To calculate the average life of a pixel, we will model each pixel as comprising 3 LEDs (R, G and B) of equal reliability. First, we will arbitrarily define LED death as a condition in which a given LED emits 50% or less of its original power. (You can pick a different definition that is most useful to you.) LED death may come gradually or suddenly. We don't propose a model in this answer for the LED degradation process although the reliability literature has many. Next, we will interpret Dr. Akbari's example LED lifetime of 5 years to mean that after 5 years, half of a sample population of LEDs are dead (using the above definition). Finally, we can calculation the probability that in a given set of 3 LEDs, one LED is dead rendering the its pixel itself non-functional or dead. To start, let's calculate the probability P(5y) that after 5 years, a 3-LED pixel is dead. Since the probability that none of the 3 LEDs is dead after 5y is (1/2)^2 = 1/8, the probability a pixel is dead after 5 years = 1 - 1/8 = 7/8. But what about after 1y or 4y or 10y? To make a guess about pixel lifetime, we need a model of LED death that contains more than just the mean lifetime. We also need a model of the distribution of lifetimes including 'infant mortality' that is hopefully screened by the manufacturer, random death due to unexpected causes such as static discharge or meteor strike and long-time wear out (which is likely where the 5y figure came from.) If we simplify the problem by suggesting that the 5y mean lifetime of an LED has a standard deviation of 1y, we can estimate the probability that an LED will be dead at age 1y or 4y or 10y. Then, we can repeat the above calculation for P(5y), P(1y), P(4y), P(10y), etc, with this information. As a caveat, what is presented above is a gross oversimplification for reliability of a system of components. We would need some sense of temperature dependence, for example. Also very useful would be a measured distribution plot of LED failure rate (sometimes referred to as a 'bathtub' curve because it is peaked at short times with infant failures, flat for a range in times dominated by random failures and peaked at some later time when long-term failures happen. And finally, a question that Dr. Akbari did not ask but which is interesting is, after what period of time would the 2MPx LCD he mentions in his question (and refers to by the shorthand 'LED') be considered dead. In the spirit of the simplified answer above, we could define LCD death as 10% (or whatever you like) of pixels in the LCD to be dead. LCD life would then be calculated as the time after which 207360 pixels (10% of all pixels) would be dead assuming that all pixels had equivalent reliability. (PS: I am assuming that I have used the same set of definitions as Dr. Akbari, namely LCD = collection of N pixels, say 2,073,600; pixel = collection of 3 LEDs (R, G, B) and LED = one of 3 light emitters within a single pixel.)

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This chart shows the cell’s output voltage over time, at different discharge rates. You can see that the expected service life is not directly proportional to power drain — halving the latter more than doubles the former. Again, with a fixed configuration, we could plan for this, but our LEDs are in motion, which doesn’t make things any easier. You’ll probably just have to come up with an informed average.

Elsewhere in the datasheet (or often printed on the cell itself, in the case of rechargeables), you may find a capacity in mAh (milliamp-hours).

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LED current use is measured in milliamps (mA). As a rule of thumb, we usually use 20 mA as a guideline for a single LED at full brightness, and each color “pixel” contains three LEDs (one each for red, green and blue), for a total of 60 mA per pixel when displaying white at full brightness. If we leave that pixel on in that state for one hour, we’ve used 60 milliamp-hours (60 mA × 1 hour = 60 mAh). If the stated battery capacity is 2100 mAh, we could expect to run that one pixel for about 35 hours continuously before the battery peters out (2100 mAh ÷ 60 mA = 35 hours).

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But single pixels are seldom left on at full brightness for hours. Usually, there’s some combination of brightness levels being mixed, some number of pixels are off entirely, and these states may change many times per second. That’s why we just use reasonable estimates, as in “On average, running this code, I think there are about ten pixels on at any given time, and the average color mix represents a brightness level of 75%.” Starting with the “60 mA per pixel” rule of thumb: 60 mA × 0.75 = 45 mA average per pixel. 45 mA × 10 pixels = 450 mA. Left to run continuously, with a 2100 mAh battery pack, 2100 mAh ÷ 450 mA = 4.66 hours.

Complicating matters further, the LED driver chips themselves use a tiny bit of current, even when the LEDs themselves are “off.” Each chip needs about 2 mA extra…for a strand of 25, it’s using about 50 mA just in this idle state. You may want to factor this into your estimation. Oh, and we forgot to mention power use for the microcontroller that’s driving all this…about 25 mA or so for an Arduino. So we’ll add about 75 mA to the above estimate: 2100 mAh ÷ 525 mA = 4 hours.

If you have a really nice multimeter with an average current recording mode, it will be your new best friend, because it’s doing this based on actual readings. But this capability is usually present only in high-end meters.

You may also want to add some “engineering overhead” to your estimate. Remember what was said about battery capacity often being idealized. So we’ll de-rate the battery by a bit, let’s assume reality is about 80% of the stated capacity: 2100 mAh × 0.8 = 1680 mAh. 1680 mAh ÷ 525 mA = 3.2 hours.

As you can see, there’s an awful lot of fudging and speculation in this process. This is why we say it’s easiest sometimes just to plug in some batteries and keep an eye on it!