Pfair scheduling of periodic tasks with allocation constraints on multiple processors
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Pfair Scheduling of Periodic Tasks with Allocation Constraints on Multiple Processors. Deming Liu and Yann-Hang Lee. Contributions. Give a polynomial-time pfair schedulability test of periodic tasks over multiple processors with arbitrary allocation constraints

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Pfair scheduling of periodic tasks with allocation constraints on multiple processors

Pfair Scheduling of Periodic Tasks with Allocation Constraints on Multiple Processors

Deming Liu and Yann-Hang Lee

Computer Science & Engineering, ASU


Contributions
Contributions Constraints on Multiple Processors

  • Give a polynomial-time pfair schedulability test of periodic tasks over multiple processors with arbitrary allocation constraints

  • Propose an approximate scheduling algorithm for fixed tasks

Computer Science & Engineering, ASU


Basics of proportionate fairness pfair
Basics of Proportionate Fairness (pfair) Constraints on Multiple Processors

  • n periodic tasks to be scheduled over m identical processors

  • Task constraint – at any time, no task can run on more than one processor

  • All input parameters of any task x are integers

    • Period px, execution time ex, release time etc.

  • Pfair (proportionate fairness)

    • Task x receives either wxt or wxt serving time in [0, t], where wx = ex / px

    • Tasks make progress in a steady rate

Computer Science & Engineering, ASU


Allocation constraints
Allocation Constraints Constraints on Multiple Processors

  • Allocation constraint

    • A task can only run on a given subset of processors

  • An instance  is defined as a tuple {T, R,{P(x)|xT }}

    • Tis a set of n periodic tasks

    • Ris a ser of m identical processors

    • Pis a allocation constraint function defined asP: Tsubsets ofRsuch that forx T, P(x) Rrepresents the subset of processors on which taskxcan be executed

Computer Science & Engineering, ASU


Schedulability with arbitrary allocation constraints
Schedulability with Arbitrary Allocation Constraints Constraints on Multiple Processors

  • For , assuming wx  1 and

  • Theorem 1. Instance is pfair schedulable if and only if there exists a set of nonnegative numbers w(x, r) (xT, rR) such that

Computer Science & Engineering, ASU


Proof by constructing a graph
Proof by Constructing a Graph Constraints on Multiple Processors

  • Proof of Theorem 1

    • Applying max-flow theory to a constructed digraph G = (V, E) representing the scheduling problem

  • V = V0V1V2V3V4V5

    • V0 = {source}.

    • V1 = {1, x, i | xT, i [0, wxL)}.

    • V2= {2, x, j | xT, j [0, L)}.

    • V3 = {3, x, j | xT, j [0, L)}.

    • V4 = {4, r, j | rR, j [0, L)}.

    • V5 = {sink}.

Computer Science & Engineering, ASU


Proof by constructing a graph1
Proof by Constructing a Graph Constraints on Multiple Processors

  • E = E0 E1 E2  E3  E4

    • E0 = {(source, 1, x, i, 1) | xT, i [0, wxL)}.

    • E1 = {(1, x, i, 2, x, j, 1) | xT, i [0, wxL), j [earliest(x, i), latest(x, i)]}

      • earliest(x, i) (latest(x, i)) is the earliest (latest) time slot at which subtask i of task x can run

    • E2 = {(2, x, j, 3, x, j, 1) | xT, j [0, L)}.

    • E3 = {(3, x, j, 4, r, j, 1) | xT, rP(x), j [0, L)}.

    • E4= {(4, r, j, sink, 1) | rR, j  [0, L)}.

Computer Science & Engineering, ASU


An example of graph construction
An Example of Graph Construction Constraints on Multiple Processors

  • An example

    • T = {x0, x1, x2, x3} and R = {r0, r1, r2}.

Computer Science & Engineering, ASU


An example of graph construction1
An Example of Graph Construction Constraints on Multiple Processors

The constructed graph for the example of scheduling problem

Computer Science & Engineering, ASU


Properties of the constructed graph
Properties of the Constructed Graph Constraints on Multiple Processors

  • Lemma 1 – A pfair schedule exists for  iff there is a flow of size of mL in G

    • If there is a fractional flow of size mL, then there is an integral flow of size mL

    • The max flow of G is mL

  • Lemma 2 – There is a flow of size mL iff there exists a set of numbers w(x, r) (xT, rR),

    • Construct a fractional flow of size mL toG [3]

    • Let w(x, r) correspond to

      • Where f(e) denotes the flow of edge e

    • The max flow of G is mL

Computer Science & Engineering, ASU


Proof of the schedulability test condition
Proof of the Schedulability Test Condition Constraints on Multiple Processors

  • Theorem 1 follows by applying Lemma 1 and Lemma 2

Computer Science & Engineering, ASU


Polynomial time pfair schedulability test
Polynomial Time Pfair Schedulability Test Constraints on Multiple Processors

  • Checking existence of the set of w(x, r) (xT, rR), can be done in O((n+m)3)

    • Similarly, construct a bipartite digraph of n+m+2 vertices [3]

    • w(x, r) is mapped to edge flow in the graph

    • Max-flow can be found by O((n+m)3)

Computer Science & Engineering, ASU


An example of polynomial time schedulability test
An Example of Polynomial Time Schedulability Test Constraints on Multiple Processors

Computer Science & Engineering, ASU


On line approximate pfair scheduling
On-Line Approximate Pfair Scheduling Constraints on Multiple Processors

  • Although schedulability test is solved, on-line pfair scheduling for any allocation constraints is still a challenge

  • Let us consider special allocation constraints, fixed tasks

    • Fixed task – A task can only run on a given processor

Computer Science & Engineering, ASU


The approximate scheduling algorithm
The Approximate Scheduling Algorithm Constraints on Multiple Processors

  • All the fixed tasks on a processor are combined to a supertask X with weight wX

  • An approximate algorithm – HPA (hierarchical pfair algorithm)

    • Global scheduling – Use PD2 algorithm [J. Anderson et al.] to schedule all migrating tasks and supertasks

    • Local scheduling – On the time slots allocated to a supertask X in global scheduling, schedule the fixed tasks in X using uniprocessor pfair scheduling algorithm

Computer Science & Engineering, ASU


Uniprocessor pfair scheduling
Uniprocessor Pfair Scheduling Constraints on Multiple Processors

  • Uniprocessor pfair scheduling – Similar to non-preemptive non-idling EDF

    • A subtask i, iN, of task x can only be eligible to run at or after time instant i/wx (the beginning of time slot i/wx )

    • At any time slot, one of eligible subtasks is selected to run according to EDF policy, where the deadline of subtask i, iN, of task x is defined as the time instant (i+1)/wx (inside time slot (i+1)/wx -1)

  • The upper (lower) limit of the windowof a subtask of a fixed task in X under HPA is not 1/wX less (greater) than the upper (lower) limit of the ideal pfair window of the subtask

    • The deviation from ideal pfair is bounded

Computer Science & Engineering, ASU


An application of pfair scheduling

Channel 1 Constraints on Multiple Processors

Session 1

Scheduler

Channel 2

Session 2

… …

… …

Session n

Channel m

Parallel packet scheduling in WDM switching networks

An Application of Pfair Scheduling

  • PFRR (parallel fair round robin) packet scheduling for switching networks

    • Multiple channels exist in links, e.g., WDM

  • n sessions share m channels

Computer Science & Engineering, ASU