Capacity and Scheduling in Heterogeneous Networks€¦ · Capacity and Scheduling in Heterogeneous...
Transcript of Capacity and Scheduling in Heterogeneous Networks€¦ · Capacity and Scheduling in Heterogeneous...
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Capacity and Scheduling in HeterogeneousNetworks
Phil Whiting
Macquarie UniversityNorth Ryde, NSW 2109
Joint Work with Sem Borst and Stephen Hanly
Monday, June 16th 2014
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Talk Summary
Mobile Radio and the Spectrum Crunch
Getting more Capacity and How much do we Have?
Utility Schedulers
Closing Remarks
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Mobiles Past
An Entrepeneur Securing a Deal using an Early Mobile Phone
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Mobiles Present
Progress toward Data, Apps - Location Based Information
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Mobiles Future
Future User having Trouble with a Hotel Booking ....
Whiting Heterogeneous Networks
![Page 6: Capacity and Scheduling in Heterogeneous Networks€¦ · Capacity and Scheduling in Heterogeneous Networks Phil Whiting Macquarie University North Ryde, NSW 2109 Joint Work with](https://reader034.fdocuments.net/reader034/viewer/2022042515/5f787ff0154a92446b264fa2/html5/thumbnails/6.jpg)
Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Talk Summary
A SnapShot Resource Allocation Problem
A Continuous LP
Capacity and Scheduling
α-fair Utility Scheduling
Stability Results
Conclusions
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Gaining Capacity using HetNets
Small cells (pico/femto) to increase frequency reuse
Place in areas of poor coverageAreas of traffic concentration - ”Hot Spots”
Adapt Network to Match Traffic Load
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
A Simplified HetNet Model
L = 4 picos - all users in range of macro and at most one picoNo Interference between Pico Cells
1
3
4
20
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Flexible Allocation
Time Share Spectrum
Macro Cell/Pico CellsUse Almost Blanking SubFrames (fine granularity)
Cell Range Expansion for Picos
Expand to cover more mobilesContract and send at Higher Rate
For following, see [1]
[1] S. Borst, S. Hanly, P. Whiting “Optimal resource allocation in HetNets”, ICC, Budapest, Hungary, 2013.
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
ABS Frames and Time Sharing
1
3
4
20
1 time unit = macro time allocation + pico time allocation
Macro time 1− f Pico time f
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Mobiles can TimeShare Macro/Pico (Split)
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Empty the Network !!
How do we Empty the Network in Minimum Time?
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Solve the Following LP!
The problem to be solved is the following linear program:
min f +L∑
l=0
Nl∑n=1
yl,nSl,n
sub
Nl∑n=1
xl,nRl,n≤ f ∀l
xl,n + yl,n ≥ Dl,n ∀l , ∀n = 1, 2, . . .Nl
f ≥ 0, xl,n ≥ 0, yl,n ≥ 0 ∀l , ∀n = 1, 2, . . .Nl
where f is the time allocated to the picocells.
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Solution Structure
ρl,n :=Rl,n
Sl,n
Order User - Decreasing in ρLarge ρ→ pico, Small ρ→ macro, = ρ→ Split
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Let’s make the Problem Continuous ...
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Continuous LP parameters
λSη (dξ) = λ (dξ), η (dξ) probability density
R`(ξ),S`(ξ) Phy. Rates Pico/Macro - Pico `
x`(ξ), y`(ξ) bit assignments at location ξ
D download file size (could be random, here fixed)
Largest λS for which network is stable?
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Continuous LP
min τ = f +L∑`=1
∫y`(ξ)
S`(ξ)λ (dξ) (1)
sub
∫x`(ξ)
R`(ξ)λ (dξ) ≤ f ∀` (2)
where,y`(ξ) = D − x`(ξ)
is the file constraint
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Optimal solutionFor some ρ1, · · · , ρL > 0,
x∗` (ξ) =
{D R`(ξ)
S`(ξ) ≥ ρ`0 R`(ξ)
S`(ξ) < ρ`(3)
f ∗ = max`
∫x∗` (ξ)
R`(ξ)dξ
If τ∗ < 1, ∃ a stable schedule · · ·
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Bundling [nT , (n + 1)T ), n ∈ N0
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Bundling Algorithm
1 B := 1, Wait until nB = 1
2 Serve bundle B, starting nBT
3 Let fBT completion slot for bundle B
3 B := B + 1, nB := max {fB ,B}4 Go to 2
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Bundling defines a D/G/1 queue, bundle delay =: Wn
τ < 1, assumptions → E [Wn] Uniformly BoundedWn satisfies Spitzer’s identity,
E [Wn] = E[
maxk≤n
S+k
]=
n∑k=1
1
kE[S+k
]Sk
.= Xk − kT , Xk duration first k bundles
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
SLLN and Stability
Any bounded, measurable v : S → <+,
1
T
NT∑n=1
v(ξn(ω))→∫S
v(ξ)λ (dξ) (4)
a.s. and in L1.
