Modelling and channel borrowing in mobile communications networks Richard J. Boucherie University of Twente Faculty of Mathematical Sciences University

Modelling and channel borrowing in Modelling and channel borrowing in mobile communications networksmobile communications networks

Richard J. BoucherieUniversity of Twente

Faculty of Mathematical Sciences

University of Twente - Stochastic Operations Research

Stochastic network analysis for the design of self optimising cellular mobile communications networks

30

Aim : dynamic channel allocationAim : dynamic channel allocation

Initial capacity C channels Not sufficient for required QoS (blocking probabilities)

Traffic jam peak requires T > C channels

How do we provide capacity?



29

C channels C channelsC channelsC channels

Borrowingbut from which neighbour?

Interaction road traffic and teletraffic (shape of traffic jam, ...)

Plan : Plan : issues relevant to solve problemissues relevant to solve problem1 Telecommunications model for QoS

Single cell : Erlang loss model - equilibrium exact results - transient : Modified Offered Load Network : Handovers - equilibrium approximate results - transient : MOL

2 Highway traffic modelFluid model for unidirectional road

3 Relation road traffic and teletrafficImplementation MOL approximation to characterise required capacity

4 Self optimising networkOn-line capacity borrowing on the basis of shape traffic jam



28

GSM : GSM : terminologyterminology • Base stations, mobile terminals, subscribers, …• Call = microwave connection (base - mobile)• Limited bandwidth• Channels:

FDMA / TDMA : Frequency / Time Division Multiple Access ~ 100 frequencies of 200 kHz (carriers) 7 or 8 channels per frequency

• Calls use single channel call arrival process call length mobility of calls

• Call blocking probabilities



27

Basic model: single cell Basic model: single cell • Assign all frequencies to single base station

• Establish connection upon request

• Base station C=100 x 7,5 = 750 channels

• With mean time 2 mins/hour, 30.000 subscribers

blocking probability E(1000,750) = 25 %

• 3.5 milj. Subscribers (KPN Mobile, medio 2000)

180 mins / year blocking probability 1%

busy hour 100 %



26

Blocking probability: Erlang loss formula (1917)

Telecommunications model: single cellTelecommunications model: single cellPoisson arrivals rate call length L mean number of channels C



25

1

0 !

)(

!

)(),(

kC

CEkC

k

C

blocked call

GSM and UMTS (single cell model)

C

Erlang loss queueErlang loss queueMarkov chain (birth-death process) for exponential call length• State space S={0,1,…,C}• Markov chain

• birth rate q(n,n+1)= • death rate q(n,n-1)=n/ • Equilibrium distribution

• Blocking probability

determined by offered load =



24

1

0 !

)(

!

)(),(

kC

CEkC

k

C

1

0 !

)(

!

)())((lim

kn

ntNPkC

k

nC

t

)0),(( ttNN CC

Insenstive!

Proof: (exponential case)Proof: (exponential case) n n+1

n/ (n+1)/

equilibrium distribution

solution global balance

detailed balance

insert distribution and rates, and normalize

Markov chain (birth-death process) for exponential call length

State space S={0,1,…,C}

Markov chain N=(N(t), t0)

birth rate q(n,n+1)= death rate q(n,n-1)=n/ Equilibrium distribution

Blocking probability

determined by offered load =



23

0)},'()'()',()({'

nnqnnnqnSn

1

0 !

)(

!

)())((lim)(

kn

ntNPnkC

k

nC

t

nnnqnnnqn ),1()1()1,()(

Erlang loss queue: offered loadErlang loss queue: offered load



22

e

nntNP

n

t !))((limunlimited capacity C=

1

0 !!))((lim

kn

ntNPkC

k

nC

t

ee ))(|)((lim CtNntNPt

Time dependent arrival process )(t

t

Lt

duut )()(

)(

!

