Stochastic Scheduling with two Processors and Arbitrary Precedence Relations

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Stochastic Scheduling with two Processors and Arbitrary Precedence Relations

Carlos Camino

August 4, 2015

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Overview

I. Introduction

II. Theoretical Foundations III. Methods

IV. Results and Discussion V. Conclusion

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Overview

I. Introduction

II. Theoretical Foundations III. Methods

IV. Results and Discussion V. Conclusion

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What is scheduling?

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What is scheduling?

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What is scheduling?

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What is scheduling?

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Literature Review

Problem Reference

P 2 | prec | C max Solved by Coffman and Graham in 1972.

P 2 | intree , exp | E [C _max ] Solved by Chandy and Reynolds in 1975.

P 2 | prec , exp | E[C max ] Subject of this thesis.

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Overview

I. Introduction

II. Theoretical Foundations III. Methods

IV. Results and Discussion V. Conclusion

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Probability Theory

Why consider exponentially distributed random variables X _i ∼ Exp(λ _i )?

I Expected value:

E [X _i ] = 1 λ i

.

I Minimum:

min{X ₁ , . . . , X _n } ∼ Exp(λ ₁ + . . . + λ _n ).

I Memorylessness:

Pr[X i > x + y |X _i > y] = Pr[X i > x ] (x, y > 0).

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Snapshots

Let m be the number of available machines. A snapshot S = (T , P, A) consists of

I a finite set T ,

I a binary relation P ⊆ T × T and

I a subset A ⊆ T ,

such that (T , P ) is a DAG, |A| ≤ m and every t ∈ A is a source in (T , P).

T (S ), P (S ) and A(S) can be defined as T , P and A, respectively.

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Example

Let m = 2. The snapshot S with

I T (S) = {1, 2, 3, 4, 5},

I P (S ) = {(1, 3), (1, 4), (2, 4)} and

I A(S) = {1}

can be represented as

S

1 2

3 4 5

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Strategies and Direct Snapshots

I A strategy σ is a function that maps a snapshot S = (T , P, A) to a snapshot σ(S) = (T , P , A ⁰ ) with A ⊆ A ⁰ .

I The resulting snapshot S ⁰ after removing a task t ∈ A from a snapshot S = (T , P , A) is called direct subsnapshot of S.

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Example

Snapshots S, S ⁰ , S ⁰⁰ with

I σ(S) = S ⁰ for some strategy σ and

I S ⁰⁰ direct subsnapshot of S ⁰ :

S

1 2

3 4 5

S ⁰

1 2

3 4 5

S ⁰⁰

2 3 4 5

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A formula for the expected makespan E [T _S ^σ ]

Assuming independent finishing times X 1 , . . . , X n ∼ Exp(1) we can derive for a given strategy σ:

E [T _S ^σ ] = E

h T _σ(S) ^σ ⁱ

= E

h T _σ(S) ^σ ⁰ + T _σ(S) ^σ ⁰⁰ ⁱ

= E

h T _σ(S) ^σ ⁰ ⁱ + E

h T _σ(S) ^σ ⁰⁰ ⁱ

= E [min{X _t | t ∈ A(σ(S))}] + ^X

S

⁰

∈D(σ(S))

E [T _S ^σ

0

] · 1

|D(σ(S ))|

= 1

|D(σ(S ))|



 1 + ^X

S

⁰

∈D(σ(S))

E [T _S ^σ

⁰

]



 ,

where D(S) is the set of all direct subsnapshots of a snapshot S .

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For m = 2 machines we obtain:

E[T _S ^σ ] =



 



 



0 if D(σ(S )) = ∅

1 + E[T _S ^σ

0

] if D(σ(S )) = {S ⁰ }

1 2 + ¹ ₂ E [T _S ^σ

0

] + ¹ ₂ E [T _S ^σ

00

] if D(σ(S )) = {S ⁰ , S ⁰⁰ } for S ⁰ 6= S ⁰⁰ .

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Overview

I. Introduction

II. Theoretical Foundations III. Methods

IV. Results and Discussion V. Conclusion

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Schedule Visualization

PROSIT (Probabilistic Scheduling Interactive Tool) allows to

I draw a snapshot and

I visualize its schedule.

