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Network Design and Planning (sq16) Analysis of offered, carried and lost traffic in circuit-switched systems

Massimo Tornatore Dept. of Electronics, Information and Bioengineering

Politecnico di Milano

Dept. Computer Science University of California, Davis

[email protected]

Traffic theory

Summary

General considerations Statistical traffic characterization Analysis of server groups Dimensioning server groups

2

Traffic theory

Summary

General considerations Definitions Parameters

Traffic characterization Analysis of server groups Dimensioning server groups

3

Traffic theory

We are dealing with circuit-switched networks with given resources/capacity System that we analyse

4

Network Basic concepts

mA B

offered traffic/ users/sources

servers

Traffic theory


3+(1) fundamental parameters A: offered load m: Service system with certain capacity P: quality of service (e.g. delay or blocking probability) F: functional characteristics (e.g. queueing discipline, routing technique, etc.)

Problems Dimensioning (synthesis, network planning)

• Given A, P (and F), find m at minimum cost/capacity Performance evaluation (analysis)

• Given A, m (and F), find P Management (traffic engineering)

• Given A and m, find F optimizing P

5

Traffic theory


For each model a statistical characterization needed for Traffic sources Server systems

Traffic sources

Service system

6

Traffic theory


Sources S traffic sources

• Generate connection requests (calls) Busy source: source engaged in a service request

• Otherwise the user is not busy or free Average number of busy sources = Average amount of offered traffic

Servers m system servers

• Satisfy requests issued by sources Busy server: server engaged in a service to a source for a time duration

requested by the source (holding time of the connection) Average number of busy servers = Average amount of carried traffic

Congestion: a connection request is not accepted ⇒ Blocked request Denied request (loss systems) Delayed request (waiting systems)

7

Traffic theory


E[θ]: average holding time of a connection Offered traffic

Λo: average rate of connection requests Ao: average number of connection requests issued in a time interval equal to

the average holding time ⇒ Ao =ΛoE[θ] = Λo / µ

Carried traffic Λs: average acceptance rate of connection requests (statistical equilibrium) As: average number of connection requests accepted in a time interval equal

to the average holding time ⇒ As = ΛsE[θ] = Λs / µ

Lost traffic Λp: average refusal rate of connection requests Ap: average number of connection requests denied in a time interval equal to

the average holding time ⇒ Ap = ΛpE[θ] = Λp / µ

Ao, As, Ap adimensional ⇒ Erlang 8

Traffic theory

How do we use queueing theory for traffic characterization?

9

Ap

Traffic theory

Summary

General considerations Traffic characterization

Statistical behaviour Modeling of offered traffic

Analysis of server groups Dimensioning server groups

10

Traffic theory

Traffic description Statistical behaviour

Relevant time instants Time of service request Time of service completion

X(t,ω) = Number of servers busy at time t of realization ω of the process

Assumptions Stationarity

• E[to,to+τ][X(t, ω)] = Aτ(t0, ω) = Aτ (ω)= A(ω)

Ergodicity • A(ω) = A

1

2

3

4

X(t) 4

3

2

1

0

´

´ ´

´

´ ´

´ ´

H 1

H 2

H 3

H 3

H 6 H 1

H 5 H 4

H 5 H 2

H 4 H 8

H 7

t 0

H 7

H 3 H 4 H 7 H 6 H 5

H 6 H 7 H 8

H 8

t 0 + τ t

11

Traffic theory

Two main parameters Holding time θ (duration of the call/request)

• It is the inverse of the service rate: E(θ)=1/µ • We will stick to the traditional assumption of negative

exponential distribution of the holding time – Simple and practical

Interarrival time T (time between the arrival of two calls) • It is the inverse of the arrival rate E(Τ)=1/λ • We will consider the traditional assumption (Poisson), as well

as two other cases (Bernoulli and Pascal)

Traffic description

12

Traffic theory

Traffic description Modelling service duration

Possible histogram of holding times and corresponding approximation though exponential distribution

Oltre 10 min.

Valor medio 1.6

1 2 3 4 5 6 7 8 9 10 11

Tempi di tenuta 0 (min)

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0

Freq

uenz

a di

pre

sent

azio

ne

Pr θ > t{ }= e−

t˜ θ

13

Freq

uenc

y of

occ

uren

ce

Holding time (min)

Avg. holding time (min)

Traffic theory

As for the interarrival time we will see three distributions: Pascal, Bernoulli, Poisson

Why are they interesting? See next slides

Traffic description Interarrival time distributions

14

Traffic theory

Traffic characterization Poisson

Parameters Ao = Λo = 30 m = 50

50 100 150 200 250 300 35010.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

15

Traffic theory

Traffic characterization Bernoulli

Parameters Ao = 30 m = 50 S = 40

50 100 150 200 250 300 35010.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

16

Traffic theory

Traffic characterization Pascal

Parameters Ao = 30 m = 50 c = 10

50 100 150 200 250 30010.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

17

Traffic theory

Traffic description How do we model the three previous traffic behaviors?

We use a birth & death process [X(t)] to represent the offered traffic Births: arrivals of service requests Deaths: service completions

In general b&d processes are characterized by two parameters same characterization for all traffic types (offered, carried, lost)

Typically, modelling simplicity suggests

In this lecture we go beyond Poisson (VMR = 1) and we also consider Smoothed traffic (VMR < 1) - Bernoulli Peaked traffic (VMR >1) - Pascal

[ ]

[ ] [ ][ ] factor) s(peakednes

)(E)(VarVMR )(Var •

)(E •

tXtXortX

tX

=

[ ] [ ]

[ ] ( ) tk

tEt

ektktXirths

eeteaths

λ

µθ

λθ

−

−−

==

==>

!)(Pr - Poisson :B •

Pr - lexponentia :D •

18

Traffic theory

Offered traffic model Assumptions

Arrival and service processes Indipendent identically distributed (IID) interarrival times IID service times Arrival and service process mutually independent Ergodicity Stationarity

19

Traffic theory

Offered traffic model Single source

Source model Two states: idle (0) or busy (1)

⇒ interarrival and service times with exponential distribution and

• λ' = conditioned average interarrival rate (idle source) • µ = conditioned average rate of service completion (busy source)

Steady-state limiting probabilities

{ }{ } tttt

tttt∆=∆→∆′=∆→

µλ

1 | )+,( in 01Pr0 | )+,( in 10Pr

source anby traffic offered11

source aby traffic offered1

rate interrival average individual 111

1

1

1

10

idlea

aq

q

qa

A

Aqq

o

o

−=

−=

−=

′=

+=

+′′

===

+′′

==→′

+=

=+′′

=+′

=

λµλ

µλα

αα

µλλ

µλ

µλµλµλ

λµλ

µλλ

µλµ

20

Traffic theory

Offered traffic model Multiple sources

Single source model ensures that the occupancy process of a source groups is markovian continuous-time and time-homogeneous with discrete states of birth & death type

In formulas, this mean that the transition probabilities can be written as

IID service times (also called source occupancy times) ⇒ µn = nµ Interbirth times described by three models

{ }{ } tnttt

tnttt

n

n∆=∆→∆′=∆→

µλ

1 sources,busy | )+,( in source a for 01Pr0 sources,busy | )+,( in source a for 10Pr

( )( ) integer] [sources] [Pascal •

sources] [Poisson •sources] [Bernoulli •

cnc

SnS

n

on

n

∞+′=′∞=Λ=′

−′=′

λλλλ

λλ

21

Traffic theory

Offered traffic model Steady state characterization

State probabilities in steady-state conditions derived by queues M/M/∞ X = Number of sources busy at the same time Ao = As = E[X] = Λo/µ

( )

( )µλααα

µλ

µλλ

µλ

′=−

−+=

==

+′′

====−

=

−

−

cnn

an

n

nSnn

nnc

pPascal

aenapPoisson

paSnaanS

pBernoulli

11

!

,...,01 1

22

Traffic theory


State probabilities in steady-state conditions derived by queues M/M/∞ X = Number of sources busy at the same time Ao = As = E[X] = Λo/µ

( ) ( )

−′+Λ=

−′+Λ=

−−

−−=

−=

+′−′

XnnX

a

caSa

cSaAX

nnnncnS

onon

oX

o

nn

~~1

111

11

1~

)()(PascalPoissonBernoulli

RVM

222

λλλλα

αα

µλσσ

αα

µλ

µµµµλλλλ

23

Traffic theory


Traffic source models Random traffic Poisson Smoothed traffic Bernoulli Peaked traffic Pascal

Limiting cases

Given E[X] and Var[X] one of the three traffic models is adopted with parameters

0andif1

withPoissonPascal •

0andifwithPoissonBernoulli •

→′∞→−

=→

→′∞→=→

λα

αλ

ccA

SSaA

o

o

RVM111RVM11

1RVM~

RVM1

2

2

2

22

2

2

−=−=−=−=

−=

−===

−=

−=

X

o

o

X

o

oX

oXo

o

Xo

o

AA

a

AA

AcXAAA

AS

PascalPoissonBernoulli

σασ

σσ

σ

24

Traffic theory

Based on the previous 3 models for traffic sources, we can analyze the behaviour of the service system according to three main cases

Behaviour of a source requesting service to a blocked system Blocked calls cleared – BCC (chiamate perdute sparite - CPS) (loss sytem)

• Source gives up Blocked calls held – BCH (chiamate perdute tenute - CPT)

• Source keeps asking for service for a time Tq; → θeff = θ - Tq Blocked calls delayed – BCD (chiamate perdute ritardate - CPR) (delay

systems) • Sources keeps asking for service indefinitely

From the traffic/sources model to the system/server model

25

Traffic theory

Summary

General considerations Traffic characterization Analysis of server groups

Behavior upon congestion Grade of service

Dimensioning server groups

26

Traffic theory

Analysis of server group Behaviour upon congestion - CPS

System with m servers X: number of busy sources n: number of users in the system

CPS (BCC) ⇒ pure loss queue - M/M/m/0 Time congestion: Sp = Pr {blocked system} Call congestion: Πp = Pr {blocked system | 1 arrival}

Poisson case: Pr{1 arrival | n = m} = Pr{1 arrival} ⇒ Sp = Πp

{ } { }{ } { }

{ }{ } { }

{ }{ }

{ }arrival 1Pr | arrival 1Pr

arrival 1PrPr | arrival 1Pr

arrival 1Prarrival 1 Prarrival 1 | Pr

PrPr

mnSmnmn

mnmX

mnmXS

p

p

p

==

===

∩====Π

====

27

Traffic theory

Analysis of server group Behaviour upon congestion - CPT


CPT (BCH) ⇒ queue with infinite servers of which m are true, the other fictitious – M/M/ ∞ Source receives either true service (X < m) for the requested time or fictitious

service (X = m) for a time Tq Server becoming idle in state X = m makes effective a fictitiuos service for a

residual time θeff = θ - Tq Time congestion: St = Pr {busy true servers} Call congestion: Πt = Pr {busy true servers | 1 arrival}

Poisson case: Pr{1 arrival | n = m} = Pr{1 arrival} ⇒ St = Πt

{ } { }{ } { } { }

{ }arr 1Prarr 1 Prarr 1 | Prarr 1 | Pr

PrPr∩≥

=≥===Π

≥===

mnmnmX

mnmXS

t

t

28

Traffic theory

Analysis of server group Behaviour upon congestion - CPR


CPR (BCD) ⇒ pure delay system - M/M/m Time congestion: Sr = Pr {blocked service} Call congestion: Πr = Pr {blocked service | 1 arrival}

Poisson case: Pr{1 arrival | n = m} = Pr{1 arrival} ⇒ Sr = Πr

{ } { }{ } { }

{ }arrival 1Prarrival 1 Prarrival 1 | Pr

PrPr∩≥

===Π

≥===mnmX

mnmXS

r

r

29

Traffic theory

Analysis of server group BCC (CPS) - m = ∞

0

2

4

6

8

10

12

14

0 5 10 15

Parameter Value

Distr. Bernoulli

Policy CPS

Sim time 100

m 15

λ 0.5

µ 0.1

s 13

Ao 11.1429

σ2 1.59184

RVM 0.142857

30

Traffic theory

Analysis of server group BCC (CPS)

0

2

4

6

8

10

12

14

0 5 10 15

Parameter Value

Distr. Bernoulli

Policy CPS

Sim time 100

m 10

λ 0.5

µ 0.1

S 13

Ao 11.1429

σ2 1.59184

RVM 0.142857

31

Traffic theory

Analysis of server group BCH (CPT)

0

2

4

6

8

10

12

14

0 5 10 15

Parameter Value

Distr. Bernoulli

Policy CPT

Sim time 100

m 10

λ 0.5

µ 0.1

S 13

Ao 11.1429

σ2 1.59184

RVM 0.142857

32

Traffic theory

Analysis of server group BCD (CPR)

0

2

4

6

8

10

12

14

0 5 10 15

Parameter Value

Distr. Bernoulli

Policy CPR

Sim time 100

m 10

λ 0.5

µ 0.1

S 13

Ao 11.1429

σ2 1.59184

RVM 0.142857

33

Traffic theory

In the next slides, the formulas for S and Π are reported for BCC, BCH and (partially) BCD

Only a subset of the proofs is requested (according to what is covered during the lecture) No need to know the other formulas, we will only check the performance

comparison on graphs It is important to be able to interpret the two graphs Note the observation about Molina’s formula

Analysis of server group Notes regarding next slides

34

Traffic theory

Analysis of server group BCC (CPS) - Bernoulli

( )

( ) ( )

( )

( ) (Engset) )ˆ,,1(ˆ,,

ˆ1

ˆ1ˆ

(source) 1ˆ1ˆ

= source); (free +1

=11

ˆˆ

),...,0( ˆ

ˆ

),...,1(),...,0(ˆ

0

0

αα

α

αλ

αα

µλ

αα

µλ

α

α

α

µµλλ

mSSmS

jS

mS

mSS

pS

aa

a

mn

jS

nS

pmnnmnnS

pp

m

j

j

m

opp

mp

ppp

m

j

j

n

nn

n

−=Π⇒

−

−

=Λ

−′=Π

=

Π−+=

ΠΠ−−=

′=

=

====−′=

∑

∑

=

=

35

Traffic theory

Analysis of server group BCC (CPS) - Bernoulli

( ) ( )

( ) ( )

( ) [ ][ ]( ) [ ][ ]

[ ][ ] ( )

[ ] ( ))ˆ,1,1()ˆ,,( 1)ˆ,1,1()ˆ,,(1RVM

1)ˆ,1,1()ˆ,,(1

)ˆ,1,1()ˆ,,(1)ˆ,,(1

)ˆ,1,1()ˆ,,(1 )ˆ,,(

ˆ

ˆ1

ˆ11

ˆ

ˆ1

ˆˆ~ˆ

22

2

20

22

0

1

0

0

0

00

αααασσ

σαα

ααασ

ααασ

α

αα

α

ααα

µλ

µ

λ

µ

−−><−−−−=≤⇒

=−≤−−−−≤

−−−−Π−=

−−−−=−=

Π=−=

−

==Π−=Π−=

−

=−=−′

==Λ

==

∑

∑

∑∑

∑

∑∑

=

=

−

=

=

=

==

mSAmSAmSAmSA

aSamSAmSASa

mSAmSAmSSa

mSAmSAmSApAk

AAAA

jS

kS

SkpSaAA

jS

kS

SASnSp

ASaA

ssss

os

oss

ssps

sssm

kkss

posop

m

j

j

m

k

km

kkppos

m

j

j

m

k

k

s

m

jjj

ooo

36

Traffic theory

Analysis of server group BCC (CPS) - Poisson

B) (Erlang

!

!)(

),...,0(

!

! ),...,1(),...,0(

0

,1

0

∑

∑

=

=

===Π=

======Λ=

m

j

j

m

ommpp

m

j

j

n

nn

on

ja

ma

AEpS

mn

ja

na

pmnnmn

µµλλ

37

Traffic theory

Analysis of server group BCC (CPS) - Poisson

( )

( ) ( ) ( ) ( )[ ][ ]( ) ( ) ( )( )( ) ( )[ ]

( )[ ] ( )

( ) ( )[ ] [ ] 1)()(111RVM

)()()()(1

)()(1

1

11

)(

)(1

!

!1

!

!

,11,1

22,11,1

2,11,1

,11,12

20

22

,1

,1

00

1

0

0

<−−=−−−=<⇒

>=<−−<

<−−=

−>=<<

−−−=−=

=

−=

−===

==

−

−−

−

=

==

−

=

=

∑

∑∑

∑∑

omomoss

os

omomooomomoo

omomoss

ssoos

sssm

kkss

omop

omom

j

jo

mo

om

j

jo

m

k

ko

om

kks

o

AEAEAmAmA

AEAEAAEAEAA

AEAEAmA

mAmAAmA

mAmAmApAk

AEAA

AEA

jA

mA

A

jAk

A

AkpA

aA

σσ

σ

σ

σ

σ

µλ

38

Traffic theory

Analysis of server group Erlang-Engset comparison

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110-4

10-3

10-2

10-1

100

Traffico offerto per servente, a

Prob

abilit

a' d

i blo

cco

∞ /1

5 / 1

∞ /2

10 / 2

5 / 2

∞ /5

20 / 510 / 5

∞ /10

20 / 10

39

Blo

ckin

g Pr

obab

ility

Offered Traffic per Server, a

Erlang

Engset

# users / # servers

Obs. 1: the comparison is performed between Erlang and Engset fixing the same offered load Ao

Obs. 2: - For small #users, Erlang provides excessive sovraestimation - For large #users, the two formulas tend to return very similar results

Traffic theory

Analysis of server group BCC (CPS) - Pascal

( )

( )

( ) ( )αα

α

αλ

αα

µλα

α

α

µµλλ

ˆ,,1ˆ,,

ˆ

ˆˆ

-1ˆ

ˆ

),...,0( ˆ1

ˆ1

),...,1(),...,0(ˆ

0

0

mcSmc

jjc

mmc

mcS

pS

mn

jjc

nnc

pmnnmnnc

pp

m

j

j

m

opp

mp

p

m

j

j

n

nn

n

+=Π

+

+

=Λ

+′=Π

=

Π=

′=

=

−+

−+

====+′=

∑

∑

=

=

40

Traffic theory

Analysis of server group BCC (CPS) - Pascal

( ) ( )

( ) ( )

( )( ) [ ][ ]

[ ] 1)ˆ,,()ˆ,1,1(1RVMn)(definitio

)ˆ,,()ˆ,1,1(1 )ˆ,,(

ˆ1

ˆ ˆ1

11

ˆˆ

~ˆ

1

220

22

0

1

0

0

0

≤≥−−++=<

−−++=−=

Π=−=

−+

+

==Π−−

=Π−=

+=+′

=

+′

==Λ

=−

=

∑

∑

∑∑

∑

=

=

−

=

=

=

αασσ

ααασ

α

αα

αα

αµ

λµ

λ

µ

λ

µαα

mcAmcA

mcAmcAmcApAk

AAAA

jjc

kkc

ckpcAA

AcAcXcpAcA

ss

os

sssm

kkss

posop

m

j

j

m

k

km

kkppos

ss

m

jjj

ooo

41

Traffic theory

Analysis of server group BCH(CPT)

( ) ( )

( )

( ) ( )( ) ( )

( ) ( )amSSamS

aak

SaS

aakSkS

aakSS

SnaanSp

SnnSnnS

/SM/M/S/Bernoulli

ppS

tt

S

mk

kSk

kSkS

mkt

S

mk

kSkt

nSnn

n

n

S

mkkk

ot

S

mkkt

,,1,,

111

1

1

,...,01,...,1,...,0

0 model Queue - •

1

1 1

1

1

−=Π

−

−=

−′

−

−′

=Π

−

=

=−

=

===−′=

Λ=Π=

∑∑

∑

∑∑

−

=

−−

−−

=

=

−

−

−

==

λ

λ

µµλλ

λ

42

Traffic theory

Analysis of server group BCH(CPT)

( ) ( )

( )

( ) ( )( )

( ) ( )αα

ααµ

αα

ααλ

αα

ααµµ

λλ

µµλλ

,,1,,

1

1

11

11

,...1,011,...2,1,...1,0

model Queue - •

(Molina) !

,...1,0!,...2,1

,...1,0 model Queue - •

1

mcSmc

kkc

ckkckc

kkcS

nnncp

nnnnc

M/M/Pascal

ekaS

nenap

nnn

M/M/Poisson

tt

mk

ck

ck

mkt

mk

ckt

cnn

n

n

mk

ak

tt

an

nn

on

+=Π

−

+=

−

−

−++′

=Π

−

−+=

=−

−+=

===+′=

∞

=Π=

======Λ=

∞

∑∑

∑

∑

∞

=

+

∞

=

∞

=

∞

=

−

−

43

Traffic theory

Analysis of server group BCD(CPR)

( )

( )

−−

=Λ

=Π

==

−

++=

=

−=

=

=−=

=

=−′=−

−−

−

=

−=

−

=

−

−

∑

∑

∑

α

λ

α

αα

α

α

µµµ

λλ

mE

pnSmSp

mE

ppS

mmk

kSp

Smnmmn

nSp

mnnSp

p

SmnmmnnSnnS

m/SM/M/m/SBernoulli

mS

mS

mkkk

or

mS

mS

mkkr

kkmS

mk

S

mnn

n

n

n

n

1,1

1

,1

10

0

0

~1

1 !

!1

,..., !

!

1,...,0

,...,1,...,1

,...,0 model Queue - •

44

Traffic theory

Analysis of server group BCD(CPR)

( ) C) (Erlang

!!

!

!!,...1, !

1

1,...,0!

1

,...1,1,...,1

,...1,0 model Queue - •

1

0

,2

11

00

0

0

amm

ma

ka

amm

ma

AEpS

amm

ma

kap

mmnmm

ap

mnn

app

mmnmmnn

nM/M/mPoisson

mkm

k

m

ommk

krr

mkm

kmn

n

n

n

n

on

−+

−===Π=

−+=

+=

−==

+=−=

=

==Λ=

∑∑

∑

−

=

∞

=

−−

=−

µµµ

λλ

45

Traffic theory

Analysis of server group BCC (CPS) – BCH (CPT) – BCD (CPR) comparison

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910-3

10-2

10-1

100

Traffico offerto per canale (erlang)

Prob

abilit

a' d

i blo

cco

/ atte

sa

Confronto tra le prestazioni di code poissoniane CPS, CPT e CPR

1

11

5

5

5

10

10

10

CPSCPTCPR

46

Blo

ckin

g/W

aitin

g Pr

obab

ility

Offered Traffic per Server, a

Obs. 1: the comparison is performed between Erlang and Engset fixing the same offered load Ao

Traffic theory

Summary

General considerations Traffic characterization Analysis of server groups Dimensioning server groups

Theoretical analysis Wilkinson’s approach Fredericks’ approach Lindberger’s approach

47

Traffic theory

Theoretical analysis Overflow servers – Finite case

Primary servers: n Overflow servers: m Sequential search

Poisson offered traffic

System described by a 2-d Markov chain

with state (j,i), j = 0,...,n; i = 0,...,m

µλσ == 2

ooA

( )[ ] ( ) ( )( )[ ] ( )

( )1

1,...,011,...,01,...,011

,00

1,,1,

1,1,,1,

1,,1,1,

=

+=+−=+++=++

−=−=

++++=++

∑∑==

−−

−+−

++−

ijm

i

n

j

mnmnmn

inininin

ijijijij

p

pppmnmippippinminjpipjppij

λλµλµλµλ

µµλµλ

48

Traffic theory


Distributions

( ) ( ) ( )

( )

( )

),...,1( 111

1

),...,1( 111

r)(Brockmeye 1

SS

!

!SS

N.B. 1!

01

10

0,

n1

n0

,1

0

n1

j0

0

mrSrv

SSa

SK

mkakrK

SixiKp

AE

iAj

A

vrv

vmAAS

vnm

rvnr

mnr

mn

rkrm

krk

xjxixi

xim

xij

onn

i

io

jo

vmo

m

vo

mr

=

−−

=

=

=

−−

−=

+−=

==

−+−

=

+

=+

+

−

=

−++

−

=

=

−

=

∑

∑

∑

∑∑

njSS

pP

SixiKpQ

n

j

ijm

ij

xnxixi

xim

xij

n

ji

===

+−==

∑

∑∑

=

−+++

−

==

per B-Erlang serversprimary•

)1( serversoverflow •

1

0,

0

10

,0

49

Traffic theory


Grade of service

( ) ( )

( ) ( )[ ]( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( )on

omn

w

ppomnn

m

nm

p

omnpponpp

m

imn

mn

n

n

oomnonopwis

omnoponow

AEAE

AA

mAES

SmS

AEmnmnSAEnnS

SS

SSAAEAEAAAiQA

AEAAAEAA

,1

,1,1

1

,1,1

0 1

0

1

0,1,1

,1,1

++

+

+

=+

+

+

+

==Π=

=+Π=+=Π=

−=−=−==

==

∑

50

Traffic theory

Overflow servers: m = ∞

Theoretical analysis Overflow servers – Infinite case

51

( )[ ] ( ) ( )( )[ ] ( )

( ) ( )

( )

!

,...2,1 !

1

!1Solution•

1

,...1,01,...1,0

1,...,011

)( chain Markov d2 •

0

0

100,

,00

1,1,,1,

1,,1,1,

hAeC

hsh

AssveC

CCC

vA

ivCp

p

ippippini

njpipjppij

hoAh

sho

h

s

Ahv

nv

nv

jv

vo

v

niij

iji

n

j

inininin

ijijijij

o

o

−

−

=

−

+

∞

=

∞

==

−+−

++−

=

=−

−+=

−

−=

=

=+++=++

=−=

++++=++

∞=−

∑

∑

∑∑

m

λµλµλ

µµλµλ

Traffic theory

Theoretical analysis Overflow servers – Infinite case

( )[ ] ( )1956) - (Wilkinson

11

(Riordan) !

serversoverflow for give of moments Factorial•groupoverflow intraffic offered=traffic carried•

2

,11,11

0

0

−+++−=⇒

==⇒=

=

=

→∞=

∑∞

=

ow

owww

onowono

nk

nkoi

ik

i

AAnAAA

AEAMAAEAMCCAQ

kikM

Qm

σ

52

Traffic theory

Analysis of channel group BCC (CPS)

