergey]

John, some additional remarks to my first answer.
First, if after using of parametric estimation (by means of MLE or Linear Regression) you will get "non-good" results (error from real data will large or Correlation will far from one) you have to suppose, that Location Parameter isn't zero. In this case instead variable L you have to use (L - Delta). To search optimal value of Delta you can perform 3-parameter optimization (both for MLE and Non-Linear Regression) ot to use "embedded" optimization (first for Delta and for fix Delta - for Lambda and Beta). Results may be essentially better.
Second, if for your situation some batteries were not replaced after failure - even in this case you can get required estimations. You only have to fix, in what events (miles) were replacement and in what events were not replacement. Sergey.

[tarik]

Sergey,
What do you mean by L and Delta and L? Is it the scale parameter (eta) and location parameter (gama) of the 2-parameter Weibull?

[Sergey]

Tarik, for L(i) I mean "miles for i-th replacement" (see, please, my first answer) and for Delta I mean Location parameter - but not for 2-parameter, rather 3-parameter Weibull with Rate(L)= Lambda*Beta*((L-Delta)^(Beta-1)). Sergey