It can be downloaded from

www.carlos-camino.de/prosit

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Demo: Schedule example 1

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Computing Expectancies

layer L _i

intermediate layer L ⁰ _i

layer L i+1

E [T _S ^σ ] = E h

T _σ(S) ^σ i

= 1 + E [T _S ^σ

0

] E [T _S ^σ ] = E h

T _σ(S) ^σ i

= ¹ ₂ + ¹ ₂ E [T _S ^σ

0

] + ¹ ₂ E [T _S ^σ

00

] S

σ(S)

S ⁰ S ⁰⁰ S

σ(S)

S ⁰

E [T _S ^σ ]

E h

T _σ(S) ^σ i

E [T _S ^σ

0

] E [T _S ^σ

00

] E [T _S ^σ ]

E h

T _σ(S) ^σ i

E [T _S ^σ

0

]

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The Coffman-Graham Algorithm

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6 7

8 9

10 11 12 13 14

15 16

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The Coffman-Graham Algorithm

χ ₀ χ ₁

χ ₂ χ ₃

1 2

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6 7

8 9

10 11 12 13 14

15 16

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Demo: Schedule example 2

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Overview

I. Introduction

II. Theoretical Foundations III. Methods

IV. Results and Discussion V. Conclusion

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Case Studies on the HLF Policy The snapshot

n .. .

has optimal makespan n in the deterministic setting and optimal expected makespan n in the probabilistic setting.

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The snapshot

n . . .

has optimal makespan ⁿ ₂ in the deterministic setting and optimal expected makespan

n+1

2 in the probabilistic setting.

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Demos: Two parallel chains and a small counterexample for HLF

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Low Level Tasks with High Priority

n

m .. .

. . .

u v

For which values of m, n would an optimal strategy . . . . . . assign u ?

. . . assign v ?

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The following snapshot L _m,n

n

m .. .

. . .

with optimal expected makespan l _m,n satisfies l _m,0 = ^m ₂ + 1, l _0,n = n + 1 and

l _m,n = 1

2 l m−1,n + 1

2 l m,n−1 + 1 2 for m, n ≥ 1.

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The following snapshot H _m,n

n

m .. .

. . .

with optimal expected makespan h _m,n satisfies h _m,0 = ^m ₂ + 2 and

h _m,n = 1

2 h m,n−1 + 1

2 l _m,n + 1 2 for m, n ≥ 1.

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The following snapshots A ^u _m,n and A ^v _m,n

n

m .. .

. . .

u v

n

m .. .

. . .

u v

with optimal expected makespans a ^u _m,n and a ^v _m,n satisfy

a ^u _m,n = 1

2 l m,n+1 + 1

2 h m,n+1 + 3

4 and a ^v _m,n = h m,n+1 + 1 2 .

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For which m, n it holds a _m,n ^u < a _m,n ^v ? For which a ^u _m,n > a ^v _m,n ?

n

m

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0 5 10 15 20 25 30

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For which m, n it holds a _m,n ^u < a _m,n ^v ? For which a ^u _m,n > a ^v _m,n ?

n

m a ^u _m,n > a ^v _m,n

a ^u _m,n < a ^v _m,n

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0 5 10 15 20 25 30

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How big is the difference a ^u _m,n − a ^v _m,n ?

n

m

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0 5 10 15 20 25 30

0.5

0

−0.5

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How big is the difference a ^u _m,n − a ^v _m,n ?

n

m

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0 5 10 15 20 25 30

0.5

0

−0.5

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A formula for l _m,n The recurrence relation

l _m,n = 1

2 l m−1,n + 1

2 l m,n−1 + 1 2 can be visualized as

1 2 1 2

1 2 1 2 1 2

. . .

. . . . . .

l _0,0 l _0,1 l 0,2

l _1,0 l _1,1 l 1,2

l _2,0 l _2,1 l 2,2

l _3,0 l _3,1 l 3,2

l _4,0 l _4,1 l 4,2

l _5,0 l _5,1 l 5,2

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Using combinatorical arguments we can derive

l m,n =

m−1

X

i=0

n + i − 1 i

! 1 2

n+i

l m−i,0

+

n−1

X

j=0

m + j − 1 j

! 1 2

m+j

l 0,n−j

+

m−1

X

i=0 n−1

X

j =0

i + j i

! 1 2

i+j+1

with l m−i ,0 = ^m−i ₂ + 1 and l 0,n−j = n − j + 1 for all (m, n) ∈ N 0 × N 0 \ {(0, 0)}.

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Using generating functions we can derive the more complete formula

l _m,n =

m

X

i=0

n + i − 1 i

! 1 2

n+i

l m−i,0

+

n

X

j=0

m + j − 1 j

! 1 2

m+j

l 0,n−j

+

m−1

X

i=0 n−1

X

j=0

i + j i

! 1 2

i+j+1

− 1

2 m+n m + n m

! l _0,0

with l m−i ,0 = ^m−i ₂ + 1 and l 0,n−j = n − j + 1 for all (m, n) ∈ N 0 × N 0 .

Stochastic Scheduling with two Processors and Arbitrary Precedence Relations