Channel group m channels Ao Poisson BCC

( )( )( )[ ]

( )( )

( )( )

( )( ) ( ) 1

11

11

11

11

11RVM •

11 •

•

11RVM •

•

1 •

2

2

2

,1

,1

2,1

22,1

>−+

+=−+

−−+=

−+

−−−+=

=−+

++−−+=

−+++−==

−+++−=

=

<−

−==

−−=−+−=

−=

s

s

s

sps

s

sppo

s

osppp

op

op

p

p

op

oppp

omop

s

somo

s

s

somossspsos

omos

AmAmAmAA

AmAmAAA

AmAAAmAA

AAmAA

A

AAmAAA

AEAAA

AmAEAA

AmAEAAAAmAAA

AEAA

σ

σ

σ

σ

σ

53

OFFERED

CARRIED

LOST

Wilkinson’s Formula

Traffic theory

Dimensioning of overflow channels Wilkinson approach

Channel group loaded by Ao, σo2 with RVM >1

Dimensioning/analysis

( )

( )oemmoepop

pp

oeoe

oeooo

oemoeo

oeoe

AEAAAmm

AAmAAA

AEAAAmA

e

e

+=Π=ΠΠ

−+++−=

=

,1

2

,1

20

or knowing or Compute 21

1

e knowing and Compute 1

σ

σ

54

Traffic theory

Dimensioning of overflow channels Erlang formula

Erlang formula

Recursive Erlang formula

Erlang formula for m real (Fortet representation)

Approximation of real m value to the closest integer Ceiling: dimensioning of overflow group Floor: dimensioning of equivalent group

( ) ( )om

i

io

mo

om AmE

iA

mA

AE ,

!

! •

0

,1 ==

∑=

( )[ ] ( )[ ] integer 1 • 11,1

1,1 mAE

AmAE om

oom += −

−−

( )[ ] ( ) real 1 •0

1,1 mdyyeAAE myA

oomo∫

∞−− +=

55

Traffic theory

Dimensioning of overflow channels Overflow traffic - Average

0 2 4 6 8 10 12 1410-2

10-1

100

101

m=1m=2m=3m=4m=5m=6m=7m=8m=9m=10m=11m=12

m=13

m=14

m=15

m=16

m=17

m=18

m=19

m=20

Formula di Erlang-B

Intensita media del traffico offerto, Ao (Erlang)

Val

ore

med

io d

el tr

affic

o di

trab

occo

, A p (Erla

ng)

56

Average offered traffic, AO (Erlang)

Calculation based on Erlang-B formula

Aver

age

lost

traf

fic, A

p (E

rlang

)

Traffic theory

Dimensioning of overflow channels Overflow traffic - Average

10 15 20 25 30 35 40 45 5010-2

10-1

100

101

102Formula di Erlang-B


Val

ore

med

io d

el tr

affic

o di

trab

occo

, A p (Erla

ng)

m=10-50

57


Aver

age

lost

traf

fic, A

p (E

rlang

)

Calculation based on Erlang-B formula

Traffic theory

Dimensioning of overflow channels Overflow traffic - Variance

0 2 4 6 8 10 12 1410-2

10-1

100

101m=1m=2m=3m=4m=5m=6m=7m=8m=9m=10m=11m=12m=13

m=14

m=15

m=16

m=17

m=18

m=19

m=20

Coda M/M/m/0


Var

ianz

a de

l tra

ffico

di t

rabo

cco

(Erla

ng2 )

58


Varia

nce

of l

ost t

raffi

c, A

p (E

rlang

)

Traffic theory

Dimensioning of overflow channels Overflow traffic - Variance

10 15 20 25 30 35 40 45 5010-2

10-1

100

101

102Coda M/M/m/0


Var

ianz

a de

l tra

ffico

di t

rabo

cco

(Erla

ng2 )

m=10-50

59


Varia

nce

of l

ost t

raffi

c, A

p (E

rlang

)

Traffic theory

Dimensioning of overflow channels Sum of multiple flows

Dimensioning of overflow traffic for n independent traffic flows Hp: statistical independence of flows A-Bi ⇒ independent overflow

traffics ⇒ Offered traffic to A-C derived directly from sum of lost traffics

60

Traffic theory

Dimensioning of overflow channels Sum of multiple flows

Numerical solution for two Wilkinson equations ⇒ Rapp approximation

∑∑==

==n

ipiopi

n

iA

1

22

1o A σσ

( ) ( )( )oemmoepop

oemoeoo

oooe

AEAAA

AEAAA

zzzA

e

e

+=Π=

==−+=

,1

,1

22 13 σσ

61

Traffic theory

Dimensioning of channel groups Fredericks model

Applicable for a traffic Ao, σo2 with arbitrary z (z = RVM <>1)

