file fragmentation over an unreliable network - bachelor thesis · 2009-10-06 · bachelor thesis...

File Fragmentationover an Unreliable

Channel

Martin Andreassonmandreas@kth.se

Introduction

Model and method

Conclusion

Acknowledgments

File Fragmentation over an UnreliableNetwork

Bachelor Thesis

Royal Institute of Technology

September 7, 2009

Channel

Introduction

Model and method

Conclusion

Acknowledgments

Imagine yourself playing an action game

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Introduction

Model and method

Conclusion

Acknowledgments

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

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Introduction

Model and method

Conclusion

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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Introduction

Model and method

Conclusion

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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Introduction

Model and method

Conclusion

Acknowledgments

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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Model and method

Conclusion

Acknowledgments

Basic model

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Introduction

Model and method

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Basic model

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Introduction

Model and method

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Large files are likely to be corrupted

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Introduction

Model and method

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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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Introduction

Model and method

Conclusion

Acknowledgments

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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Introduction

Model and method

Conclusion

Acknowledgments

Fragmentation prevents corruption

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Introduction

Model and method

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Packets however, carry an overhead

This makes too small packet sizes inefficient

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Model and method

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Thesis

What is the optimal fragment size?

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Model and method

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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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Introduction

Model and method

Conclusion

Acknowledgments

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)

Channel

Introduction

Model and method

Conclusion

Acknowledgments

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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Introduction

Model and method

Conclusion

Acknowledgments

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

Channel

Introduction

Model and method

Conclusion

Acknowledgments

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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Introduction

Model and method

Conclusion

Acknowledgments

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)

Channel

Introduction

Model and method

Conclusion

Acknowledgments

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)

Channel

Introduction

Model and method

Conclusion

Acknowledgments

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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Introduction

Model and method

Conclusion

Acknowledgments

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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Introduction

Model and method

Conclusion

Acknowledgments

Minimizing J

J = MLM + φ

LM + φ

) = LLM + φ

LM F̄

(LM + φ

) := L·g(

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Introduction

Model and method

Conclusion

Acknowledgments

Minimizing J

J = MLM + φ

LM + φ

) = LLM + φ

LM F̄

(LM + φ

) := L·g(

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Introduction

Model and method

Conclusion

Acknowledgments

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Introduction

Model and method

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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

Conclusion

Acknowledgments

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

Channel

Introduction

Model and method

Conclusion

Acknowledgments

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

Channel

Introduction

Model and method

Conclusion

Acknowledgments

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

Channel

Introduction

Model and method

Conclusion

Acknowledgments

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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Conclusion

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Acknowledgments

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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