File Fragmentation over an Unreliable Network - Bachelor Thesis · 2009-10-06 · Bachelor Thesis...

File Fragmentationover an Unreliable

Channel

Martin [email protected]

Introduction

Model and method

Conclusion

Acknowledgments

File Fragmentation over an UnreliableNetwork

Bachelor Thesis

Royal Institute of Technology

September 7, 2009

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Imagine yourself playing an action game

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There is always a risk of getting caught, which increasesby time

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AcknowledgmentsIf you get caught, you have to start over where you lastsaved

But saving very frequently would cost too much time

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AcknowledgmentsIf you get caught, you have to start over where you lastsaved

But saving very frequently would cost too much time

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Basic model

A file is sent over an unreliable network

Large files are more likely to be corrupted than small

If a file is corrupted, the whole file needs to be resent

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Basic model

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Basic model

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Large files are likely to be corrupted

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Basic model

To prevent file losses and heavy-tailes, large files can befragmented

Smaller fragments are less likely to be corrupted

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Basic model

To prevent file losses and heavy-tailes, large files can befragmented

Smaller fragments are less likely to be corrupted

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Fragmentation prevents corruption

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Packets however, carry an overhead

This makes too small packet sizes inefficient

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Thesis

What is the optimal fragment size?

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Mathematical model

xi + φ denotes the packet size (xi data size, φ overheadsize)

Transmission will succeed if Ai ≥ xi + φ (Ai randomvariable with distribution f)

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Mathematical model

xi + φ denotes the packet size (xi data size, φ overheadsize)

Transmission will succeed if Ai ≥ xi + φ (Ai randomvariable with distribution f)

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Model and method

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Expected transmission time

J =M−1∑i=0

xi + φ

F̄ (xi + φ)=

M−1∑i=0

h(xi ) F̄ (x) = P(Ai > x) (1)

Want to minimize J over M and xi

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Expected transmission time

J =M−1∑i=0

xi + φ

F̄ (xi + φ)=

M−1∑i=0

h(xi ) F̄ (x) = P(Ai > x) (1)

Want to minimize J over M and xi

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Minimizing J

The KKT condition for (1) is

h′(xi ) =1

F̄ (xi + φ)+ (xi + φ)

f (xi + φ)(F̄ (xi + φ)

)2 = λ

M−1∑i=0

xi = L (2)

If the failure rate (failure risk) f (x)/F̄ (x) is increasing:

The optimization over xi is convex

The solution is x∗i = x∗ for all i (packet sizes are equal)

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Minimizing J

h′(xi ) =1

F̄ (xi + φ)+ (xi + φ)

f (xi + φ)(F̄ (xi + φ)

)2 = λ

M−1∑i=0

xi = L (2)

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Minimizing J

h′(xi ) =1

F̄ (xi + φ)+ (xi + φ)

f (xi + φ)(F̄ (xi + φ)

)2 = λ

M−1∑i=0

xi = L (2)

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Minimizing J

h′(xi ) =1

F̄ (xi + φ)+ (xi + φ)

f (xi + φ)(F̄ (xi + φ)

)2 = λ

M−1∑i=0

xi = L (2)

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Minimizing J

Seek minimizer M∗ by inserting x∗ = L/M into (1):

J = MLM + φ

LM + φ

) = LLM + φ

LM F̄

(LM + φ

) := L·g(

Under the assumption that the failure rate f (x)/F̄ (x) isincreasing, g(x) is unimodal

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Minimizing J

J = MLM + φ

LM + φ

) = LLM + φ

LM F̄

(LM + φ

) := L·g(

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Minimizing J

J = MLM + φ

LM + φ

) = LLM + φ

LM F̄

(LM + φ

) := L·g(

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Bounding x∗

Definition

a := argminx

(g(x))

Theorem

If L > a:

2≤ a

1 + a/L≤ x∗(L) ≤ min

1− a/L

Corollary

limL→∞

x∗(L) = a

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Bounding x∗

Definition

a := argminx

(g(x))

Theorem

If L > a:

2≤ a

1 + a/L≤ x∗(L) ≤ min

1− a/L

Corollary

limL→∞

x∗(L) = a

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Introduction

Model and method

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Bounding x∗

Definition

a := argminx

(g(x))

Theorem

If L > a:

2≤ a

1 + a/L≤ x∗(L) ≤ min

1− a/L

Corollary

limL→∞

x∗(L) = a

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Introduction

Model and method

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Bounding x∗

Definition

a := argminx

(g(x))

Theorem

If L > a:

2≤ a

1 + a/L≤ x∗(L) ≤ min

1− a/L

Corollary

limL→∞

x∗(L) = a

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Conclusion

Under the assumption that the failure rate f (x)/F̄ (x) isincreasing,

Constant fragmentation is optimal

The fragment sizes are unique

If the file size and distribution f are known, the file canbe pre-fragmented

For large filesizes L, the optimal fragment size isindependent of L

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Steven H. Low, Jayakrishnan Nair (Caltech) andLachlan Andrew (Swinburne)

Henrik Sandberg and Karl Henrik Johansson (KTH)

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File Fragmentation over an Unreliable Network - Bachelor Thesis · 2009-10-06 · Bachelor Thesis...

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Transcript of File Fragmentation over an Unreliable Network - Bachelor Thesis · 2009-10-06 · Bachelor Thesis...