Server Scheduling in the Lp norm

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Server Scheduling in the Lp norm

• Nikhil Bansal (CMU)
• Kirk Pruhs (Univ. of Pittsburgh)

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Scheduling
Provide service such that users are satisfied
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Motivation

• Single Machine
• Arbitrary arrival or release time ( rj)
• Arbitrary processing requirement or size ( pj)

t0 (r1)
r2
r3
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Motivation
Single Machine Arbitrary arrival or release time
( rj) Arbitrary processing requirement or size (
pj)
t0 (r1)
r2
r3
c1
c2
c3
Job 1 preempted
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Flow Time

• Completion time cj
• Flow time fj cj-rj (time user waits)

t0 (r1)
r2
r3
c1
c2
c3
Flow Time of Job 1
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Flow Time

• Completion time cj
• Flow time fj cj-rj (time user waits)

t0 (r1)
r2
r3
c1
c2
c3
Flow Time of Job 3
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Stretch
Stretch (i) Flow time (i) / Size (i)
Bender Chakrabarti
Muthukrishnan98
Jobs willing to tolerate flow time proportional
to size Also known as normalized flow time Each
job contributes equally
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Known Results

• Minimize total flow time (?i fi) L1 norm
• Optimal Online algorithm
• Work on job with smallest remaining proc. time
  (SRPT)
• SRPT is 2 competitive for total stretch
  Gehrke, Muthukrishnan, Rajaraman, Shaheen99
• Concern Big jobs could be stuck! Starvation!
• Another end Minimize maximum flow time L1 norm
• First Come First Served (FCFS) is optimal
• But bad average performance Smalls stuck behind
  big

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Balancing average and maximum

• Lp norms Penalize outliers, good average
  performance
• Online algorithms for minimizing
• Flow time (i.e. (?j fjp)1/p)
• Stretch (i.e. (?j sjp)1/p)
• Lp norms, previously studied
• Load balancing Awerbuch Azar Grove 95,
  Alon Azar Woeginger
  Yadid 97, Avidor Azar Sgall 01
• Completion time scheduling Epstein,Sgall99

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Lower Bound
SRPT, FCFS optimal algorithms for p1, 1
No no(1) competitive randomized alg for Lp norms
of flow time and stretch, for 1ltplt1
Size L
Suppose p2
L3 jobs of size 1
Online Cost of big (L3)2 L6 Opt L3 Smalls
delayed by L O(L5)
By time L3 , If do not finish size L job, bad!
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Lower Bound
SRPT, FCFS optimal algorithms for p1, 1
No no(1) competitive randomized alg for Lp norms
of flow time and stretch, for 1ltplt1
Optional Stream of jobs
Size L
L3 jobs of size 1
L5 jobs of size 1/L2
By time L3 , If do not finish size L job, bad!
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Results
Resource Augmentation Kalyanasundaram, Pruhs
95 Online has more resources s-speed,
c-competitive maxI Online(I,s) / Opt(I,1) c

• Thm SRPT, SJF (Shortest Job First)
• are 1? speed O(1/?) competitive

No starvation unless close to peak capacity
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Interpreting Resource Augmentation
Performance
Optimal
Load
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Interpreting Resource Augmentation
Online
Performance
Optimal
Load
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Proof Sketch

• Lp norm of flow time as weighted Flow Time
  problem
• Age(j,t) 0 if tltrj or
  tgtcj
• t-rj if rj lt t
  lt cj
• Observation fj2/2 ?t age(j,t)
• ?j fj2 /2 ?t ?j alive at
  time t age(j,t)

Proof idea Show, at all times total age of
alive jobs lt O(1/? ) total age under Opt
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Proof Idea

• If job J of size x delayed for t units under SJF
• Then either,
• Had lot of work of size lt x before arrival of J
• Lot of work arriving continuously since J
  arrived.
• In either case, Opt can be shown to have
  sufficiently many old unfinished jobs,

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Non-Clairvoyant Scheduling

• Non-Clairvoyant Model Motwani Phillips Torng
  94
• Scheduler does not know size of a job
• Learns size only when job finishes.
• Cannot do things like Shortest Job First, SRPT
• What can we do?
• FCFS, Round Robin (time-sharing),

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Measuring Performance

• Resource Augmentation s-speed, c-competitive
• Online non-clairvoyant (s,I)
• Opt offline clairvoyant (1,I)

c
Max I
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The Algorithm (MLF)

• Multi-Level Feedback (MLF)
• Used in Windows NT, Unix
• Levels L0,L1,L2, job enters L0 first
• In Li, receive 2i amount of work,
• then promoted to Li1
• Work on level i, iff no job in level 0 .. i-1

L2
L1
L0
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Previous Results (Non-Clairvoyant)

• L1 norm of Flow Time
• MLF 1? speed, O(1/?) competitive
  Kalyanasundaram, Pruhs 95
• L1 norm of Stretch
• MLF 1? speed, O(1/?4 log2 B) competitive B.,
  Dhamdhere, Konemann, Sinha 03
• Any algorithm is 1? speed ?(log B/?) competitive
• B ratio of maximum to minimum job size

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Our Results

• MLF is 1? speed , O(1/?4) competitive
• for all norms of flow time
• MLF is 1? speed, O(1/?4 log2 B) competitive
• for all norms of stretch

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Analysis
Generic technique reduce MLF to SJF (Shortest
Job First) Reduces a non-clairvoyant problem
into a clairvoyant one.

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Final Result

• Round Robin At any time, share processor equally
• Considered to be fair (each job treated equally)
• But not good according to the Lp norm criteria
• Round Robin is not 1? speed no(1) competitive
• (for sufficiently small ?)

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Open Problems

• 1) Offline case totally open
• NP-Hard?
• Non-trivial approximation algorithms?
• 2) Multiple machines
• 3) Other notions to deal with tradeoff between
  average vs. max performance

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• Thank You!

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An example
Size L
Goal Minimize ? fi2 (i.e. p2)

a job of size 1 arrives for n time units
If n small ( lt L2), work on the small jobs If n
big ( gt L2), at time L2, shift to finish the
big job
Good algorithm A combination of SRPT FCFS
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MLF
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MLF
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MLF
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MLF
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MLF
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MLF
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MLF
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MLF
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Connection between MLF and SJF
Instance J 2i-1 Instance J 1,2,4,,2i-1
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MLF and SJF
Instance J 2i-1 Instance J 1,2,4,,2i-1
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MLF and SJF
Instance J 2i-1 Instance J 1,2,4,,2i-1
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MLF and SJF
Instance J 2i-1 Instance J 1,2,4,,2i-1
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MLF and SJF
MLF works on smallest level SJF works on smallest
job
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MLF works on a job in level i ? SJF works
on 2i size copy of same job
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MLF and SJF
MLF works on smallest level SJF works on smallest
job
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MLF works on a job in level i ? SJF works
on 2i size copy of same job
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Analysis Idea

• 2 Main ideas
• 1) MLF(J) can be viewed as SJF (J)
• 2) We know that SJF(J) ¼ Opt(J), so MLF(J) ¼
  Opt(J)

previous clairvoyant result
Opt(J)

Fairly general technique, usually allows us to
reduce a non-clairvoyant problem into a
clairvoyant one.