In the past few months we were approached by several customers regarding reliability calculations for arrays. While the actual arrays were different (a LED matrix for a variable message road sign, and a sensor array for an IR camera), the goal was the same:
Calculate the distribution of time to array failure.
An array failure occurs whenever a pixel in the array (A) is failed, and any two near pixels are also failed. An example is shown in Figure 1. Figure 2 show an example of 2 additional pixels B and C.
Figure 1: Example of failure region near pixel A
Figure 2: Example of an array failure: Pixels A, B and C are failed
As shown in figure 2: for each failed pixel (A), additional failure of two pixels (B & C) out of the eight neighbors of A causes an array failure.
A grouped failure of pixels is critical for both the road sign and the IR camera. In the case of the road sign, this might mean that a letter in the sign cannot be read by the driver. In the case of the camera, a dark spot will appear in the image. Similar definitions of critical failure apply to other systems as well: Surveillance cameras, satellites and communication towers with overlapping coverage.
In order to enumerate all the failure possibilities one would have to consider all possible positions of failed parts in the array, and the distance between failed parts. This is not a practical method. Instead, a simulation was used. A program was created which does the following:
1. Calculate the mean time to next pixel failure (this depends on the number of operating pixels). A constant failure rate was assumed for each pixel in the array
2. Randomly choose the pixel that failed
3. Check if the latest pixel failure causes an array failure. If array failure does not occur, repeat steps 1-3
4. Repeat simulation 10,000 times and find the distribution of operation times until the array failure
Results are summarized in Table 1:
Resolution Mean time to array failure in units of single pixel MTBF Mean number of pixel failures to array failure
|Resolution||Mean time to array failure in units of single pixel MTBF||Mean number of pixel failures to array failure|
Table 1: Results of simulations regarding arrays of different sizes. For each array 10,000 simulations were conducted and the mean values are given in the table.
Consider an IR camera with resolution of 320 x 200. Each pixel has a MTBF of 1,000,000 hours. There are 64,000 pixels in the array, therefore the first pixel is expected to fail after 106/64,000=15.625 operation hours. Using table 1, the mean operation time to critical failure is found to be 8,530 hours i.e. almost 1 year of continuous operation.
Note that the distribution of time to array failure is not exponential. Figure 3 presents a histogram which corresponds to the time to array failure in the 320 x 200 case: