[PPT] - Network Design and Planning (sq16) Analysis of offered, carried and PowerPoint Presentation

SLIDE 1

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

tornator@elet.polimi.it

SLIDE 2

Traffic theory

Summary

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

2

SLIDE 3

Traffic theory

Summary

 General considerations

Definitions
Parameters

 Traffic characterization  Analysis of server groups  Dimensioning server groups

3

SLIDE 4

Traffic theory



We are dealing with circuit-switched networks with given resources/capacity



System that we analyse

4

Network

Basic concepts

m

A B

ffered traffic/

users/sources servers

SLIDE 5

Traffic theory

Network

Basic concepts

 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

SLIDE 6

Traffic theory

Network

Basic concepts

 For each model a statistical characterization needed for

Traffic sources
Server systems

Traffic sources Service system

6

SLIDE 7

Traffic theory

Network

Basic concepts

 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

SLIDE 8

Traffic theory

Network

Basic concepts

 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

SLIDE 9

Traffic theory



How do we use queueing theory for traffic characterization?

9

Ap

SLIDE 10

Traffic theory

Summary

 General considerations  Traffic characterization

Statistical behaviour
Modeling of offered traffic

 Analysis of server groups  Dimensioning server groups

10

SLIDE 11

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

´ ´ ´ ´ ´ ´ ´ ´

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 H

7

H

3

H

4

H

7

H

6

H

5

H

6

H

7

H

8

H

8

t + τ t

11

SLIDE 12

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

SLIDE 13

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 Frequenza di presentazione

Pr θ >t

{ }=e

−t ˜ θ

13 Frequency of occurence Holding time (min)

Avg. holding time (min)

SLIDE 14

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

SLIDE 15

Traffic theory

Traffic characterization

Poisson

 Parameters

Ao = Λo = 30
m = 50

50 100 150 200 250 300 350 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00

15

SLIDE 16

Traffic theory

Traffic characterization

Bernoulli

 Parameters

Ao = 30
m = 50
S = 40

50 100 150 200 250 300 350 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00

16

SLIDE 17

Traffic theory

Traffic characterization

Pascal

 Parameters

Ao = 30
m = 50
c = 10

50 100 150 200 250 300 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00

17

SLIDE 18

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 ) ( Var VMR ) ( Var

)

( E

t

X t X

r

t X t X =

[ ]

[ ] ( )

t k t E t

e k t k t X irths e e t eaths

λ µ θ

λ θ

− − −

= = = = > ! ) ( Pr

Poisson

: B

Pr
l

exponentia : D

18

SLIDE 19

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

SLIDE 20

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

{ } { }

t t t t t t t t ∆ = ∆ → ∆ ′ = ∆ → µ λ 1 | ) + , ( in 1 Pr | ) + , ( in 1 Pr source an by traffic

ffered

1 1 source a by traffic

ffered

1 rate interrival average individual 1 1 1

1 1 1 1

idle a a q q q a A A q q

−

= − = − = ′ = + = + ′ ′ = = = + ′ ′ = = → ′ + = = + ′ ′ = + ′ = λ µ λ µ λ α α α µ λ λ µ λ µ λ µ λ µ λ λ µ λ µ λ λ µ λ µ

20

SLIDE 21

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
f 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

{ } { }

t n t t t t n t t t

n n

∆ = ∆ → ∆ ′ = ∆ → µ λ 1 sources, busy | ) + , ( in source a for 1 Pr sources, busy | ) + , ( in source a for 1 Pr

( ) ( )

integer] [ sources] [ Pascal

sources]

[ Poisson

sources]

[ Bernoulli

c

n c S n S

n

n

n

∞ + ′ = ′ ∞ = Λ = ′ − ′ = ′ λ λ λ λ λ λ

21

SLIDE 22

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

( ) ( )

