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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

Computer Science & Engineering, ASU

Contributions

- 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)

- 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 constraint
- A task can only run on a given subset of processors
- An instance is defined as a tuple {T, R,{P(x)|xT }}
- Tis a set of n periodic tasks
- Ris a ser of m identical processors
- Pis a allocation constraint function defined asP: Tsubsets 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

- 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) (xT, rR) such that

Computer Science & Engineering, ASU

Proof by Constructing a Graph

- Proof of Theorem 1
- Applying max-flow theory to a constructed digraph G = (V, E) representing the scheduling problem
- V = V0V1V2V3V4V5
- V0 = {source}.
- V1 = {1, x, i | xT, i [0, wxL)}.
- V2= {2, x, j | xT, j [0, L)}.
- V3 = {3, x, j | xT, j [0, L)}.
- V4 = {4, r, j | rR, j [0, L)}.
- V5 = {sink}.

Computer Science & Engineering, ASU

Proof by Constructing a Graph

- E = E0 E1 E2 E3 E4
- E0 = {(source, 1, x, i, 1) | xT, i [0, wxL)}.
- E1 = {(1, x, i, 2, x, j, 1) | xT, 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) | xT, j [0, L)}.
- E3 = {(3, x, j, 4, r, j, 1) | xT, rP(x), j [0, L)}.
- E4= {(4, r, j, sink, 1) | rR, j [0, L)}.

Computer Science & Engineering, ASU

An Example of Graph Construction

- An example
- T = {x0, x1, x2, x3} and R = {r0, r1, r2}.

Computer Science & Engineering, ASU

An Example of Graph Construction

The constructed graph for the example of scheduling problem

Computer Science & Engineering, ASU

Properties of the Constructed Graph

- 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) (xT, rR),
- 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

- Theorem 1 follows by applying Lemma 1 and Lemma 2

Computer Science & Engineering, ASU

Polynomial Time Pfair Schedulability Test

- Checking existence of the set of w(x, r) (xT, rR), 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

Computer Science & Engineering, ASU

On-Line Approximate Pfair Scheduling

- 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

- 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 – Similar to non-preemptive non-idling EDF
- A subtask i, iN, 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, iN, 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

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

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