υT` (ω).
=1
T
NT∑n=1
x`(ξn(ω))
R`(ξn(ω))
is UI, ` = 1, · · · , L.
→ fT (ω) = max`υT` (ω)
is UI so that E [fT ]→ f ∗
E [fT ] +L∑`=0
E
[1
T
NT∑n=1
y`(ξn)
S`(ξn)
]→ τ∗ < 1
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
A schedule π is clearing if departure time Dπn (ω) <∞, a.s., ∀n
Prop (Hanly, W.)
Let τ∗ be optimal solution to the LP. If τ∗ < 1, ∃ a clearing schedule πwith ergodic properties.Also define Sπn (ω) := sojourn time nth mobile, then π satisifies,
E [Sπn (ω)] < S <∞ (5)
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Converse Holds as Well!
Continuous LP τ ∗ > 1 → No Stable Schedule
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Let π be any clearing schedule. Define V πT (ω) to be network time needed
to clear mobiles arriving in [0,T ]
Prop (Hanly, W.)
Let τ∗ be the solution to the continuous LP. Suppose that τ∗ > 1 thenthere is a fixed constant η > 0, such that for all π
lim infT
V πT (ω)
T= 1 + η
almost surely.
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Proof Sketch I
Arrivals in [0,T ] supposed to arrive at time 0. Apply discrete LP with
outcome V(LP)T (ω)
Prop
∀ω and for all clearing schedule π,
lim infT
V(LP)T (ω)
T≤ lim inf
T
V πT (ω)
T(6)
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Proof Sketch II: Discretise Arrivals using Rate Ratios ρ`
Given ε > 0, choose intervals,
N(`,n)T arrivals in interval n for pico ` mean m`(n)
For all 0 < δ < 1/2 there exists In,` > 0
P{
1
TN
(`,n)T 6∈ [(1− δ)m`(n), (1 + δ)m`(n)]
}≤ e−TIn,` (7)
Borel-Cantelli implies ∃TE all arrivals close to expectation, ∀T > TE
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Proof Sketch III
Finite set A of rate ratio policies,
lim infT
V(LP)T
T≥ lim inf
Tinfa∈A
V aT
T− LεD
R(8)
= infa∈A
lim infT
V aT
T− LεD
R(9)
≥ (1 + η)− LεD
R(10)
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Utility Scheduling and Stability
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Modelling Assumptions
Discrete set - location k in cell ` - (k , `), k = 1, · · · ,Kl
Unit exponential files
Independent Poisson streams, λ(`)k > 0
Physical Rates R(`)k pico, S
(`)k macro
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Modelling Assumptions
Discrete set - location k in cell ` - (k , `), k = 1, · · · ,Kl
Unit exponential files
Independent Poisson streams, λ(`)k > 0
Physical Rates R(`)k pico, S
(`)k macro
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Modelling Assumptions
Discrete set - location k in cell ` - (k , `), k = 1, · · · ,Kl
Unit exponential files
Independent Poisson streams, λ(`)k > 0
Physical Rates R(`)k pico, S
(`)k macro
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Modelling Assumptions
Discrete set - location k in cell ` - (k , `), k = 1, · · · ,Kl
Unit exponential files
Independent Poisson streams, λ(`)k > 0
Physical Rates R(`)k pico, S
(`)k macro
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Time Sharing
Time Sharing Vector (a,b)
Feasibility constraints,
K (`)∑k=1
a(`)k +
L∑m=0
K (m)∑k=1
b(m)k ≤ 1, ∀`. (11)
with throughput,
T(`)k = a
(`)k R
(`)k + b
(`)k S
(`)k (12)
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Time Sharing
Time Sharing Vector (a,b)
Feasibility constraints,
K (`)∑k=1
a(`)k +
L∑m=0
K (m)∑k=1
b(m)k ≤ 1, ∀`. (11)
with throughput,
T(`)k = a
(`)k R
(`)k + b
(`)k S
(`)k (12)
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Processor Sharing Model for a HetNet
M
1
2
3
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Stability Region Λ0
T .= {(a,b) : (a,b) , feasible} ,
Λ.
