CPU scheduling policy: what result to run when.

Work fetch policy: when to contact scheduling servers, which projects to contact, and how much work to task for.

CPU scheduling

The CPU scheduling policy aims to achieve the following goals (in decreasing priority):

Maximize CPU utilization.
Enforce resource shares. The ratio of CPU time allocated to projects that have work, in a typical period of a day or two, should be approximately the same as the ratio of the user-specified resource shares. If a process has no work for some period, it does not accumulate a 'debt' of work.
Satisfy result deadlines if possible.
Reschedule CPUs periodically. This goal stems from the large differences in duration of results from different projects. Participant in multiple projects will expect to see their computers do work on each of these projects in a reasonable time period.
Minimize mean time to completion. For example, it's better to have one result from a project complete in time T than to have two results simultaneously complete in time 2T.

A result is 'active' if there is a slot directory for it. There can be more active results than CPUs.

Debt

The notion of 'debt' is used to respect the resource share allocation for each project. The debt to a project represents the amount of work (in CPU time) we owe it. Debt is decreased when CPU time is devoted to a project. We increase the debt to a project according to the total amount of work done in a time period scaled by the project's resource share.

For example, consider a system participating in two projects, A and B, with resource shares 75% and 25%, respectively. Suppose in some time period, the system devotes 25 minutes of CPU time to project A and 15 minutes of CPU time to project B. We decrease the debt to A by 25 minutes and increase it by 30 minutes (75% of 25 + 15). So the debt increases overall. This makes sense because we expected to devote a larger percentage of the system resources to project A than it actually got.

The choice of projects for which to start result computations can simply follow the debt ordering of the projects. The algorithm computes the 'anticipated debt' to a project (the debt we expect to owe after the time period expires) as it chooses result computations to run.

A sketch of the CPU scheduling algorithm

This algorithm is run:

Whenever a CPU is free
Whenever a new result arrives (via scheduler RPC)
Whenever it hasn't run for MV seconds, for some scheduling period MV

We will attempt to minimize the number of active result computations for a project by dynamically choosing results to compute from a global pool. When we allocate CPU time to project, we will choose already running tasks first, then preempted tasks, and only choose to start a new result computation in the last resort. This will not guarantee the above property, but we hope it will be close to achieving it.

Decrease debts to projects according to the amount of work done for the projects in the last period.
Increase debts to projects according to the projects' resource shares.
Let the anticipated debt for each project be initialized to its current debt.
Repeat until we decide on a result to compute for each processor:
1. Choose the project that has the largest anticipated debt and a ready-to-compute result.
2. Decrease the anticipated debt of the project by the expected amount of CPU time.
Preempt current result computations, and start new ones.

Pseudocode

data structures:
ACTIVE_TASK:
    double cpu_time_at_last_sched
    double current_cpu_time
    scheduler_state:
        PREEMPTED
        RUNNING
    next_scheduler_state    // temp
PROJECT:
    double work_done_this_period    // temp
    double debt
    double anticipated_debt // temp
    RESULT next_runnable_result

schedule_cpus():

foreach project P
    P.work_done_this_period = 0

total_work_done_this_period = 0
foreach task T that is RUNNING:
    x = T.current_cpu_time - T.cpu_time_at_last_sched
    T.project.work_done_this_period += x
    total_work_done_this_period += x

foreach P in projects:
    P.debt += P.resource_share * total_work_done_this_period
            - P.work_done_this_period

expected_pay_off = total_work_done_this_period / num_cpus

foreach P in projects:
    P.anticipated_debt = P.debt

foreach task T
    T.next_scheduler_state = PREEMPTED

do num_cpus times:
    // choose the project with the largest anticipated debt
    P = argmax { P.anticipated_debt } over all P in projects with runnable result
    if none:
        break
    if (some T (not already scheduled to run) for P is RUNNING):
        T.next_scheduler_state = RUNNING
        P.anticipated_debt -= expected_pay_off
        continue
    if (some T (not already scheduled to run) for P is PREEMPTED):
        T.next_scheduler_state = RUNNING
        P.anticipated_debt -= expected_pay_off
        continue
    if (some R in results is for P, not active, and ready to run):
        Choose R with the earliest deadline
        T = new ACTIVE_TASK for R
        T.next_scheduler_state = RUNNING
        P.anticipated_debt -= expected_pay_off

foreach task T
    if scheduler_state == PREEMPTED and next_scheduler_state = RUNNING
        unsuspend or run
    if scheduler_state == RUNNING and next_scheduler_state = PREEMPTED
        suspend (or kill)

foreach task T
    T.cpu_time_at_last_sched = T.current_cpu_time

Work fetch policy

The CPU scheduler is at its best when projects always have runnable results. When the CPU scheduler chooses to run a project without a runnable result, we say the CPU scheduler is 'starved'.

