Re: [PATCH 03/15] sched/fair: Add lag based placement

[Peter Zijlstra <peterz@infradead.org>]

On Wed, Oct 11, 2023 at 08:00:22PM +0800, Abel Wu wrote:
> On 5/31/23 7:58 PM, Peter Zijlstra Wrote:
> >   		/*
> > +		 * If we want to place a task and preserve lag, we have to
> > +		 * consider the effect of the new entity on the weighted
> > +		 * average and compensate for this, otherwise lag can quickly
> > +		 * evaporate.
> > +		 *
> > +		 * Lag is defined as:
> > +		 *
> > +		 *   lag_i = S - s_i = w_i * (V - v_i)
> > +		 *
> > +		 * To avoid the 'w_i' term all over the place, we only track
> > +		 * the virtual lag:
> > +		 *
> > +		 *   vl_i = V - v_i <=> v_i = V - vl_i
> > +		 *
> > +		 * And we take V to be the weighted average of all v:
> > +		 *
> > +		 *   V = (\Sum w_j*v_j) / W
> > +		 *
> > +		 * Where W is: \Sum w_j
> > +		 *
> > +		 * Then, the weighted average after adding an entity with lag
> > +		 * vl_i is given by:
> > +		 *
> > +		 *   V' = (\Sum w_j*v_j + w_i*v_i) / (W + w_i)
> > +		 *      = (W*V + w_i*(V - vl_i)) / (W + w_i)
> > +		 *      = (W*V + w_i*V - w_i*vl_i) / (W + w_i)
> > +		 *      = (V*(W + w_i) - w_i*l) / (W + w_i)
> > +		 *      = V - w_i*vl_i / (W + w_i)
> > +		 *
> > +		 * And the actual lag after adding an entity with vl_i is:
> > +		 *
> > +		 *   vl'_i = V' - v_i
> > +		 *         = V - w_i*vl_i / (W + w_i) - (V - vl_i)
> > +		 *         = vl_i - w_i*vl_i / (W + w_i)
> > +		 *
> > +		 * Which is strictly less than vl_i. So in order to preserve lag
> 
> Maybe a stupid question, but why vl'_i < vl_i? Since vl_i can be negative.

So the below doesn't care about the sign, it simply inverts this
relation to express vl_i in vl'_i:

> > +		 * we should inflate the lag before placement such that the
> > +		 * effective lag after placement comes out right.
> > +		 *
> > +		 * As such, invert the above relation for vl'_i to get the vl_i
> > +		 * we need to use such that the lag after placement is the lag
> > +		 * we computed before dequeue.
> > +		 *
> > +		 *   vl'_i = vl_i - w_i*vl_i / (W + w_i)
> > +		 *         = ((W + w_i)*vl_i - w_i*vl_i) / (W + w_i)
> > +		 *
> > +		 *   (W + w_i)*vl'_i = (W + w_i)*vl_i - w_i*vl_i
> > +		 *                   = W*vl_i
> > +		 *
> > +		 *   vl_i = (W + w_i)*vl'_i / W

And then we obtain the scale factor: (W + w_i)/W, which is >1, right?

As such, that means that vl'_i must be smaller than vl_i in the absolute
sense, irrespective of sign.