writeback: dirty position control

writeback: dirty position control

bdi_position_ratio() provides a scale factor to bdi->dirty_ratelimit, so
that the resulted task rate limit can drive the dirty pages back to the
global/bdi setpoints.

Old scheme is,
                                          |
                           free run area  |  throttle area
  ----------------------------------------+---------------------------->
                                    thresh^                  dirty pages

New scheme is,

  ^ task rate limit
  |
  |            *
  |             *
  |              *
  |[free run]      *      [smooth throttled]
  |                  *
  |                     *
  |                         *
  ..bdi->dirty_ratelimit..........*
  |                               .     *
  |                               .          *
  |                               .              *
  |                               .                 *
  |                               .                    *
  +-------------------------------.-----------------------*------------>
                          setpoint^                  limit^  dirty pages

The slope of the bdi control line should be

1) large enough to pull the dirty pages to setpoint reasonably fast

2) small enough to avoid big fluctuations in the resulted pos_ratio and
   hence task ratelimit

Since the fluctuation range of the bdi dirty pages is typically observed
to be within 1-second worth of data, the bdi control line's slope is
selected to be a linear function of bdi write bandwidth, so that it can
adapt to slow/fast storage devices well.

Assume the bdi control line

	pos_ratio = 1.0 + k * (dirty - bdi_setpoint)

where k is the negative slope.

If targeting for 12.5% fluctuation range in pos_ratio when dirty pages
are fluctuating in range

	[bdi_setpoint - write_bw/2, bdi_setpoint + write_bw/2],

we get slope

	k = - 1 / (8 * write_bw)

Let pos_ratio(x_intercept) = 0, we get the parameter used in code:

	x_intercept = bdi_setpoint + 8 * write_bw

The global/bdi slopes are nicely complementing each other when the
system has only one major bdi (indicated by bdi_thresh ~= thresh):

1) slope of global control line    => scaling to the control scope size
2) slope of main bdi control line  => scaling to the writeout bandwidth

so that

- in memory tight systems, (1) becomes strong enough to squeeze dirty
  pages inside the control scope

- in large memory systems where the "gravity" of (1) for pulling the
  dirty pages to setpoint is too weak, (2) can back (1) up and drive
  dirty pages to bdi_setpoint ~= setpoint reasonably fast.

Unfortunately in JBOD setups, the fluctuation range of bdi threshold
is related to memory size due to the interferences between disks.  In
this case, the bdi slope will be weighted sum of write_bw and bdi_thresh.

Given equations

        span = x_intercept - bdi_setpoint
        k = df/dx = - 1 / span

and the extremum values

        span = bdi_thresh
        dx = bdi_thresh

we get

        df = - dx / span = - 1.0

That means, when bdi_dirty deviates bdi_thresh up, pos_ratio and hence
task ratelimit will fluctuate by -100%.

peter: use 3rd order polynomial for the global control line

CC: Peter Zijlstra <a.p.zijlstra@chello.nl>
Acked-by: Jan Kara <jack@suse.cz>
Signed-off-by: Wu Fengguang <fengguang.wu@intel.com>