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Scheduling Using Timed Automata

Selected Topics in Algorithms and Complexity (CSE960). Scheduling Using Timed Automata. Borzoo Bonakdarpour Wednesday, April 13, 2005. Based on. Y. Abdeddaim, O. Maler, Job-shop scheduling using timed automata , 2001

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Scheduling Using Timed Automata

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  1. Selected Topics in Algorithms and Complexity (CSE960) SchedulingUsingTimed Automata Borzoo Bonakdarpour Wednesday, April 13, 2005

  2. Based on • Y. Abdeddaim, O. Maler, Job-shop scheduling using timed automata, 2001 • Y. Abdeddaim, O. Maler, Scheduling under uncertainty using timed automata, 2003 • Y. Abdeddaim, O. Maler, Task graph scheduling using timed automata, 2002 • Krac, Wi, Decidable and undecidable problem in scheduling analysis using timed automata, 2004

  3. Outline • Preliminaries: • Timed automata • Job-Shop Scheduling • Job-Shop Scheduling Using Timed Automata • Scheduling under Uncertainty • Task Graph Scheduling • Decidable and Undecidable Problems • Discussion & Future Work

  4. PART I TIMED AUTOMATA

  5. Preliminaries Timed automata is traditional finite state automata equipped with clock variables and timing constraints. It has been accepted as the standard formalism to model real- time programs. Initially, it was introduced for model checking and verification, but has been recently used for program synthesis and scheduling.

  6. Clock Variables For a set X of clocks, the set  (X) of clock constraintsg is defined by the grammar: g := x  c | c  x | x < c | c < x | g  g where x  X and c  Q is a rational number. A clock valuation is a function v : X  R v(x)is the value of clock variable x. All clock values increase with the same speed. u = v[Y := 0] for Y  X is a clock valuation for X such that x  Y: v(x) = 0 Agrees with v over the rest of the clocks.

  7. g, [] s0 s1 Timed Automata Formally, a timed automaton is a tuple <V, V0, VF, X, E> where V : set of locations, V0 : set of initial locations, VF : set of final locations, X : set of clocks, E (V   (X)  2X  V ) is a set of transitions. <s0, g, , s1> : • The stateof a real-time program is a pair (s, v) such that s is a location and v is a clock valuation for X.

  8. s0 x = 3 y = 1 s0 x = 4 y = 2 t = 1 s1 x = 0 y = 1.2 s0 x = 3 y = 1 [x := 0] Timed Automata (cont.) t 0 • Location switch: (s, v)  (s', v') where v' = v[:=0] Types of transitions: Elapse of time: (s, v)  (s, v + t1) • A run of the automaton is a finite sequence of transitions: •  = (s0, v0)  (s1, v1)  …  (sn, vn) t1 tn t2

  9. (y = 1)? (x < 1)? y := 0 (x < 1)? (x > 1)? Example s2 s0 s1 s3 Question: is s3 reachable via s2?

  10. PART II JOB-SHOP SCHEDULING

  11. Job-Shop Scheduling • M is a set of machines • Each jobJ is a triple (k, , d) where • k  N, the number of tasks •  : {1..k}  M, assignment of machines to tasks • d : {1..k}  N, duration of every task • A job-shop specification is a set J = {J1, J2, …, Jn} of jobs Ji= (ki, i, di) • Assumptions: • A job can wait arbitrary amount of time between two tasks • No preemption

  12. Schedules • A feasible schedule for J is a relation S  J  K T so that (i, j, t)  S indicates that job Ji is busy doing jth task at time t, which satisfies: • Ordering: (i, j, t)  S and (i, j', t')  S then j < j' implies t < t' • Covering & No preemption: For all i, j the set {t: (i, j, t)  S} is nonempty and in the form [r, r + d] and d  di(j) • Mutual exclusion: (i, j, t)  S and (i', j', t)  S then i(j)  i’(j') • The length of a schedule is the maximal t over all (i, j, t)  S • The optimal job-shop scheduling problemis to find a minimal length. It is known to be NP-hard.

