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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
1
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
4
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)
5
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).
8
Outline
1 Application and Queueing Model
2 Tail Asymptotics
3 Tail Optimal Rule
4 Tail Optimality vs. Mean Optimality
5 Conclusion
9
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
12
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
13
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
14
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
15
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
16
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
17
Outline
1 Application and Queueing Model
2 Tail Asymptotics
3 Tail Optimal Rule
4 Tail Optimality vs. Mean Optimality
5 Conclusion
18
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)
19
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)]
20
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∗ ]
21
Outline
1 Application and Queueing Model
2 Tail Asymptotics
3 Tail Optimal Rule
4 Tail Optimality vs. Mean Optimality
5 Conclusion
22
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
23
Enjoy The Visit in Eindhoven...
Unfiled Notes Page 1
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