Optimal File Splitting for Wireless Networks withConcurrent Access

Gerard Hoekstra1, Rob van der Mei2,Yoni Nazarathy3, Bert Zwart4

NET-COOP 2009,EURANDOM, Eindhoven

1CWI Amsterdam, Thales Nederland B.V.

2CWI Amsterdam, VU University Amsterdam

3CWI Amsterdam, EURANDOM, Mechanical Engineering TU/e

4CWI Amsterdam, VU University Amsterdam, EURANDOM, Georgia Institute of Technology

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Outline

1 Application and Queueing Model

2 Tail Asymptotics

3 Tail Optimal Rule

4 Tail Optimality vs. Mean Optimality

5 Conclusion

2

Outline

1 Application and Queueing Model

2 Tail Asymptotics

3 Tail Optimal Rule

4 Tail Optimality vs. Mean Optimality

5 Conclusion

3

Idea: Use Several Wireless Links for File Transfer

Example StoryPolice vehicle equipped with several camerasPhotos are taken and files are to be uploaded to headquartersVehicle has several slow wireless linksExample: GSM, SATCOM, MANETUpon file transfer, FTP opens several concurrent connectionsone on each linkGoal: minimize file transfer delay by utilizing all links

Similar Stories (perhaps)Jobs arriving to web-server farms are splitDownloading of file fragments in peer to peer networks

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Model a Link as a PS Queue

Processor Sharing Abstraction of A LinkOn the processor sharing of file transfers in wireless LANs(Hoekstra, van der Mei, 2009)

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How to Split the File?

Some Options1 Don’t split. Instead route (e.g. JSQ)2 Split (to 2 pieces):

– Transmit piece 1 from first byte "forward"– Transmit piece 2 from last byte "backwards"

3 Use fixed static splitting:decide on α = (α1, . . . , αN) and always use it

4 . . .

In this work, we follow option 3

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Queueing Model

N processor sharing queues, rates c1, . . . , cN

N background Poisson file streams, λ1, . . . , λN

1 foreground Poisson stream, λ0

I.I.D. file sizes per stream, B0,B1, . . . ,BN , means β0, β1, . . . , βN

Foreground files split into fragments:Splitting rule: α = (α1, . . . , αN), fragment i is of size αiB0

ρi := λiβi . Assume αiρ0 + ρi < ci

Assume steady state

Of interest:- Sojourn time of foreground files (maximum of fragments)- Choosing a "good" splitting rule α

Related Work: "JSQ to PS Farm", Gupta, Harchol Balter, Sigman, Whitt (2007), "Joining Games", Altman, Ayesta, Prabhu (2008),

Chen, Marden, Wierman (2009), "Fork-Join FCFS Tail Asymptotics", Lelarge (2008, 2009)

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M/G/1-PS Refresher

Denote:V ≡ sojourn time of a job arriving to steady stateB̃ ≡ service time random variableR(x) :=

∫ x0

11+Q(t)dt

Property 1

E [V |B̃] = B̃c−ρ

Property 2

P(V > x) = P(B̃ > R(x))

Property 3, Reduced Load Approximation (RLA) (Zwart, Boxma 2000)

Assume P(B̃ > x) = L(x)x−ν , ν > 1, L(·) slowly varying. Then,

P(V > x) ∼ P(B̃ > (c − ρ)x)

Property 4 (Multi-Class RLA)Class 1 regularly varying as above. Class 2 has 1 + ε moments.

P(V1 > x) ∼ P(B̃1 > (1− ρ)x).

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Outline

1 Application and Queueing Model

2 Tail Asymptotics

3 Tail Optimal Rule

4 Tail Optimality vs. Mean Optimality

5 Conclusion

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Main Result

Reminder of our Story

1 Foreground file B0 arrives to steady state

2 File splits to N fragments αiB0, with sojourn Vi

3 Sojourn time of whole file: Mα := max{V1, . . . ,VN}

Theorem (Reduced Load Equivalence)Assume,

(i) E [B1+εi ] <∞, i = 1, . . . ,N

(ii) P(B0 > x) = L(x)x−ν , ν > 1, L(·) slowly varyingThen,

P(Mα > x) ∼ P(B0 > γαx),

with,γα = min

i=1,...,Nγi , with γi =

ci − ρi − αiρ0

αi

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Proof, cont.