  13. J1 J2 (a) m1 J1 J2 J1 m2 m1 9 J1 (b) m2 12 Example • M = {m1, m2} • J1 = (m1, 4), (m2, 5) • J2 = (m1, 3)

  14. m1 m2 m1 m2 J1 J1 m1 m2 m1 m2 J2 J2 m1 m1 J3 J3 (a) (b) Laziness • A schedule S is lazy at task j of job i if immediately before starting that task there an interval in which both the job and the corresponding machine are idle.

  15. Job-shop Timed Automata • Constructing Job Timed Automaton (Ai)for each job Ji: • One clock that represents the elapse of time for each task. • Two states for each task j such that (j) = m • m: indicates that the task is waiting to start on machine m • m: indicates that the task is executing on machine m • Timing constraintsis based on the duration of tasks

  16. J2 c1 := 0 c2 := 0 c1 4 c2 3 f c1 := 0 m1 m1 m2 m1 m1 m2 c1 5 f Example J1 M = {m1, m2} J1 = (m1, 4), (m2, 5) J2 = (m1, 3)

  17. Job-shop Timed Automata (cont.) • Constructing mutual exclusion composition (A) of the job timed automata • An n-tuple q = (q1, …, qn)  (M  M  {f})n is conflicting if it contains two components qa= qb = m  M • We compose the individual job timed automata such that in the final timed automaton • Does not contain conflicting states • Every transition of A satisfies the properties of Ai for all i

  18. J2 c2 := 0 c2 3 f J1 c2 := 0 c2 3 f m1 c1 := 0 c1 := 0 c1 := 0 m1, m1, m1 m1, f c1 4 c1 4 c1 4 c2 := 0 c2 3 m1 f c1 := 0 c1:= 0 c1 := 0 c1 := 0 m1 m2 m1 c2 := 0 c2 3 m1 m2, m1 m1, m2, m1, m1, m2 m1 m1 m2, m1 m1 m1 m2, m2, m1 m2, f f c1 5 c1 5 c1 5 c1 5 c2 := 0 c2 3 m2, f f, f f, Example

  19. Schedules and Runs • A run  is complete if it starts at (s, 0) and ends in f. • Theorem1: If A is a job-shop timed automaton for J then: • For every complete run  of A, its associated schedule Scorresponds to a feasible schedule for J. • For every feasible schedule S for J, there is a run  of A such that S = S and if S is non-lazy so is . • Theorem2: The optimal job-shop scheduling problem can be reduced to the problem of finding the shortest non-lazy path in a timed automata

  20. Job-shop Timed Automata (cont.) • Finding the shortest non-lazy path • The job-shop timed automaton is acyclic • Reachability problem for cyclic timed automata is PSPACE-complete. • Reachability problem for acyclic timed automata is NP-complete. • There exist efficient algorithms that find the shortest path in a timed automaton.

  21. m1 f m2 m1 m1 m1 m1 m1 m2, m1 m1, m1 m1, m2, m1 m1, m2, m1 m2 m2, m2, m1 m2, f f f, m1 f, f f, Example L(S1) = 9 L(S2) = 12 J2 c2 := 0 c2 3 f J1 c2 := 0 c2 3 m1 f c1 := 0 c1 := 0 c1 := 0 m1, m1, f c1 4 c1 4 c1 4 c2 := 0 c2 3 c1 := 0 c1 := 0 c1 := 0 c1 := 0 c2 := 0 c2 3 c1 5 c1 5 c1 5 c1 5 c2 := 0 c2 3

  22. PART III SCHEDULING UNDER UNCERTAINTY

  23. Scheduling Under Uncertainty • The duration of tasks is within an interval [l, u] • Example: J1 = (m1, 10) , (m3, [2, 4]) , (m4, 5) J2 = (m2, [2, 8]), (m3, 7) • How can we design a scheduling policy? • Follow the worst case schedule in both time and ordering • Follow only the ordering of tasks as soon as a machine is available. • Design a scheduling strategy.