= ∪{T (a,b) : (a,b) ∈ T }
Then,Λ0
.= {λ : ∃ε > 0,λ+ ε ∈ Λ}
Stable scheduler exisits iff λ ∈ Λ0
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Continuous Time Markov Processes
N(t).
=(N(0)(t), · · · ,N(L)(t)
)∈∏
`NK (`)
0 =: N
Arrivals, rate λ(`)k ,(
N(0), · · · ,N(L))→(N(0), · · · ,N(L)
)+(
0, · · · , e(`)k , · · · , 0
)Departures policy θ, in state n ∈ N , rate T
(aθ(n),bθ(n)
)(N(0), · · · ,N(L)
)→(N(0), · · · ,N(L)
)−(
0, · · · , e(`)k , · · · , 0
)
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
A Static Utility Optimization Problem
U(a,b).
=L∑`=0
∑k
N(`)k Uα
(T
(`)k (a,b)
N(`)k
)(13)
α-fair utilitiesUα(·) = (1− α)−1x1−α, α ∈ (0,∞)
For solution to above,, see [2]
[2] S. Borst, S. Hanly, P. Whiting “Throughput Utility Optimization in HetNets”, VTC, Dresden, Germany, 2013.
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Prop (Hanly, W.)
Suppose λ ∈ Λ0. Then∀α > 0 the Markov Process defined by α-fairscheduling is positive recurrent so that
P {N(t) = N} → πα(N) as t →∞ (14)
Moreover limiting α moments exist; that is, for all (k , `),
Eπα[(
N(`)k
)α]<∞ (15)
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Proof Sketch
As demonstrated in [3]
L(N).
=L∑`=0
K (`)∑k=1
{λ
(`)k
}−α {N(`)k
}1+α
(1 + α)(16)
is a Lypuanov function
Let N(n) jump chain sequence of the uniformized Markov process, then,
L(N(n))
has supermart. property outside a compact set.
[3] T. Bonald and L. Massoulie“Impact of Fairness on Internet Performance”, ACM SIGMETRICS Performance Evaluation Review,
Vol. 29, No. 1, pp 82–91, 2001.
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Example: Proportional Fair Scheduler
UN.
=L∑`=0
K (`)∑k=1
N(`)k log
T(`)k
N(`)k
(17)
Quadratic Lypuanov function L,
L(N).
=1
2
L∑`=0
K (`)∑k=1
{N
(`)k
}2
λ(`)k
(18)
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Numerical Results
Whiting Heterogeneous Networks
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Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Conclusions
Traffic Capacity Determined by LP
Fixed Schedule Stable
Estimate ηEstimate R`(ξ), S`(ξ)Infer Capacity
Results extend to more general networks
α-fair Utility Scheduler maximally stable
Equilibrium Moments shown to exist depending on α
Results extend to periodic schedulers
Whiting Heterogeneous Networks
![Page 45: Capacity and Scheduling in Heterogeneous Networks€¦ · Capacity and Scheduling in Heterogeneous Networks Phil Whiting Macquarie University North Ryde, NSW 2109 Joint Work with](https://reader034.fdocuments.net/reader034/viewer/2022042515/5f787ff0154a92446b264fa2/html5/thumbnails/45.jpg)
Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Conclusions
Traffic Capacity Determined by LP
Fixed Schedule Stable
Estimate ηEstimate R`(ξ), S`(ξ)Infer Capacity
Results extend to more general networks
α-fair Utility Scheduler maximally stable
Equilibrium Moments shown to exist depending on α
Results extend to periodic schedulers
Whiting Heterogeneous Networks
![Page 46: Capacity and Scheduling in Heterogeneous Networks€¦ · Capacity and Scheduling in Heterogeneous Networks Phil Whiting Macquarie University North Ryde, NSW 2109 Joint Work with](https://reader034.fdocuments.net/reader034/viewer/2022042515/5f787ff0154a92446b264fa2/html5/thumbnails/46.jpg)
Why do we need HetNets?Preliminaries
A Continuous LPConverse
Utility Scheduling - Preliminariesα-fair Utility Scheduling
Thanks!
Whiting Heterogeneous Networks