µ λ α α α µ λ µ λ λ µ λ ′ = −         − + = = = + ′ ′ = = = = −         =

− − c n n a n n n S n n

n n c p Pascal a e n a p Poisson p a S n a a n S p Bernoulli 1 1 ! ,..., 1

1

22

SLIDE 23

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

( ) ( )

      − ′ + Λ =       − ′ + Λ = − − − − = − = + ′ − ′ X n n X a c a Sa c Sa A X n n n n c n S

n
n
X
n

n

~ ~ 1 1 1 1 1 1 1 ~ ) ( ) ( Pascal Poisson Bernoulli

RVM

2 2 2

λ λ λ λ α α α µ λ σ σ α α µ λ µ µ µ µ λ λ λ λ

23

SLIDE 24

Traffic theory

Offered traffic model

Steady state characterization

 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

and if 1 with Poisson Pascal

and

if with Poisson Bernoulli

→

′ ∞ → − = → → ′ ∞ → = → λ α α λ c c A S Sa A

RVM

1 1 1 RVM 1 1 1 RVM ~ RVM 1

2 2 2 2 2 2 2

− = − = − = − = − = − = = = − = − =

X

X
X
X
X
A

A a A A A c X A A A A S Pascal Poisson Bernoulli σ α σ σ σ σ

24

SLIDE 25

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

SLIDE 26

Traffic theory

Summary

 General considerations  Traffic characterization  Analysis of server groups

Behavior upon congestion
Grade of service

 Dimensioning server groups

26

SLIDE 27

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 1 Pr | arrival 1 Pr arrival 1 Pr Pr | arrival 1 Pr arrival 1 Pr arrival 1 Pr arrival 1 | Pr Pr Pr m n S m n m n m n m X m n m X S

p p p

= = = = = ∩ = = = = Π = = = =

27

SLIDE 28

Traffic theory

Analysis of server group

Behaviour upon congestion - CPT

 System with m servers

X: number of busy sources
n: number of users in the system

 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 1 Pr arr 1 Pr arr 1 | Pr arr 1 | Pr Pr Pr ∩ ≥ = ≥ = = = Π ≥ = = = m n m n m X m n m X S

t t

28

SLIDE 29

Traffic theory

Analysis of server group

Behaviour upon congestion - CPR

 System with m servers

X: number of busy sources
n: number of users in the system

 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

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

SLIDE 35

Traffic theory

Analysis of server group

BCC (CPS) - Bernoulli

( )

( ) ( )

(Engset) ) ˆ , , 1 ( ˆ , , ˆ 1 ˆ 1 ˆ (source) 1 ˆ 1 ˆ = source); (free + 1 = 1 1 ˆ ˆ ) ,..., ( ˆ ˆ ) ,..., 1 ( ) ,..., ( ˆ α α α α λ α α µ λ α α µ λ α α α µ µ λ λ m S S m S j S m S m S S p S a a a m n j S n S p m n n m n n S

p p m j j m

p

p m p p p p m j j n n n n

− = Π ⇒         −         − = Λ − ′ = Π = Π − + = Π Π − − = ′ = =                 = = = = − ′ =

∑ ∑

= =

35

SLIDE 36

Traffic theory

Analysis of server group

BCC (CPS) - Bernoulli

( ) ( )

( ) [ ] [ ]

( )

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

) ˆ , 1 , 1 ( ) ˆ , , ( 1 ) ˆ , 1 , 1 ( ) ˆ , , ( 1 RVM 1 ) ˆ , 1 , 1 ( ) ˆ , , ( 1 ) ˆ , 1 , 1 ( ) ˆ , , ( 1 ) ˆ , , ( 1 ) ˆ , 1 , 1 ( ) ˆ , , ( 1 ) ˆ , , ( ˆ ˆ 1 ˆ 1 1 ˆ ˆ 1 ˆ ˆ ~ ˆ

