Smiths Rule In Stochastic Scheduling Caroline Jagtenberg Uwe - - PowerPoint PPT Presentation

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Smith’s Rule In Stochastic Scheduling

Caroline Jagtenberg Uwe Schwiegelshohn Marc Uetz

Utrecht University Dortmund University University of Twente

Aussois 2011

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

The (classic) setting

Problem n jobs, nonpreemptive, processing times pj and weights wj m identical, parallel machines Cj = completion time of job j goal: minimize total weighted completion time, wjCj P| | wjCj (thanks JKL) Complexity The problem is (strongly) NP-hard

[Bruno et al. 1974]

PTAS exists

[Skutella and Woeginger, 2000]

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

WSPT a.k.a. Smith’s rule a.k.a. Photographer’s Rule

WSPT Schedule jobs in order of non-increasing ratios wj/pj Performance On 1 machine WSPT is optimal

[Smith, 1956]

For identical, parallel machines WSPT is a 1+

√ 2 2

≈ 1.207- approximation; this is tight

[Kawaguchi and Kyan, 1986]

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Step to stochastic scheduling

Stochastic Scheduling processing times P = (P1, . . . , Pn) unknown in advance Pj’s are random variables, known distribution solution no schedule, but scheduling policy Π for any policy Π: wjCj(Π, P) is a random variable Minimize expected performance E( wj Cj(Π, P))

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Complexity of (general) stochastic scheduling

In general, optimal policies are NP-hard to find Calculating the objective value of a given policy can be # P complete

[Hagstrom 1988]

Optimal policy may require deliberate idleness

[U. 2003]

Question Does it become (significantly) easier if we restrict e.g. to only exponentially distributed processing times, i.e., Pj ∼ exp(λj)? i.e., Pj’s are memory-less, P[Pj > x + t | Pj > t] = P[Pj > x] Open Problem

1 Does there exist an optimal policy without deliberate idleness? Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Intuition

Quote ”Scheduling: Theory, Algorithms, and Systems”

[Pinedo, 2002]

Example: P||Cmax is NP-hard for deterministic scheduling, but for Pj ∼ exp(λj), LEPT is optimal

[Weiss and Pinedo, 1980]

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Most natural & simple scheduling policy: WSEPT

WSEPT or Smith’s rule Greedily schedule jobs in order of decreasing wj/E(Pj) = wj λj. Facts about WSEPT for minimizing E[ wjCj] For one machine WSEPT is optimal

[Rothkopf, 1966]

For parallel machines WSEPT is optimal if ordering exists w. w1 ≥ ... ≥ wn and w1λ1 ≥ ... ≥ wnλn

[K¨ ampke, 1987]

For parallel machines WSEPT is a (2 − 1

m)-approximation

[M¨

• hring, Schulz, U. 1999]

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Our (Counterintuitive?) Result

Theorem Performance of WSEPT is not better than 1.243 OPT. That is, ∃ instances where in expectation E[ wjC WSEPT

j

] > 1.243 E[ wjC OPT

j

] Counterintuition: This is even worse than WSPT in deterministic scheduling, which is at most 1.207 OPT. Proof Follows from analysis and adaptation of the instance given by Kawaguchi and Kyan.

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Kawaguchi & Kyan (deterministic) example

x ·m big jobs with pj = wj = p, n ·m small jobs with pj = wj = 1

n

Left schedule: wjCj = (p2xm ) + ( 1

2 1 1−x m ) + o(1)

Right schedule: wjCj = ((1 + p)pxm ) + ( 1

2m ) + o(1)

p = 1 + √ 2 and x =

1 2+ √ 2 gives a maximal ratio of 1+ √ 2 2

≈ 1.207

Marc Uetz Smith’s Rule in Stochastic Scheduling

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Stochastic version of Kawaguchi & Kyan example

x · m i.i.d. big jobs each with Pj ∼ exp(λ), and wj := E[Pj] = 1 λ := p n · m i.i.d. small jobs each with Pj ∼ exp(n), and wj := E[Pj] = 1 n

Marc Uetz Smith’s Rule in Stochastic Scheduling

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Scheduling m jobs with Pj ∼ exp(λ)

