Optimal Preemptive Scheduling for General Target Functions

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Optimal Preemptive Scheduling for General Target
Functions

• Leah Epstein
• The Interdisciplinary Center, Herzliya, Israel
• Tamir Tassa
• The Open University, Ramat Aviv, Tel Aviv, Israel
  
• Ben Gurion University, Beer Sheva, Israel

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How to cook three pancakes in 3 minutes

• Each pancake has to be cooked for one minute on
  each side
• We may use two frying pans
• Non-preemptive cooking - Cooking of a pancake
  must be done continuously
• Preemptive cooking - Cooking of a pancake may be
  stopped and resumed at a later stage

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The non-preemptive optimal offline schedule
The preemptive optimal offline schedule
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Non-preemptive scheduling

• A set of m uniformly related machines
• Machine i has speed si
• A set of jobs J
• Job j has weight pj which is its processing time
  on a unit speed machine
• Time to run job j on machine i is pj / si
• This is called the load of job j on machine i
• Each job is assigned to one of the machines

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Load versus Weight

• The weight of a machine i is the sum of weights
  of jobs assigned to it
• Notation
• The weight of a machine i is the sum of loads of
  jobs assigned to it
• Notation
• We have

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Goal functions

• The goal is to minimize a function of machine
  completion times
• A standard goal function is Minimizing the
  makespan (the maximum completion time, i.e. the
  maximum load of any machine)

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Jobs Machines
1 2 4
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An optimal non-preemptive schedule
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Preemptive scheduling

• Each job is cut into pieces and each piece is
  assigned to one of the machines
• Two parts of the same job cannot run
  simultaneously
• The goal is to minimize a function of machine
  completion times
• Makespan is again a common goal function

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Jobs Machines
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An optimal preemptive schedule
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Optimal preemptive scheduling to minimize makespan

• Was studied by Horvath, Lam and Sethi, 1977
• gave the first optimal polynomial time
  algorithm
• They give an exact solution, but with a large
  number of preemptions
• Gonzalez and Sahni, 1978 reduce the number of
  preemptions to 2(m-1)
• Shachnai, Tamir and Woeginger 2002, also study
  the problem with respect to the number of
  preemptions.

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Optimality for various goal functions

• Is the optimal makespan solution optimal for any
  goal function?
• For any nice goal function?
• Symmetric
• Monotone
• Convex
• Examples norms (in particular the
  norm), threshold functions.

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• are the completion times
• The norm for
• The threshold function
• also known as Extensible Bin Packing

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Applications

• The norm measures the balance in the
  schedule, we would like most completion times not
  to vary to much
• The threshold function gives the cost in the case
  where some running time of the machines was
  bought in advance, and needs to be paid for no
  matter if it is used or not

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The norm and the threshold function versus
makespan
The norm cost is
The threshold function cost with c1 is 12
6
10
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The norm versus makespan
The norm cost is
16/3 12/3 8/3 4/3 0
5/3
13/6
8/3
3/2
6
10
12
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The threshold function versus makespan
The threshold function cost with c1 is 8.25
2.25
4 3 2 1 0
2.5
1.5
1
1
6
10
12
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Properties of nice functions

• Idle time (holes) is not necessary
• The completion times are non-decreasing (I.e. a
  faster machine has a larger (or equal) completion
  time) than a slower machine.
• This is not true for general functions

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Non-symmetric functions

• Speeds 1 and 2
• One job of size 5
• An optimal schedule must have a hole

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Conditions on Load and Weight

• Let Wm be the sum of all job weights, then
• Assume that
• Let Wk be the sum of the k largest job weights,
• for some then
• i.e. the amount (weight) assigned to the k
  fastest machines is at least the weight of the
  largest k jobs

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The mathematical program

• Let be the bounded polyhedron of
  all non-negative machine weights
• that satisfy the conditions from the previous
  slide
• The optimal value of the goal function is

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An optimal solution

• Two independent steps
• Finding the completion times by solving a
  mathematical program
• scheduling the jobs given the completion times
• For special cases as the norm and the
  threshold function we solve the mathematical
  program using relatively simple algorithms

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Solving the assignment problem (given the
solution of the math. Program)

• Sort the jobs according to size (starting from
  largest)
• We define two functions
• Red function time
• Green function speed
• (also identifies the machine, if there are
  identical speeds, we need to add a third
  function)
• Given the completion times and speeds
• Speed 1, completion time 10 interval 0,10
• Speed 0.8, completion time 8 interval 10,18
• Speed 0.6, completion time 6 interval 18,24
• Speed 0.3, completion time 3 interval 24,27

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Green function (speed)
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Red function time
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Intervals which are valid for removal
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Future work

• Given the completion times, is it necessary to
  sort the jobs to assign them? (no)
• Simple solution of the mathematical program for
  other goal functions