2 2 2 2 2 2 1

α α α α σ σ σ α α α α α σ α α α σ α α α α α α α µ λ µ λ µ − − > < − − − − = ≤ ⇒ = − ≤ − − − − ≤ − − − − Π − = − − − − = − = Π = − =                 − = = Π − = Π − =                 − = − = − ′ = = Λ = =

∑ ∑ ∑ ∑ ∑ ∑ ∑

= = − = = = = =

m S A m S A m S A m S A a Sa m S A m S A Sa m S A m S A m S Sa m S A m S A m S A p A k A A A A j S k S S kp Sa A A j S k S S A S n S p A Sa A

s s s s

s
s

s s s p s s s s m k k s s p

s
p

m j j m k k m k k p p

s

m j j m k k s m j j j

36

SLIDE 37

Traffic theory

Analysis of server group

BCC (CPS) - Poisson

B) (Erlang ! ! ) ( ) ,..., ( ! ! ) ,..., 1 ( ) ,..., (

, 1

∑ ∑

= =

= = = Π = = = = = = = Λ =

m j j m

m

m p p m j j n n n

n

j a m a A E p S m n j a n a p m n n m n µ µ λ λ

37

SLIDE 38

Traffic theory

Analysis of server group

BCC (CPS) - Poisson

( )

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

( ) [ ] ( ) [ ] ( )

( ) ( ) [ ]

[ ]

1 ) ( ) ( 1 1 1 RVM ) ( ) ( ) ( ) ( 1 ) ( ) ( 1 1 1 1 ) ( ) ( 1 ! ! 1 ! !

, 1 1 , 1 2 2 , 1 1 , 1 2 , 1 1 , 1 , 1 1 , 1 2 2 2 2 , 1 , 1 1

< − − = − − − = < ⇒ > = < − − < < − − = − > = < < − − − = − = = − =                 − = = = = =

− − − − = = = − = =

∑ ∑ ∑ ∑ ∑

m
m
s

s

s
m
m
m
m
m
m
s

s s s

s

s s s m k k s s

m
p
m
m

j j

m
m

j j

m

k k

m

k k s

A

E A E A m A m A A E A E A A E A E A A A E A E A m A m A m A A m A m A m A m A p A k A E A A A E A j A m A A j A k A A kp A a A σ σ σ σ σ σ µ λ

38

SLIDE 39

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 1 10-4 10-3 10-2 10-1 100

Traffico offerto per servente, a Probabilita' di blocco

∞ /1 5 / 1 ∞ /2 10 / 2 5 / 2 ∞ /5 20 / 5 10 / 5 ∞ /10 20 / 10

39

Blocking Probability Offered Traffic per Server, a Erlang Engset # users / # servers

Obs. 1: the comparison is

performed between Erlang and Engset fixing the same

ffered load Ao
Obs. 2:
For small #users, Erlang

provides excessive sovraestimation

For large #users, the two

formulas tend to return very similar results

SLIDE 40

Traffic theory

Analysis of server group

BCC (CPS) - Pascal

( ) ( ) ( ) ( )

α α α α λ α α µ λ α α α µ µ λ λ ˆ , , 1 ˆ , , ˆ ˆ ˆ

1

ˆ ˆ ) ,..., ( ˆ 1 ˆ 1 ) ,..., 1 ( ) ,..., ( ˆ m c S m c j j c m m c m c S p S m n j j c n n c p m n n m n n c

p p m j j m

p

p m p p m j j n n n n

+ = Π         +         + = Λ + ′ = Π = Π = ′ = =         − +         − + = = = = + ′ =

∑ ∑

= =

40

SLIDE 41

Traffic theory

Analysis of server group

BCC (CPS) - Pascal

( ) ( )

( ) ( ) [ ] [ ] [ ]

1 ) ˆ , , ( ) ˆ , 1 , 1 ( 1 RVM n) (definitio ) ˆ , , ( ) ˆ , 1 , 1 ( 1 ) ˆ , , ( ˆ 1 ˆ ˆ 1 1 1 ˆ ˆ ~ ˆ 1