The work fetch policy has the following goal:

Maintain sufficient work so that the CPU scheduler is never starved.
More specifically: given a 'connection period' parameter T (days), always maintain sufficient work so that the CPU scheduler will not be starved for T days, given average work processing rate. The client should contact scheduling servers only about every T days.
Don't fetch more work than necessary, given the above goals. Thus, try to maintain enough work that starvation will occur between T and 2T days from now.

When to get work

At a given time, the CPU scheduler may need as many as

min_results(P) = ceil(ncpus * P.resource_share)

results for project P to avoid starvation.

Let

ETTRC(RS, k)
[estimated time to result count]

be the amount of time that will elapse until the number of results in the set RS reaches k, given CPU speed, # CPUs, resource share, and active fraction. (The 'active fraction' is the fraction of time in which the core client is running. This statistic is continually updated.)
Let

ETTPRC(P, k) = ETTRC(P.runnable_results, k)
[estimated time to project result count]

Then the amount of time that will elapse until starvation is estimated as

min { ETTPRC(P, min_results(P)-1) } over all P.

It is time to get work when, for any project P,

ETTPRC(P, min_results(P)-1) < T

where T is the connection period defined above.

Computing ETTPRC(P, k)

First, define estimated_cpu_time(S) to be the total FLOP estimate for the results in S divided by single CPU FLOPS.

Let ordered set of results RS(P) = { R_1, R_2, ..., R_N } for project P be ordered by increasing deadline. The CPU scheduler will schedule these results in the same order. ETTPRC(P, k) can be approximated by

estimated_cpu_time(R_1, R_2, ..., R_N-k) / avg_proc_rate(P)

where avg_proc_rate(P) is the average number of CPU seconds completed by the client for project P in a second of (wall-clock) time:

avg_proc_rate(P) = P.resource_share * ncpus * 'active fraction'.

How much work to get

To only contact scheduling servers about every T days, the client should request enough work so that for each project P:

ETTPRC(P, min_results(P)-1) >= 2T.

More specifically, we want to get a set of results REQUEST such that

ETTRC(REQUEST, 0) = 2T - ETTPRC(P, min_results(P)-1).

Since requests are in terms of CPU seconds, we make a request of

estimated_cpu_time(REQUEST) = avg_proc_rate * (2T - ETTPRC(P, min_results(P)-1).

A sketch of the work fetch algorithm

The algorithm sets P.work_request for each project P and returns an 'urgency level':

NEED_WORK_IMMEDIATELY
    CPU scheduler is currently starved (may not have idle CPU)
NEED_WORK
    Will starve within T days
DONT_NEED_WORK
    otherwise

It can be called whenever the client can make a scheduler RPC.

urgency = DONT_NEED_WORK
For each project P
1. Let RS(P) be P's runnable results ordered by increasing deadline.
2. Let S = ETTPRC(RS(P), min_results(P)-1).
3. If S < T
  1. If S == 0: urgency = NEED_WORK_IMMEDIATELY
  2. else: urgency = max(urgency, NEED_WORK)
4. P.work_request = max(0, (2T - S) * avg_proc_rate(P))
If urgency == DONT_NEED_WORK
1. For each project P: P.work_request = 0
Return urgency

The mechanism for actually getting work checks if a project has a non-zero work request and if so, makes the scheduler RPC call to request the work.

Pseudocode

data structures:
PROJECT:
    double work_request

avg_proc_rate(P):
    return P.resource_share * ncpus * time_stats.active_frac

ettprc(P, k):
    results_to_skip = k
    est = 0
    foreach result R for P in order of DECREASING deadline:
        if results_to_skip > 0:
            results_to_skip--
            continue
        est += estimated_cpu_time(R) / avg_proc_rate(P)
    return est

compute_work_request():
    urgency = DONT_NEED_WORK
    foreach project P:
        project_active_frac = active_frac * P.resource_share
        P.work_request = 0
        est_time_to_starvation = ettprc(P, min_results(P)-1)

        if est_time_to_starvation < T:
            if est_time_to_starvation == 0:
                urgency = NEED_WORK_IMMEDIATELY
            else:
                urgency = max(NEED_WORK, urgency)
        P.work_request =
            max(0, (2*T - est_time_to_starvation)*avg_proc_rate(P))

    if urgency == DONT_NEED_WORK:
        foreach project P:
            P.work_request = 0

    return urgency

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