Stability and rate of convergence ofresource allocation within a cellular

network

Aaron Pim

12 October 2017

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Presentation outline

• Definitions• RB allocation• Equations• Existence of a solution• Formulation 1: Newton’s method• Formulation 2: continuous approximation• Formulation 3: least squares error• Comparisons• Self-organising networks

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Example network

Cell 4

Cell 1

Cell 3Cell 2

7

8

6

12

53

4

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Definitions 1/3

NotationNumber of UEs = nNumber of cells = m.

I define the following sets• C denotes the set of all cells.• U denotes the set of all UEs.• Q := {1, . . . ,n}• U|c ⊂ U is the set of all UE’s connected to cell c.

DefinitionThe proportion of the bandwidth allocated to uj is xj .

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Definitions 2/3

NotationLet uj ∈ U be connected to a cell then I will denote that cellby c ∈ C, otherwise I will denote a general cell by c ∈ C

• Sj denotes the SINR from cell c to UE uj .• Ij is the unwanted signals in the channel between c

and uj , in Watts.• σ2 is the internal noise that is added to the system, in

Watts.• Ψj is the signal strength of uj and c connection, in

Watts.

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Definitions 3/3

NotationP ∈ Rn×m is the pathloss matrix between the UEs and cells.A ∈ {0,1}n×m is the cell-to-UE mapping matrix.

P =

0 p1,2 p1,3 p1,40 p2,2 p2,3 p2,4

p3,1 0 p3,3 p3,4p4,1 0 p4,3 p4,4p5,1 p5,2 0 p5,4p6,1 p6,2 p6,3 0p7,1 p7,2 p7,3 0p8,1 p8,2 p8,3 0

A =

1 1 0 0 0 0 0 00 0 1 1 0 0 0 00 0 0 0 1 0 0 00 0 0 0 0 1 1 1

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Abstract network diagram.

Cell 1

UE 1 UE 2

Cell 4

UE 7 UE 6UE 8

Cell 2

UE 3 UE 4

Cell 3

UE 5

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RB allocation in one timeslot

Cell 11

UE 1

2 3 4

UE 2

5 6 7

Cell 21 2

UE 3

3

UE 3

4 5

UE 46

UE 4

7

UE 4

Cell 31

UE 5

2

UE 5

3

UE 5

4

UE 5

5 6 7

Cell 41

UE 6

2 3

UE 7

4

UE 7

5 6

UE 8

7

UE 8

Frequency

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Probability of resource block collisionNotationP(uj ↔ c) is the probability that a resource block isassigned to uj by c and is also assigned to some other UEby cell c.

I assume that the resource blocks are allocated uniformlyat random across the entire bandwidth. Hence:

P(uj ↔ c) =Proportion of bandwidth assigned to uj ×Proportion of bandwidth assigned by c

P(uj ↔ ci) = xj(A x)i

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Interference

The unwanted signal generated by a cell ci which interfereswith the channel between uj and c is given by:

Intf(uj , ci) = Channel gain(uj , ci)× Prob. of collision

Then the total inference from other cells received by UE ujis the sum of the interferences.

Ij := xj

m∑i=1

Pi,j(A x)i

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SINR

NotationThe symbols � and � denote the Hadamard product anddivision respectively.

The signal to noise + interference vector is given by:

S(x) = Ψ� (x�PA x +σ2)

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SINR to spectral efficiency

The spectral efficiency is the channel capacity per unitbandwidth.

UnitsSpectral efficiency has units b/s/Hz = b, which is acounting variable and hence dimensionless.

I denote function f : Rn → R to be the single spectralefficiency function. I then vectorise this into the functionF : Rn → Rn which I define to be:

F (S) = [f (S1), . . . , f (Sn)]T

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SINR to spectral efficiency

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Optimisation

I define the vector of data demands to be d. I seek thesolution to the fixed point problem:

x = d�F (Ψ� (x�PA x +σ2)) x ∈ (0,1]n

This can be written as a iterative method:

xt+1 = d�F (Ψ� (xt �PA xt +σ2))

Using this formula in conjunction with the trust regionmethod, I will refer to the autonomous simulation.

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Existence of a solution

Parameters• f - sigmoid• σ2 = 0• k = PA x�Ψ

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Existence of a solution

Consider the Shannon spectral efficiency, using bounds onlogarithms I define a sufficient condition for non-existence.

If ∃i ∈ Q such that:

di ln(2)(PA x)i > Ψi

Then there does not exist a solution.

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Stability of a solution

Assuming a solution exists then I was able to prove that:• Such a point is asymptotically stable.• This point is unique.

This is for a solution that exists within the regionx ∈ (0,∞)n, meaning this physically has little meaninghowever if the model can be adapted to another systemthen it would be useful.

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Other Bounds

There is a lower bound on the rate of convergence:

di ln(2)σ2 − 1 6 xi .

Through this a lower bound for existence of a solutionwithin the trust region [0,1]n can be derived.

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Formulation 1: Newton’s method

We seek the root of the following equation using Wolfeconditions and a trust region:

G(x) = x�F (Ψ� (x�PA x +σ2))− d x ∈ (0,1]n

The Jacobian is given by:

JG(x)i,j :=

f (Si(x))− (PA x)i

(f ′(Si(x))

S2i xiΨi

)if i = j

−PAi,j f ′(Sj(x))S2

j x2j

Ψjif i 6= j

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Formulation 1: Newton’s method

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Formulation 1: Newton’s method

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Formulation 1: Newton’s method

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Formulation 2: Continuous approximate

Using an approximation, I am able express the problem interms of an autonomous ODE.

x = d�F (Ψ� (x�PA x +σ2))− x

This has no exact solution and hence I use Runge-Kuttamethods to solve this system.

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Formulation 2: Continuous approximate

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Formulation 3: Least squares error

I wish to minimise the proportion of bandwidth allocated tothe UEs such that the data demands are met.

x∗ = argmin(||x ||2) such that G(x∗)i > 0,x∗ ∈ [0,1]n, ∀i ∈ Q

If the Shannon spectral efficiency is being considered thenG(x) is a concave function and hence feasible region isconvex. Therefore this problem may be solved as a convexoptimisation problem.

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Formulation 3: Least squares errorProblemThe least squares model makes the assumption that all ofthe data demands can be met, that is usually not the case.

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Comparisons

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Self-organising networks

Implementing a Newton method on clusters within thenetwork would be the best way to have the network be selforganising, due to the speed of convergence.Theinformation that would need to be shared.• Proportion of the bandwidth that each cell assigns to a

UE.• The positions of each cell (Fixed)• The positions of each UE (Dynamic)• The data demands of each UE.

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Thank you for your time.

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