Lemma Say we start at time t = 0 m i.i.d. jobs with Pj ∼ exp(λ), the expected number of available machines at time t is at least f (t) := m(1 − e−tλ) − 1 . Interpretation

Marc Uetz Smith’s Rule in Stochastic Scheduling

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Behaviour of parallel jobs with Pj ∼ exp(λ)

When scheduling in parallel m jobs with i.i.d. processing times Pj ∼ exp(λ), the first completion is expected at time 1/(mλ). As Pj’s are memory less, E[Pj − t | Pj > t] = E[Pj] = 1/λ, the second completion is expected time 1/((m − 1)λ) later. etc., so jth completion is expected at time tj =

j

• i=1

1 (m − i + 1)λ Using H(m) =

m

• i=1

1 i ≥ ln(m) + 0.58, find that tj ≤ 1

λln( m m−j ),

so # free machines at t: ≥ ⌊m(1 − e−tλ)⌋ ≥ m(1 − e−tλ) − 1

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Stochastic version of (worst case) WSPT schedule

Remember Kawaguchi and Kyan’s (worst case) schedule Machines finish processing short jobs “more or less” at t = 1 E[difference] ≤ 1

n

m−1

i=1 1 i ≈ 0 (as we have n > m)

Each long job completes in expectation at time ≈ (1 + 1

λ)

Hence, E[ wjCj] ≈ to the deterministic case.

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Stochastic version of (optimal) WSPT schedule

The expected optimal schedule of the stochastic variant: Contribution of long jobs is the same as in the deterministic case. What about the small jobs? Compute time T such that T

0 f (t)dt ≥ total expected processing

volume of small jobs. How? Numerically, T = 1.2933 suffices.

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

We can now approximate location of small jobs. But how much do they contribute to the objective value E[ wjCj] ?

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Contribution of Small Jobs

Lemma Consider nT jobs with i.i.d. processing times Pj ∼ exp(n) and weights wj = 1/n, scheduled on a single machine. Then for all ε > 0 there exists n large enough so that E[

j wjCj] ≤

T t dt + ε . Proof. E[

j wjCj] = 1 n nT 1/n+T 2

= 1

2T 2 + 1 2nT =

T

0 t dt + 1 2nT

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Contribution of Small Jobs

Generalization We can generalize this lemma for parallel machines. Let m(t) be the number of machines available at time t, then E[

j wjCj] ≤

T m(t) t dt + ε

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Comparing the objective values E[ wjCj]

Ingredients

1 long jobs’ contribution same as in deterministic case 2 machines w. small jobs finish at ≈ equal times (“sand”) 3

for OPT

• # available machines ≥ f (t) = m(1 − e−tλ) − 1

small jobs contribute E[

j wjCj] ≤

T

0 f (t) t dt

Putting all that together, we get WSEPT is an α-approximation, with α ≥

E(P

j wjCj[B])

E(P

j wjCj[A])

≥ 1.229 (n → ∞, m → ∞)

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

The result

Optimizing over # and length E[Pj] of the long jobs The result above was for

1 2+ √ 2 m ≈ 0.29 m long jobs with

E[Pj] = 1 + √ 2 ≈ 2.4 Taking for example: 0.43 m long jobs with E[Pj] ≈ 1.8 yields α > 1.243 Theorem For jobs with exponentially distributed processing times, WSEPT is no better than a 1.243 - approximation.

Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End

Conclusions

What we’ve found With Pj ∼ exp(λj), WSEPT can be factor > 1.243 away from

• ptimal policy (in expectation); worse than tight bound for

deterministic scheduling, ≈ 1.207 [→ WAOA 2010 proceedings] Open Problems

2 Instance(s) where WSEPT performs even worse? 3 I’d rather go and improve the upper bound (2 − 1/m) ! 4 Stochastic scheduling for Pj ∼ exp(λj), hard at all? 5 And the complexity of computing E[

j wjC WSEPT j

]?

Marc Uetz Smith’s Rule in Stochastic Scheduling

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Smith’s Rule = Photographer’s Rule

Group photos. . . Put short and important people first

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