Resource augmentation and on-line scheduling on multiprocessors

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Resource augmentation and on-line scheduling on
multiprocessors

• Phillips, Stein, Torng, and Wein. Optimal
  time-critical scheduling via resource
  augmentation. STOC (1997). Algorithmica (to
  appear).

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Background on-line algorithms

• Optimization problems given problem instance I,
  algorithm A obtains a value valA(I) -- goal is to
  maximize this value
• On-line algorithms vs an optimal off-line/
  clairvoyant algorithm (OPT)
• Competitive ratio of on-line algorithm A
• min all I ( valA(I)/ valOPT(I) )
• Goal Design an on-line algorithm with largest
  competitive ratio

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Background hard-real-time scheduling

• The on-line problem
• Instance I J1, J2, ..., Jn of jobs
• Each job Jj (rj, pj, dj)
• arrives at instant ri
• needs to execute for pi units...
• by a deadline at instant di
• Job Ji is revealed at instant ri
• Difficult to formulate as an optimization problem
  -- all deadlines must be met!
• In uniprocessor systems, we dodged this issue
• EDF/ LL are optimal algorithms (always meet all
  deadlines)
• EDF/ LL are on-line algorithms...
• ... with competitive ratio one

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Hard-real-time scheduling multiprocessors

• No optimal (in the EDF/LL sense) on-line
  algorithm exists
• Must still meet all deadlines...So, give the
  on-line algorithm extra resources (more/ faster
  processors)
• This paper asks how much extra resources do EDF/
  LL need, in order to meet all deadlines for sets
  of jobs known to be feasible on m processors?
• The answers
• EDF/ LL meet all deadlines if processors are (2
  - 1/m) times as fast
• No on-line algorithm can meet all deadlines if
  processors are lt 1.2 times as fast
• EDF cannot always meet all deadlines if
  processors are (2 - 1/m - ?) times as
  fast, for any ? gt 0

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Why we care

• Our (RTS) task systems
• usually pre-specified (e.g., periodic tasks/
  sporadic tasks)
• on-lineness usually not an issue
• exception overload scheduling (later)
• Well do feasibility analysis (does a schedule
  exist?)
• If feasible, well use the results in this paper
• choose an algorithm (usually, EDF)
• overallocate resources as mandated by these
  results
• sleep well, knowing that the system performs as
  expected
• Why choose feasibility analysis (versus
  schedulability analysis with chosen algorithm)?
• provably competitive performance translates to
  approximation guarantees

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Model and definitions

• Instance I J1, J2, ..., Jn of jobs
• Each job Jj (rj, pj, dj)
• arrives at instant ri
• needs to execute for pi units...
• by a deadline at instant di
• If I is feasible on m processors, an s-speed
  on-line algorithm will meet all deadlines on m
  processors each s times as fast
• (Thus, EDF is a (2 - 1/m)-speed algorithm)

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Digression An example of how wed use these
results
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Scheduling periodic tasks - taxonomy
Periodic task system ? ?1, ?2,..., ?n ?i
(Ti, Ci),
RM EDF
LL/ Pfair
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Remember this? (last class)

• RM-US(1/4)
• all tasks ?i with (Ti/ Ci gt 1/4) have highest
  priorities
• for the remaining tasks, rate-monotonic
  priorities
• Lemma Any task system satisfying
• (SUM ?j ? j ?? Ci /Ti) ? m/4 and
• (ALL ?j ? j ?? Ci /Ti) ? 1/4
• is successfully scheduled using RM-US(1/4)
• Theorem Any task system satisfying
• (SUM ?j ? j ?? Ci /Ti) ? m/4
• is successfully scheduled using RM-US(1/4)

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A new (job-level static priority) scheduling
algorithm

• EDF-US(1/2)
• If Ci/Ti ? 0.5, then jobs of ?i get EDF
  priority
• If Ci/Ti gt 0.5, then jobs of ?i get highest
  priority
• (EDF implementation set deadline to -?)
• Lemma Any task system satisfying
• (SUM ?j ? j ?? Ci /Ti) ? m/2 and
• (ALL ?j ? j ?? Ci /Ti) ? 1/2
• is successfully scheduled using EDF-US(1/2)
• Theorem Any task system satisfying
• (SUM ?j ? j ?? Ci /Ti) ? m/2
• is successfully scheduled using EDF-US(1/2)

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Scheduling periodic tasks w/ migration
RM-US(1/4) EDF-US(1/4)
Pfair
25 50
100
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Back to the results in this paper...(faster
processors)
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The big insight

• Definitions
• A(j,t) denotes amount of execution of job j by
  Algorithm A until time t
• A(I,t) SUM j ?I A(j,t)
• The crucial question Let A be any busy
  (work-conserving) scheduling algorithm executing
  on m processors of speed ? ? 1. What is the
  smallest ? such that at all times t, A(I, t) ?
  A(I,t) for any other algorithm A executing on m
  speed-1 processors?
• Lemma 2.6 ? turns out to be (2 - 1/m)
• Use Lemma 2.6, and an individual algorithms
  scheduling rules, to draw conclusions regarding
  these algorithms

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The oh-so-important lemma 2.6
Lemma Let I be an input instance, t ? 0 any
time-instant. For any busy algorithm A using
(2-1/m)-speed machines, A(I,t) ? A(I, t) for any
algorithm A using 1-speed machines.

• Proof by contradiction
• Suppose there are time instants at which this is
  not true
• Let ? i ? t ? A(I,t) lt A(I,t) and A(i,t)
  lt A(i,t)
• Let j be the job with the earliest release time
  rj in ?
• Let to be the earliest time instant at which
• A(I,to) lt A(I,to) Eq (1)
• A(j,to) lt A(j,to) Eq (2)

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EDF is a (2 - 1/m)-speed algorithm
Instance I J1, J2, ..., Jn job Jj (rj, pj,
dj) is feasible on m procs Wlog, assume that di ?
di1 for all i Let Ik J1, J2, ..., Jk Proof
Induction on k Base EDF on m (1 - 2/m)-speed
procs meets all deadlines for I1, .., Im IH EDF
on m (1 - 2/m)-speed procs meets all deadlines
for I1, .., Ik Were considering Ik1.

• Let Qk1 ? Ik1 denote the jobs in Ik1 with
  deadlines at dk1
• (Ik1 \ Qk1) is Iq for some q ? k
• By IH, EDF on m (1 - 2/m)-speed procs meets all
  deadlines for Iq
• BY definition of EDF, EDF(Ik1) is identical to
  EDF(Iq) on jobs of Iq -- thus, all deadlines in
  Iq are met in EDF(Ik1)
• By Lemma 2.6, EDF(Ik1,dk1) ? OPT(Ik1, dk1)
• Since OPT meets all deadlines at dk1, so must
  EDF on m (1 - 2/m)-speed procs