  24. m1 m3 m4 J2 10 14 m2 m3 J1 21 8 (a) m1 m3 m4 m1 m3 m4 J2 J2 10 (2, 8) m2 10 m3 m2 m3 J1 J1 21 (b) 19 (c) Example (4, 8) J1 = (m1, 10) , (m3, [2, 4]) , (m4, 5) J2 = (m2, [2, 8]), (m3, 7)

  25. m1 m3 m4 m1 m3 m4 J1 J1 10 (4, 4) m2 m3 10 m2 m3 J2 J2 20 (c) 21 (b) Example J1 = (m1, 10) , (m3, [2, 4]) , (m4, 5) J2 = (m2, [2, 8]), (m3, 7)

  26. Dynamic Scheduling • What was the problem?! • Instead of following a static schedule, whenever a task terminates, we reschedule the residual problem. • Example (4, 8): J1 = (m1, 10) , (m3, [2, 4]) , (m4, 5) J2 = (m2, [2, 8], 7) After terminations of m2 in J2 after 4 time: J1 ' = (m1, 6) , (m3, [2, 4]) , (m4, 5) J2' = (m3, 7)

  27. h(f, v) = 0 h(q, v) = min {t + h(q', v'): (q, v)  (q, v+t1)  (q', v') t 0 Uncertain Job-shop Automata • Generate job automata • Construct mutual exclusion composition • In the beginning of a run reset a global clock • Upon termination of a task formulate the residual problem by calculating the length of runs from the current state (q, v) to (f, v) using backward reachability. This can be done in polynomial time:

  28. 7-c2+h(m4, f, c1+7- c2, ) 5-c1+h(f, m3, , c2+5- c1) min{ 7 c1 if c2  c1 2 5  c1if c2 c1 2 min{ Optimal Strategies for Timed Automata h(f, f, , ) = 0 h(m4, f, c1, ) = 5 – c1 h(m3, f, , c2) = 7 – c2 h(m4, m3, c1, c2) =

  29. Final Analysis • Question: Is this a game?! • Theorem: The problem of finding optimal strategies for job-shop scheduling under uncertainty is solvable using timed automata reachability algorithms.

  30. PART VI TASK GRAPH SCHEDULING

  31. P2 P1 16 2 P6 P3 P4 2 6 16 P7 P5 2 8 Task Graph Scheduling • In task graph scheduling,we need to schedule tasks on a limited number of machines, while respecting the precedence constraints.

  32. c2 := 0 c1 := 0 c2= 16 c1= 1 p1 p2 c3 := 0  p1 c3= 6 p1 p2 p1 p2 p1 p3 p1 TGS Using Timed Automata • Generating the automata for each task

  33. p1 p4 f p5 p3 f p7 f p2 p6 p1 p5 p2 p3 p6 p7 p4 TGS Using Timed Automata (cont.) • Constructing chain covers H = {H1, H2, …, Hk} of (P, ) • Each chain Hi is linearly ordered • Hi Hj =  for all i  j • ikHi = P c1 := 0  p1 c1 := 0 c1= 2 c1 := 0  p3 c1= 16 c1= 16 c2 := 0 c2= 6 c2 := 0  p4 c2= 6 c2= 2 c2 := 0 c3 := 0  p1 c2= 2

  34. TGS Using Timed Automata (cont.) • Constructing mutual exclusion composition • Avoid global states with number of active tasks more the number of available machines • Find the shortest path

  35. Decidable and Undecidable Problems • A large class of schedulability problems are decidable • It has been shown that the schedulability is undecidable if three conditions hold: • The execution times of tasks are uncertain. • A task can preempt another task. • A task can announce its completion time.

  36. Open Problems • Modeling other types scheduling (minimum makespan, minimum completion time) using timed automata • Designing approximation algorithms

  37. Questions & Comments

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