1. Computer Science & Engineering, ASU1/17 Pfair Scheduling of Periodic Tasks with Allocation Constraints on Multiple Processors Deming Liu and Yann-Hang Lee

2. Computer Science & Engineering, ASU2/17 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

3. Computer Science & Engineering, ASU3/17 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 p x, execution time e x, release time etc.  Pfair (proportionate fairness)  Task x receives either  w x t  or  w x t  serving time in [0, t], where w x = e x / p x  Tasks make progress in a steady rate

4. Computer Science & Engineering, ASU4/17 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 }}  T is a set of n periodic tasks  R is a ser of m identical processors  P is a allocation constraint function defined as P : T  subsets of R such that for x  T, P(x)  R represents the subset of processors on which task x can be executed

5. Computer Science & Engineering, ASU5/17 Schedulability with Arbitrary Allocation Constraints  For , assuming w x  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

6. Computer Science & Engineering, ASU6/17 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 = V 0  V 1  V 2  V 3  V 4  V 5  V 0 = {source}.  V 1 = {  1, x, i  | x  T, i  [0, w x L)}.  V 2 = {  2, x, j  | x  T, j  [0, L)}.  V 3 = {  3, x, j  | x  T, j  [0, L)}.  V 4 = {  4, r, j  | r  R, j  [0, L)}.  V 5 = {sink}.

7. Computer Science & Engineering, ASU7/17 Proof by Constructing a Graph  E = E 0  E 1  E 2  E 3  E 4  E 0 = {(source,  1, x, i , 1) | x  T, i  [0, w x L)}.  E 1 = {(  1, x, i ,  2, x, j , 1) | x  T, i  [0, w x L), 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  E 2 = {(  2, x, j ,  3, x, j , 1) | x  T, j  [0, L)}.  E 3 = {(  3, x, j ,  4, r, j , 1) | x  T, r  P(x), j  [0, L)}.  E 4 = {(  4, r, j , sink, 1) | r  R, j  [0, L)}.

8. Computer Science & Engineering, ASU8/17 An Example of Graph Construction  An example  T = {x 0, x 1, x 2, x 3 } and R = {r 0, r 1, r 2 }. TaskPeriodExecution time WeightAllocation constraint x0x0 530.6 r 0, r 1 x1x1 540.8 r 0, r 1 x2x2 540.8No x3x3 540.8No

9. Computer Science & Engineering, ASU9/17 An Example of Graph Construction The constructed graph for the example of scheduling problem

10. Computer Science & Engineering, ASU10/17 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 to G [3]  Let w ( x, r ) correspond to  Where f ( e ) denotes the flow of edge e  The max flow of G is mL

11. Computer Science & Engineering, ASU11/17 Proof of the Schedulability Test Condition  Theorem 1 follows by applying Lemma 1 and Lemma 2

12. Computer Science & Engineering, ASU12/17 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 )

13. Computer Science & Engineering, ASU13/17 An Example of Polynomial Time Schedulability Test

14. Computer Science & Engineering, ASU14/17 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

15. Computer Science & Engineering, ASU15/17 The Approximate Scheduling Algorithm  All the fixed tasks on a processor are combined to a supertask X with weight w X  An approximate algorithm – HPA (hierarchical pfair algorithm)  Global scheduling – Use PD 2 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

16. Computer Science & Engineering, ASU16/17 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/w x  (the beginning of time slot  i/w x  )  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)/ w x  (inside time slot  ( i +1)/ w x  -1)  The upper (lower) limit of the window of a subtask of a fixed task in X under HPA is not  1 / w X  less (greater) than the upper (lower) limit of the ideal pfair window of the subtask  The deviation from ideal pfair is bounded

17. Computer Science & Engineering, ASU17/17 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 Parallel packet scheduling in WDM switching networks Session 1 Session 2 Session n Scheduler Channel 1 … Channel 2 Channel m