Real situation Group of m channels Births: individual with general interarrival distribution

Model z integer, z>1 Births : in groups of z, with exponential interarrival distribution (Aog = σog

2 = Ao) Deaths: in groups of z z channel groups with m / z channels each

62

1 arrival z arrivals

General

Exponential

Traffic theory

Dimensioning of channel groups Fredericks model

Hp: z channels requested per birth, resulting in one channel requested per group

63

Traffic theory

Fredericks model Analysis

Offered traffic in z flows has the first two moments of the real traffic Hp: very small loss probability X = Total number of busy channels in the z groups Y = Number of busy channels in each group

Ao = zAoz , As = zAsz , Ap = zApz Global loss probability = Individual group loss probability

Fredericks model hold also for z real values and for z < 1

=Π

zAE o

zmp ,1

[ ] [ ]

[ ] [ ][ ] [ ] 222

ooog

oog

og

zAz

AzYVarzXVar

Az

AzYzEXE

zYXz

AYVarYE

σ====

===

=

==

64

( )os AA ≅

Traffic theory

Fredericks model Analysis

−−=

−−=

−

−==

−=

−==

−+++−=

−+++−==

=

==

ps

ss

spz

ssz

o

zmozszszs

o

zmo

o

zm

oszs

op

opp

ozpz

ozpzpzpzp

o

zmo

o

zm

opzp

AzA

AmzA

zAmA

zAzA

zm

zAEAAzz

zAEA

zAE

zAzzAA

AAmzA

zA

zA

AAzm

AAAzz

zAEA

zAE

zAzzAA

1

11

1

11

2,1

2222

,1,1

2222

,1,1

σσ

σσ

65

Traffic theory

Loss per flow Lindberger model

Offered traffic = sum of n flows Wilkinson-Fredericks models only give global average loss Lindberger model gives loss per flow

n flows statistical independent zi = σoi

2/Aoi arbitrary (i =1,...,n) Arrivals of “heavy” calls, each requesting

zi channels, with exponential interarrivals Equivalent to receiving a Poisson traffic

Aoi /zi (one request per time) on a group of m/zi servers ⇒ Same Fredericks equations if

• Aoi, σoi2 non-Poisson traffic replaced by Aozi = σozi

2 = Aoi/ zi • Each request of the new flow i occupies zi channels out of the m

total channels

66

A o1 /z 1 A on /z n A oi /z i

A p1, σ p1 2 A pn, σ pn 2 A pi, σ pi 2

X 1 z 1

X i z i

X n z n

m

Traffic theory

Lindberger model Analysis

Xi : number of accepted requests for i-th flow X : total number of busy channels It is proven that

Loss probability Complex computation of distribution π ⇒

approximation of π such that

flows) of indip. (stat.

1

1

2

2

2

2

n

A

A

A

Azz

oin

i

n

ioi

oi

oi

o

o

oi

oi

i

p

pi

∑

∑

=

=

===Π

Π

σ

σ

σ

σ

( ) { }( ) ) gives( 1,...,

formula Erlang dgeneralize

!

1,...,Pr,...,

1

1111

Gkk

k

A

GkXkXkk

n

i

kozn

innn

ii

=

====

∑

∏=

π

π

iin

izXX ∑

==

1

{ }ipi zmX −>=Π Pr

67

A o1 /z 1 A on /z n A oi /z i

A p1, σ p1 2 A pn, σ pn 2 A pi, σ pi 2

X 1 z 1

X i z i

X n z n

m

Traffic theory

Dimensioning/analysis of channel groups Overall equations

( ) ( )222

22

22

,1

1

11

111

esimo RivoloGlobale

pip

ssiisip

s

sss

pioisipos

oio

ppipi

op

oppp

pioipipop

ippi

o

zmp

AA

mAAzAzA

AmzA

AAAA

zA

AAmzA

zA

zA

AAAAzz

zAE

i

−+=

−−=

Π−=Π−=

−+=

−+++−=

Π=Π=

Π=Π

=Π

−

σσ

σσ

σσ

p

pp

oi

oii

o

oA

zA

zA

z222

σσσ

===

68

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