Calculating the load average in the freebsd kernel

[Kannan Varadhan]

Hello,

I am staring at the code in kern/kern_synch.c that calculates the load
average of the system, and I cannot fully understand how the freebsd version
works.  Specifically, it looks as:

/*
 * Constants for averages over 1, 5, and 15 minutes
 * when sampling at 5 second intervals.
 */
static fixpt_t cexp[3] = {
        0.9200444146293232 * FSCALE,    /* exp(-1/12) */
        0.9834714538216174 * FSCALE,    /* exp(-1/60) */
        0.9944598480048967 * FSCALE,    /* exp(-1/180) */
};

...

/*
 * Compute a tenex style load average of a quantity on
 * 1, 5 and 15 minute intervals.
 * XXXKSE   Needs complete rewrite when correct info is available.
 * Completely Bogus.. only works with 1:1 (but compiles ok now :-)
 */
static void
loadav(void *arg)
{
        int i, nrun;
        struct loadavg *avg;

        nrun = sched_load();
        avg = &averunnable;

        for (i = 0; i < 3; i++)
                avg->ldavg[i] = (cexp[i] * avg->ldavg[i] +
                    nrun * FSCALE * (FSCALE - cexp[i])) >> FSHIFT;

 ...

And elsewhere, FSCALE is defined as 1<<FSHIFT, and FSHIFT is 11.

Focusing only the formula, then

avg->ldavg[i] = (cexp[i] * avg->ldavg[i] +
                 nrun * FSCALE * (FSCALE - cexp[i])) >> FSHIFT;
                        ^^^^^^
Why do we have that extra FSCALE multiplier in the second term?  If I do
some logical simplifications, this seems to get me:

(\alpha * FSCALE * ldavg[I] + nrum * FSCALE * FSCALE (1 - \alpha)) >> FSHIFT

I.e.

\alpha * ldavg[I] + nrun * FSCALE * (1 - \alpha)

What am I missing in this arithmetic?

Thanks,

Kannan