Proof:(

P(max(V1,V2) > x) ∼ P(B0 > min(γ1, γ2)x))

Assume min(γ1, γ2) = γ1.

P(Mα>x)P(B0>γαx)

=P(B0>

R1(x)

α1)

P(B0>γ1x)+

P(B0>γ2x)P(B0>γ1x)

(P(B0>

R2(x)

α2)

P(B0>γ2x)−

P(B0>max(R1(x)

α1,

R2(x)

α2))

P(B0>max(γ1,γ2)x)

).

P(B0>γ2x)

P(B0>γ1x)=

L(γ2x)

L(γ1x)

(γ2γ1

)−ν→

(γ2γ1

)−ν,

Each queue is a 2-class queue: P(B0 >Ri (x)αi

) ∼ P(B0 > γi x) (Zwart 1999)

Remaining to show: P(B0 > max( R1(x)α1

,R2(x)α2

)) ∼ P(B0 > max(γ1, γ2)x)

Use Guillemin, Robert, Zwart, 2004. Thm1: Let limx→∞ S(x)/x = γ a.s.and take B̃ independent regularly varying. Then P(B̃ > S(x)) ∼ P(B̃ > γx),

if we can find a c, such that P(S(x) ≤ cx

)= o

(P(B̃ > x)

)In our case B̃ = B0 and S(x) = max

(R1(x)α1

,R2(x)α2

)P(max

(R1(x)α1

,R2(x)α2

)≤ cx) ≤ P( R1(x)

α1≤ cx) = o

(P(B0 > x)

)Last step is as in analysis of Guillemin, Robert, Zwart for M/G/1

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Outline

1 Application and Queueing Model

2 Tail Asymptotics

3 Tail Optimal Rule

4 Tail Optimality vs. Mean Optimality

5 Conclusion

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The Rule To Use

α∗i :=ci − ρi∑N

j=1(cj − ρj)

InterpretationsMinimizes asymptotic tail (as we show in next slide)"Split proportional to free capacity"Equating conditional (on B0) mean sojourn times in PS queues.Set,

α∗1B0

c1 − ρ1 − α∗1ρ0=

(1− α∗1)B0

c2 − ρ2 − (1− α∗1)ρ0

and solve for α∗1

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Minimizing the Tail

Since we have P(Mα > x) ∼ P(B0 > γαx), let’s optimize γα:maxα

mini=1,...,N

(ci − ρi

αi

)− ρ0

s.t.N∑

i=0

αi = 1, α ≥ 0.

The unique solution is given by α∗ =( ci−ρi∑N

j=1(cj−ρj )

)Example for N = 2:

0 Α* 1Α

c1 - Ρ1

1 - Α-Ρ0

c2 - Ρ2

Α-Ρ0

ΓΑ

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Simulated Examples

Some ExamplesN = 2, c1 = c2 = 1, β0 = β1 = β2 = 1Parameterize by ρ, κ, η:

ρ =λ0 + λ1 + λ2

2, κ =

1− λ1

1− λ2, η =

λ0

λ1 + λ2.

