Scheduling Using Timed Automata

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

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

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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

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PART I

• TIMED AUTOMATA

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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.

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Clock Variables

• For a set X of clocks, the set ? (X) of clock
  constraints g is defined by the grammar
• g x ? c c ? x x lt c c lt 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 vY 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.

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Timed Automata

• Formally, a timed automaton is a tuple ltV, V0,
  VF, X, Egt 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.

lts0, g, ?, s1gt

• The state of a real-time program is a pair (s,
  v) such that s is a location and v is a clock
  valuation for X.

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Timed Automata (cont.)

• Types of transitions
• Elapse of time (s, v) ? (s, v t1)

t
0

• Location switch (s, v) ? (s', v') where v'
  v?0

• A run of the automaton is a finite sequence of
  transitions
• ? (s0, v0) ? (s1, v1) ? ? (sn, vn)

t1
tn
t2
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Example
s2
s3
s1
s0
Question is s3 reachable via s2?
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PART II

• JOB-SHOP SCHEDULING

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Job-Shop Scheduling

• M is a set of machines
• Each job J 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

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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 lt j' implies t lt 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 problem is to
  find a minimal length. It is known to be NP-hard.

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Example

• M m1, m2
• J1 (m1, 4), (m2, 5)
• J2 (m1, 3)

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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.

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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 constraints is based on the duration of
  tasks

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Example
J1
M m1, m2 J1 (m1, 4), (m2, 5) J2 (m1, 3)
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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

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Example
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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

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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.

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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 ? 3
c2 0
c1 ? 5
c1 ? 5
c1 ? 5
c1 ? 5
c2 0
c2 ? 3
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PART III

• SCHEDULING UNDER
• UNCERTAINTY

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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.

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Example
(4, 8)
J1 (m1, 10) , (m3, 2, 4) , (m4, 5) J2 (m2,
2, 8), (m3, 7)
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Example
J1 (m1, 10) , (m3, 2, 4) , (m4, 5) J2 (m2,
2, 8), (m3, 7)
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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)

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Uncertain Job-shop Automata

1. Generate job automata
2. Construct mutual exclusion composition
3. In the beginning of a run reset a global clock
4. 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

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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)
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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.

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PART VI

• TASK GRAPH SCHEDULING

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Task Graph Scheduling

• In task graph scheduling, we need to schedule
  tasks on a limited number of machines, while
  respecting the precedence constraints.

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TGS Using Timed Automata

1. Generating the automata for each task

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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?k Hi P

c1 0 ? p1
c1 0
c1 2
c1 16
c1 0 ? p3
c1 16
c2 0
c2 6
c2 6
c2 0 ? p4
c2 2
c2 0
c3 0 ? p1
c2 2
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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

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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.

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Open Problems

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

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• Questions Comments