I have a situation where an event is supposed to occur every x minutes for a number of different sites (each site could be configured for a different time interval x). From time to time the event may fail.

I'd like to be able to predict an interval of time that I should worry if an event hasn't occurred. (e.g. for site A 1 missed event indicates a problem that needs attention, but site B misses a lot, so I wouldn't worry about 5 missed events).

To make this a little more concrete. Let's say that I have 3 stores that are supposed to upload their sales data every 30 minutes. If store 1 misses an upload, it may be a temporary problem and they'll be able to upload at the next 30 minute interval. This is especially true if this happens frequently for store 1.

On the other hand store 2 may almost always upload at the 30 minute interval mark, so that one failure indicates a problem that someone needs to look at.

Assuming I have a lot of historical data on how frequently each store fails uploading, how would I go about creating a model for each store based on their history that I can use to determine an estimated prediction of when each store should next upload their data?

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    $\begingroup$ Should this be tagged survival analysis instead of time-series? $\endgroup$ – Jason Morgan Jan 10 '12 at 0:43
  • $\begingroup$ As @JasonMorgan notes, survival models are appropriate for this task. Presumably, you want a parametric repeated-event model and then you would track the value of the deviance residual for a specific store. $\endgroup$ – gung - Reinstate Monica Jan 10 '12 at 4:58

The answer can be offered by survival analysis which, when used in engineering, is called reliability analysis. I understand that you need to estimate fault occurrence cross-time so you might consider this article on assessing product Reliability.

Models you might want to consider include:

  1. Power low (Duane) model
  2. Homogeneous Poisson Process model or if it comes to survival analysis:
  3. Cox-Regression Analysis:

A good reference that covers these is the book Survival Analysis Using S.

A second approach is from perspective of processes control and quality control. A theoretical point of view is in in the book Reliability Modeling, Analysis And Optimization.

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