Polynomial-Time Pfair Scheduling with Allocation Constraints on Multiple Processors

1. Pfair Scheduling of Periodic Tasks with Allocation Constraints on Multiple Processors Deming Liu and Yann-Hang Lee Computer Science & Engineering, ASU

2. 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

3. 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

4. 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)|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

5. 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) (xT, rR) such that Computer Science & Engineering, ASU

6. 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 = 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

7. Proof by Constructing a Graph • 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

8. An Example of Graph Construction • An example • T = {x0, x1, x2, x3} and R = {r0, r1, r2}. Computer Science & Engineering, ASU

9. An Example of Graph Construction The constructed graph for the example of scheduling problem Computer Science & Engineering, ASU

10. 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) (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

11. Proof of the Schedulability Test Condition • Theorem 1 follows by applying Lemma 1 and Lemma 2 Computer Science & Engineering, ASU

12. Polynomial Time Pfair Schedulability Test • 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

13. An Example of Polynomial Time Schedulability Test Computer Science & Engineering, ASU

14. 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

15. 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

16. Uniprocessor Pfair Scheduling • 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

17. Channel 1 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