)())(( t

n

en

tntNP

Modified Offered Load approximation ))(|)(())(( CtNntNPntNP C

1

0 !!))((lim

kn

ntNPkC

k

nC

t

Capacity C

L

Offered load

Telecommunication model: single cellTelecommunication model: single cell

So know we know

how to compute blocking probabilities

in a single cell with static users

(assuming that the network consists of a single cell)



21

C channels

Telecommunications network modelTelecommunications network modelGSM - divide channels (F/TDMA) over cells



20

Capacity = number of available channels

- fixed channel assignment (FCA)- dynamic channel assignment (DCA)- cells not regular- interference constraints - capacity known in advance

Telecommunications network modelTelecommunications network modelmultiple cells i=1,…,D

Poisson arrivals rate (i)call length mean (i)number of channels C(i)

number of calls in cell i n(i)



19

Basic networkn(i) C(i)

Layered network:n(i) C(i)+C(0)n(i)+n(j) C(i)+C(j)+C(0)n(i)+n(j)+n(k) C(i)+C(j)+C(k)+C(0)

i

j

k

}:{ CAnnS State space

C(0)

Telecommunications network modelTelecommunications network modelMarkov chain (loss network) for exponential call length• State space

• Markov chain (i) • birth rate q(n,n+e(i))= (i)• death rate q(n,n-e(i))=n(i)/(i) n(i)/(i)

• Equilibrium distribution

• offered load (i)= (i) (i)



18

)!(

)()!(

)())((lim

)(

1

)(

1 ini

ini

ntNPinD

iSn

inD

i

C

t

}:{ CAnnS

Loss network

)0),(( ttNN CC

Telecommunications network modelTelecommunications network model• Equilibrium distribution

• Blocking or loss probabilities

• Computation: - recursive methods, inversion Laplace transforms (exact) - asymptotic methods (normal approx, large deviations) - Monte Carlo summation



17

)!(

)()!(

)())((lim

)(

1

)(

1 ini

ini

ntNPinD

iSn

inD

i

C

t

)!(

)()!(

)())((lim

)(

1

)(

1 ini

ini

TtNPEinD

iSn

inD

iTni

C

ti

i

Telecommunications network modelTelecommunications network modelTypical capacity allocation problems

• Dimensioning problem:

For given offered load (i), there are C(i) channels required at cell i for loss probability < 1%Find colouring with C(i) colours at edge i

• Call admission problem:For given colouring, find load such that loss <1 %

When load changes then a new colouring is required



16

Telecommunications network modelTelecommunications network model



15

)!(

)()!(

)()),,()((lim

)(

1

)(

11 inin

nntNPinD

iSn

inD

iD

C

t

equilibrium distribution

Truncation of infinite capacity case Infinite capacity Poisson distribution remains valid for time-dependent load (i,t)

Modified Offered Load approximation

Depends only on load (i)= (i) (i)

(i) (i)

e

inntNP

inD

i )!(

)())((

)(

1

(i,t)(i,t) (i,t)(i,t)

e

in

inD

iSn )!()( )(

1

=C

i

j

(i)

p(i,j)

(i)p(i,0)Telecommunications network modelTelecommunications network model



14

handovers

Infinite capacity Poisson distribution for time-dependent load (i,t)

exponential holding times

Poisson arrivals rate (i )time in cell mean (i )call termination rate (i ) p(i,0 )handover rate (i ) p(i,j )

-1

-1

Offered load

,t

,t

1

1

1 )()(),()()()(

iiijpjjiD

j

0 ,tdt

id )(,t ,t ,t ,t

,t,t

,t,t,t,t

e

ine

inntNP

inD

iSn

inD

i

C

)!()(

)!()(

))(()(

1

)(

1

(i,t)(i,t) (i,t)(i,t)

Modified Offered Load approximation

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0

0,002

0,004

0,006

0,008

0,01

0,012

0,014

0,016

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0

0,002

0,004

0,006

0,008

0,01

0,012

0,014

0,016

0,018

0,02

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,008

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0

0,5

1

1,5

2

2,5

3

3,5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20



13

Poisson arrivals rate (i+1,t+1 )= (i, t)time in cell mean (i )call termination probaility p(i,0)=0.8 resp. 0.2handover probability p(i,i+1)=1-p(i,0)

load cell 1 cell 3

load cell 1 cell 3

Telecommunication model: networkTelecommunication model: network

So know we know

how to compute blocking probabilities

in a network with users moving among the cells

(assuming that users move according to a Markov chain)