Examples:System ρ κ η Distribution 0 Distribution 1 Distribution 2 (λ0, λ1, λ2) α∗

1 0.5 1.5 0.5 Pareto 3 Pareto 3 Pareto 3 ( 13 ,

15 ,

715 ) 0.6

2 0.5 1.5 0.5 Pareto 3 Deterministic Deterministic as System 1 -3 0.5 1.5 0.5 Pareto 3 Exponential Exponential as System 1 -4 0.5 1.5 0.5 Pareto 3 Exponential Deterministic as System 1 -5 0.5 1.5 0.5 Deterministic Deterministic Deterministic as System 1 -6 0.5 1.5 0.5 Erlang 2 Erlang 2 Erlang 2 as System 1 -7 0.5 1.5 0.5 Exponential Pareto 3 Erlang 2 as System 1 -8 0.5 2.0 0.5 Pareto 3 Pareto 3 Pareto 3 ( 1

3 ,19 ,

59 ) 2

39 0.5 1.0 0.5 Exponential Exponential Exponential ( 1

3 ,13 ,

13 ) 0.5

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Finite Tail Optimality vs. Asymptotic Optimality

Simulation runs of System 4.Search for α∗(x) = argminα P(Mα > x),x = 1,2,3,5,8,11,17,25,35,48,64,85,115,160,210,270,350,500Compare to α∗ = limx→∞ α

∗(x) = 0.6Plot also E [Mα] (to see argminαE [Mα])

0.2 0.4 0.6 0.8 1.0Α

5

10

15

- log PHMΑ>xL

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More Graphs...

Error ≡ P(Mα∗>x)−P(Mα∗(x)>x)

P(Mα∗(x)>x)

Heavy Tailed Foreground Light Tailed Foreground

0 50 100 150 200x0.00

0.05

0.10

0.15

0.20Error

System 1

System 2

System 3

System 4

System 8

0 10 20 30 40 50 60 70x0

2

4

6

8

10Error

System 6

System 7

System 5

CommentsFor heavy tailed foreground files: Approximation is good formoderate xFor light tailed foreground files: Our α∗ is not tail optimal

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Outline

1 Application and Queueing Model

2 Tail Asymptotics

3 Tail Optimal Rule

4 Tail Optimality vs. Mean Optimality

5 Conclusion

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Optimization of E [Mα]

Plots of E [Mα], for Systems 1-9.

0.4 0.5 0.6 0.7 0.8Α1.0

1.2

1.4

1.6

1.8

2.0Mean

Systems 1-7

System 8

System 9

Note: The dots are plots of E [MJSQ].

Some Observations:minα E [Mα] ≈ E [Mα∗ ]

Near insensitivity, similar to the JSQ results (Gupta et. al.)

E [Mα∗ ] < E [MJSQ] (not surprising)

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Discussion on minα E [Mα] ≈ E [Mα∗], 1’st Justification

Reason 1

Denote R(x) := mini=1,...,NRi (x)αi

Observe R(x)x → γα and R−1(x)

x → 1γα

, a.s.We have, P(Mα > x) = P(B > R(x))

Denote by M(b) the sojourn time of a foreground file of size bWe have that M(b) = R−1(b).Define µ(b) := E [M(b)]

We have µ(b)b → 1

γαas b →∞

Thus, for large b: µ(b) ≈ bγα

Thus selecting α such that γα is maximal minimizes µ(b) when bis large. and approximately minimizes the unconditional sojourntime E [M] = EB[µ(B)]

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Discussion on minα E [Mα] ≈ E [Mα∗], 2’nd Justification

General SettingSome "stochastic model" parameterized by scalar α

α yields 1− Fα(x). µα =∫∞

0 Fα(u)du

LemmaAssume that Fα(x) is unimodal in α and that Fα(x) and µα are differentiablein α, then there exists an x > 0 such that

argminαµα = argminαFα(x).

In Our ModelWe saw empirically that α∗(x) does not vary much with x

Combining with the above lemma, we get minα E [Mα] ≈ E [Mα∗ ]

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Outline

1 Application and Queueing Model

2 Tail Asymptotics

3 Tail Optimal Rule

4 Tail Optimality vs. Mean Optimality

5 Conclusion

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Some Unanswered Questions

Tail asymptotics for light tails of foreground (harder)Mathematical basis for minα E [Mα] ≈ E [Mα∗ ]

Mathematical basis of "Near-Insensitivity"Tail Asymptotics for JSQ

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Enjoy The Visit in Eindhoven...

Unfiled Notes Page 1

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