writeback: dirty rate control

It's all about bdi->dirty_ratelimit, which aims to be (write_bw / N)
when there are N dd tasks.

On write() syscall, use bdi->dirty_ratelimit
============================================

    balance_dirty_pages(pages_dirtied)
    {
        task_ratelimit = bdi->dirty_ratelimit * bdi_position_ratio();
        pause = pages_dirtied / task_ratelimit;
        sleep(pause);
    }

On every 200ms, update bdi->dirty_ratelimit
===========================================

    bdi_update_dirty_ratelimit()
    {
        task_ratelimit = bdi->dirty_ratelimit * bdi_position_ratio();
        balanced_dirty_ratelimit = task_ratelimit * write_bw / dirty_rate;
        bdi->dirty_ratelimit = balanced_dirty_ratelimit
    }

Estimation of balanced bdi->dirty_ratelimit
===========================================

balanced task_ratelimit
-----------------------

balance_dirty_pages() needs to throttle tasks dirtying pages such that
the total amount of dirty pages stays below the specified dirty limit in
order to avoid memory deadlocks. Furthermore we desire fairness in that
tasks get throttled proportionally to the amount of pages they dirty.

IOW we want to throttle tasks such that we match the dirty rate to the
writeout bandwidth, this yields a stable amount of dirty pages:

        dirty_rate == write_bw                                          (1)

The fairness requirement gives us:

        task_ratelimit = balanced_dirty_ratelimit
                       == write_bw / N                                  (2)

where N is the number of dd tasks.  We don't know N beforehand, but
still can estimate balanced_dirty_ratelimit within 200ms.

Start by throttling each dd task at rate

        task_ratelimit = task_ratelimit_0                               (3)
                         (any non-zero initial value is OK)

After 200ms, we measured

        dirty_rate = # of pages dirtied by all dd's / 200ms
        write_bw   = # of pages written to the disk / 200ms

For the aggressive dd dirtiers, the equality holds

        dirty_rate == N * task_rate
                   == N * task_ratelimit_0                              (4)
Or
        task_ratelimit_0 == dirty_rate / N                              (5)

Now we conclude that the balanced task ratelimit can be estimated by

                                                      write_bw
        balanced_dirty_ratelimit = task_ratelimit_0 * ----------        (6)
                                                      dirty_rate

Because with (4) and (5) we can get the desired equality (1):

                                                       write_bw
        balanced_dirty_ratelimit == (dirty_rate / N) * ----------
                                                       dirty_rate
                                 == write_bw / N

Then using the balanced task ratelimit we can compute task pause times like:

        task_pause = task->nr_dirtied / task_ratelimit

task_ratelimit with position control
------------------------------------

However, while the above gives us means of matching the dirty rate to
the writeout bandwidth, it at best provides us with a stable dirty page
count (assuming a static system). In order to control the dirty page
count such that it is high enough to provide performance, but does not
exceed the specified limit we need another control.

The dirty position control works by extending (2) to

        task_ratelimit = balanced_dirty_ratelimit * pos_ratio           (7)

where pos_ratio is a negative feedback function that subjects to

1) f(setpoint) = 1.0
2) df/dx < 0

That is, if the dirty pages are ABOVE the setpoint, we throttle each
task a bit more HEAVY than balanced_dirty_ratelimit, so that the dirty
pages are created less fast than they are cleaned, thus DROP to the
setpoints (and the reverse).

Based on (7) and the assumption that both dirty_ratelimit and pos_ratio
remains CONSTANT for the past 200ms, we get

        task_ratelimit_0 = balanced_dirty_ratelimit * pos_ratio         (8)

Putting (8) into (6), we get the formula used in
bdi_update_dirty_ratelimit():

                                                write_bw
        balanced_dirty_ratelimit *= pos_ratio * ----------              (9)
                                                dirty_rate

Signed-off-by: Wu Fengguang <fengguang.wu@intel.com>