12

C channelsC channels

Highway traffic model



11

0),(),(),(

t

txvtx

t

tx

x

tx

T

vvtxv

Tx

txvtxv

t

txv ee

),(

2)(),(

1),(),(

),(

Travelling with speed v

(x,t)Cell 0 Cell 1 Cell 2

Density of traffic (x,t)

speed of traffic v(x,t)

System of differential equations (fluid model)

Location xtime t

Relation road traffic and teletraffic



10


(x,t)

Cell 0 Cell 1 Cell 2


density of calls in cell i (i,t)

Location xtime t

Mean call length (extends over the cells!)

arrival rate per unit traffic mass

Relation road traffic and teletraffic



9


(x,t)Cell 0 Cell 1 Cell 2


density of calls in cell i (i,t)

Theorem: Offered Load model where

Depends on call length only through its mean

Location xtime t

dxtxtiicellx

),(),(

),()(

1 )!(

),())(( ti

inD

i

ein

tintNP

Self optimising network (GSM - FCA)Self optimising network (GSM - FCA)



8


(x,t)Cell 0 Cell 1 Cell 2 Cell 0 Cell 1 Cell 2

C

C(0,t) C(2,t)

C(1,t)

Modified Offered Load approximation:

Under FCA distribution factorises over the cells

blocking probability in cell i

To guarantee QoS: solve C(i,t) such that E < 1%

1

0 !

)),((

!

)),(()),,((

k

tititiE

k

k

C

C(i,t)C(i,t)

C(i,t)

C(i,t)C

C

C

But where do we get the capacity




7



C

C(0,t) C(2,t)

C(1,t)

Borrowing from the left neighbour:

Let be the capacity that cell s+1 borrows from cell s at time t.

The family of functions is borrowing from the left if

at most 2 cells are borrowing at each time, and

Borrowing from the right neighbour: similar

tstsCththC ss ),,()()( 1

)(ths,...2,1),( sths




6



C

C(0,t) C(2,t)

C(1,t)

We can completely characterise on the basis of C(i,t)

i.e. on the basis of the road traffic information (x,t)

borrow from the left if (x,t) steeper on the left right if (x,t) steeper on the right

road traffic prediction 10 mins ahead of timesufficient for channel re-allocation

)(ths




5

0

5

-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1

a)cells

traffic k(x ,0)

0

0.5

1

1.5

2

0 4 8 12 16 20

b)time

Diagonal traffic jam moving along road with constant speed

Estimated blocking probabilitiesin cell 1 over time reach 1.87%above 1% for considerable time




4

0

20

0 2 4 6 8 10 12 14 16 18 20

b)time

d 1.2 (t )

0

20

0 2 4 6 8 10 12 14 16 18 20

a)time

d1.0 (t )

Demand for capacity in cell 1for blocking probabilities of 1% and 1.2%obtained from MOL approximation

C(1,t) C(1,t)




3

0

1.2

3 5 7 9 11

b)%

cal

ls b

lock

ed

.

time

n+i (t )-i (t+2)20

blocking

0

1

2 4 6 8 10

a)

% c

alls

blo

cked

.

time

n+h(t )-h(t-2)

blocking

chan

nels

.

20

chan

nels

.

borrowing from right borrowing from left

Estimated blocking probabilities




2

0

0.5

1

1.5

2

2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5time

% c

alls

lost

No

Left

Right

Estimation from discrete event simulation

We have realised: dynamic channel allocationWe have realised: dynamic channel allocation

Initial capacity C channels Not sufficient for required QoS (blocking probabilities)

Traffic jam peak requires T > C channels

How do we provide capacity?



29

C channels C channelsC channelsC channels

Borrowingbut from which neighbour?

Interaction road traffic and teletraffic (shape of traffic jam, ...)

Concluding remarksConcluding remarks

• QoS constraints require sufficient capacity

• estimation of QoS (blocking probabilities)

• borrowing based on road traffic information

• shape of traffic density determines strategy

• feasible solution: 10 mins ahead of time prediction

sufficient for modern networks



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Documents

Modelling and channel borrowing in mobile communications networks Richard J. Boucherie University of Twente Faculty of Mathematical Sciences University