Online Scheduling of Precedence Constrained Tasks

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Online Scheduling of Precedence Constrained Tasks

• Yumei Huo
• http//web.njit.edu/yh23
• yh23_at_njit.edu
• Department of Computer Science
• New Jersey Institute of Technology

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Outline

• Introduction
• Three basic scheduling algorithms
• Online Scheduling of Precedence Constrained Tasks
• Four nonpreemptive scheduling problems
• Three preemptive scheduling problems
• Future Work

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Outline

• Introduction
• Three basic scheduling algorithms
• Online Scheduling of Precedence Constrained Tasks
• Four nonpreemptive scheduling problems
• Three preemptive scheduling problems
• Future Work

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Coffman-Graham algorithm(P2pj1, precCmax and
P2pj1, prec ?Ci )

• Step 1. Labeling
• Step 2. Scheduling

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Coffman-Graham algorithm (cont.)

• Comparing two decreasing sequences of positive
  integers by lexicographical order
• Example
• (8, 6, 4, 3) lt (8, 6, 5) and (9, 8, 6) lt (9, 8,
  6, 4, 3).

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Example of Coffman-Graham algorithm
(2,1)
(1)
(3)
(4)
(1)
(2)
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Hus algorithm(Ppj1, intreeCmax and Ppj1,
outtreeCmax )

• Step 1. Labeling
• Step 2. Scheduling

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Hus algorithm---Level

• Definition The level of a job i with no
  immediate successor is its processing time pi.
  The level of a job with immediate successor(s) is
  its processing time plus the maximum level of its
  immediate successor(s).

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Example for Hus algorithm
(16)
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Muntz-Coffman algorithm (P2pmtn, prec Cmax,
Ppmtn, intreeCmax and Ppmtn, outtreeCmax )

• Assign one processor each to the jobs at the
  highest level. If there is a tie among y jobs
  (because they are at the same level) for the last
  x (x lt y) processors, then assign x/y processor
  to each of these y jobs. Whenever either of the
  two events below occurs, reassign the processors
  to the unexecuted portion of the unfinished tasks
  according to the above rule.
• Event 1 A task is completed.
• Event 2 We reach a point where, if we
  were to continue the present assignment, we would
  be executing some tasks at a lower level at a
  faster rate than other tasks at a higher level.

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Example for Muntz-Coffman algorithm
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Outline

• Introduction
• Three basic scheduling algorithms
• Online Scheduling of Precedence Constrained Tasks
• Four nonpreemptive scheduling problems
• Three preemptive scheduling problems
• Future Work

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Online Scheduling

• Tasks are released, along with their constraints,
  at various different times.The scheduler
  schedules tasks with no future information.

• We say that an online scheduling algorithm is
  optimal if it always produces a schedule with the
  minimum Cmax, i.e., a schedule as good as any
  schedule produced by any scheduling algorithm
  with full knowledge of future releases of tasks.

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Classical Scheduling Problems with polynomial
optimal algorithms

• Ppj1, intreeCmax and Ppj1, outtreeCmax ---
  Hus algorithm
• P2pj1, precCmax and P2pj1, prec ?Ci ---
  Coffman-Graham algorithm
• P2pmtn, prec Cmax, Ppmtn, intreeCmax and
  Ppmtn, outtreeCmax --- Muntz-Coffman algorithm
• PpmtnCmax --- McNaughtons Rule

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Online version of these problems

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax
• Ppj1, intreei is released at riCmax
• Ppj1, outtreei is released at riCmax
• P2pj1, preci is released at riCmax
• Preemptive scheduling problems
• P2pmtn, preci is released at riCmax
• Ppmtn, intreei is released at riCmax
• Ppmtn, outtreei is released at riCmax
• Ppmtn, rjCmax (K.S.Hong and J.Y-T.Leung, 1988)

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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P2pjp, chainsi are released at riCmax
(Counterexample)
M2 pj2
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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Ppj1, intreei is released at riCmax
(Counterexample)
M3
Tree1 released at r10
S1
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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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P2pj1, preci is released at riCmax

• Algorithm A
• Whenever new tasks arrive, do
• t the current time
• U the set of tasks active (i.e., not finished)
  at time t
• Call the Coffman-Graham algorithm to reschedule
  the tasks in U

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Example
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Our result

• Theorem 2.1.4 Algorithm A is optimal for
  P2pj1, preci is released at riCmax.
• Moreover, the schedule produced by Algorithm A
  has the largest number of tasks completed at any
  time instant t.
• (Note Algorithm A is optimal for P2pj1, preci
  is released at ri ?Cj)

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Ppj1, outtreei is released at riCmax

• Algorithm B
• Whenever new tasks arrive, do
• t the current time
• U the set of tasks active (i.e., not finished)
  at time t
• Call Hu's algorithm to reschedule the tasks in
  U

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Example
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Example
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Our result

• Theorem 2.1.5 Algorithm B is optimal for Ppj1,
  outtreei is released at riCmax.
• Moreover, the schedule produced by Algorithm B
  has the largest number of tasks completed at any
  time instant t.
• (Note Algorithm B is optimal for Ppj1,
  outtreei is released at ri ?Cj)

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Ppj1, pmtn, intreei is released at riCmax
(Counterexample)
M3
Tree1 released at r10
S1
S2
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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, pj1, intreei is released at riCmax----
  no optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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P2 pmtn,preci is released at riCmax

• Algorithm C
• Whenever new tasks arrive, do
• t the current time
• U the set of tasks active (i.e., not finished)
  at time t
• Call Muntz-Coffman algorithm to reschedule the
  tasks in U

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Example
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Example
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Our result

• Theorem 2.2.4 Algorithm C is an optimal online
  algorithm for P2 pmtn,preci is released at
  riCmax.

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Our results

• Nonpreemptive scheduling problems
• P2pjp, chainsi are released at riCmax-----no
  optimal algorithm can possibly exist
• Ppj1, intreei is released at riCmax-----no
  optimal algorithm can possibly exist
• P2pj1, preci is released at riCmax ---- there
  is optimal algorithm
• Ppj1, outtreei is released at riCmax-----
  there is optimal algorithm
• Preemptive scheduling problems
• Ppmtn, intreei is released at riCmax---- no
  optimal algorithm can possibly exist
• P2 pmtn, preci is released at riCmax ---- there
  is optimal algorithm
• Ppmtn, outtreei is released at riCmax---- there
  is optimal algorithm

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Ppmtn, outtreei is released at riCmax

• Theorem 2.2.6 Algorithm C is an optimal online
  algorithm for Ppmtn, outtreei is released at
  riCmax.

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Outline

• Introduction
• Three basic scheduling algorithms
• Online Scheduling of Precedence Constrained Tasks
• Four nonpreemptive scheduling problems
• Three preemptive scheduling problems
• Future Work

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Future Work (Continued Work)

• Online scheduling problems
• Competitive Ratio
• Objective function mean flow time

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Thank you!
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Application of Online Scheduling

• Industrial Applications
• Multiuser operating systems such as Unix and
  Windows
• Web servers
• Database servers
• Load balancers sitting in front of server farms
• (Pruhs K., Sgall J., and Torng E., 2004 )