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How do you get rid of Hz when calculating MIPS?

I'm learning computer structure. I have a question about MIPS, one of the ways to calculate CPU execution time.

The MIPS formula is as follows.

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And if the clock rate is 4 GHz and the CPI is 1. I think MIPS is 4,000hz.

Because It's 4 * 10^9 * Hz / 1 * 10^6.

I don't know if it's right to leave units of Hz.

Solution

  • For any quantity, it's important to know what units it's in. As well as a scale factor (like a parsec is many times longer than an angstrom), units have dimensions, and this is fundamental (at least for physical quantities like time; it can get less obvious when you're counting abstract things).

    Those example units are both units of length so they have the same dimensions; it's physically meaningful to add or subtract two lengths, or if we divide them then length cancels out and we have a pure ratio. (Obviously we have to take care of the scale factors, because 1 parsec / 1 angstrom isn't 1, it's 3.0856776e+26.) That is in fact why we can say a parsec is longer than an angstrom, but we can't say it's longer than a second. (It's longer than a light-second, but that's not the only possible speed that can relate time and distance.)

    1 m/s is not the same thing as 1 kg, or as dimensionless 1.

    Time (seconds) is a dimension, and we can treat instructions-executed as another dimension. (I'll call it I since there isn't a standard SI unit for it, AFAIK, and one could argue it's not a real physical dimension. That doesn't stop this kind of dimensional analysis from being useful, though.)

    (An example of a standard count-based unit is the mole in chemistry, a count of particles. It's an SI base unit.)

    Counts of clock cycles can be treated as another dimension, in which case clock frequency is cycles / sec rather than just s-1. (Seconds, s, are the base SI unit of time.) If we want to make sure we're correctly cancelling it out in both sides, that's a useful approach, especially when we have quantities like cycles/instruction (CPI). Thus cycle time is s/c, seconds per cycle.

    Hz has dimensions of s-1, so if it's something per second we should not use Hz, if something isn't dimensionless. (Clock frequencies normally are given in Hz, because "cycles" aren't a real unit in physics. That's something we're introducing to make sure everything cancels properly).

    MIPS has dimensions of instructions / time (I / s), so the factors that contribute to it must cancel out any cycle counts. And we're not calling it Hz because we're considering "instructions" as a real unit, thus 4000 MIPS not 4000 MHz. (And MIPS is itself a unit so it's definitely not 4000 Hz MIPS; if it made sense to combine units that way, that would be dimensions of I/s2, which would be an acceleration not a speed.).

    From your list of formulas, leaving out the factor of 10^6 (that's the M in MIPS, just a metric prefix in front of Instructions Per Sec, I/s)

    • instructions / total time obviously works without needing any cancelling.

    • I / (c * s / c) = I / s after cancelling cycles in the denominator

    • (I * c/s) / (I * c/I) cancel the Instructions in the denominator:
      (I * c/s) / c cancel the cycles:
      (I * 1/s) / 1 = I/s

    • (c/s) / (c/I) cancel cycles:
      (1/s) / (1/I) apply 1/(1/I) = I reciprocal of reciprocal
      (1/s) * I = I / s

    All of these have dimensions of Instructions / Seconds, i.e. I/S or IPS. With a scale factor of 106, that's MIPS.

    BTW, this is called "dimensional analysis", and in physics (and other sciences) it's a handy tool to see if a formula is sane, because both sides must have the same dimensions.

    e.g. if you're trying to remember how position (or distance-travelled) of an accelerating object works, d = 1/2 * a * t^2 works because acceleration is distance / time / time (e.g. m/s^2), and time-squared cancels out the s^-2 leaving just distance. If you mis-remembered something like 1/2 a^2 * t, you can immediate see that's wrong because you'd have dimensions of m / s^4 * s = m / s^3 which is not a unit of distance.

    (The factor of 1/2 is not something you can check with dimensional analysis; you only get those constant factors like 1/2, pi, e, or whatever from doing the full math, e.g. taking the derivative or integral, or making geometric arguments about linear plots of velocity vs. time.)