2 2 2 2 1

≤≥ − − + + = < − − + + = − = Π = − =         − +         + = = Π − − = Π − = + = + ′ =       + ′ = = Λ = − =

∑ ∑ ∑ ∑ ∑

= = − = = =

α α σ σ α α α σ α α α α α α µ λ µ λ µ λ µ α α m c A m c A m c A m c A m c A p A k A A A A j j c k k c c kp c A A A c A c X c p A c A

s s

s

s s s m k k s s p

s
p

m j j m k k m k k p p

s

s s m j j j

41

SLIDE 42

Traffic theory

Analysis of server group

BCH(CPT)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

a m S S a m S a a k S a S a a k S k S a a k S S S n a a n S p S n n S n n S /S M/M/S/ Bernoulli p p S

t t S m k k S k k S k S m k t S m k k S k t n S n n n n S m k k k

t

S m k k t

, , 1 , , 1 1 1 1 1 ,..., 1 ,..., 1 ,..., model Queue

1

1 1 1 1

− = Π −         − = − ′ −         − ′ = Π −         = = −         = = = = − ′ = Λ = Π =

∑ ∑ ∑ ∑ ∑

− = − − − − = = − − − = =

λ λ µ µ λ λ λ

42

SLIDE 43

Traffic theory

Analysis of server group

BCH(CPT)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

α α α α µ α α α α λ α α α α µ µ λ λ µ µ λ λ , , 1 , , 1 1 1 1 1 1 ,... 1 , 1 1 ,... 2 , 1 ,... 1 , model Queue

(Molina)

! ,... 1 , ! ,... 2 , 1 ,... 1 , model Queue

1

m c S m c k k c c k k c k c k k c S n n n c p n n n n c M/M/ Pascal e k a S n e n a p n n n M/M/ Poisson

t t m k c k c k m k t m k c k t c n n n n m k a k t t a n n n

n

+ = Π −         + = − −         − + + ′ = Π −         − + = = −         − + = = = = + ′ = ∞ = Π = = = = = = = Λ = ∞

∑ ∑ ∑ ∑

∞ = + ∞ = ∞ = ∞ = − −

43

SLIDE 44

Traffic theory

Analysis of server group

BCD(CPR)

( ) ( )

        − − = Λ = Π         = =         −         + + =        =         − =         =    = − = = = − ′ = −

− − − = − = − = − −

∑ ∑ ∑

α λ α α α α α µ µ µ λ λ m E p n S m S p m E p p S m m k k S p S m n m m n n S p m n n S p p S m n m m n n S n n S m/S M/M/m/S Bernoulli

m S m S m k k k

r

m S m S m k k r k k m S m k S m n n n n n n 1 , 1 1 , 1 1

~ 1 1 ! ! 1 ,..., ! ! 1 ,..., ,..., 1 ,..., 1 ,..., model Queue

44

SLIDE 45

Traffic theory

Analysis of server group

BCD(CPR)

( )

C) (Erlang ! ! ! ! ! ,... 1 , ! 1 1 ,..., ! 1 ,... 1 , 1 ,..., 1 ,... 1 , model Queue

1

, 2 1 1

a m m m a k a a m m m a A E p S a m m m a k a p m m n m m a p m n n a p p m m n m m n n n M/M/m Poisson

m k m k m

m

m k k r r m k m k m n n n n n

n

− + − = = = Π =         − + =        + = − = =    + = − = = = = Λ =

∑ ∑ ∑

− = ∞ = − − = −

µ µ µ λ λ

45

SLIDE 46

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 0.9 10-3 10-2 10-1 100

Traffico offerto per canale (erlang) Probabilita' di blocco / attesa

Confronto tra le prestazioni di code poissoniane CPS, CPT e CPR 1 1 1 5 5 5 10 10 10

CPS CPT CPR

46

Blocking/Waiting Probability Offered Traffic per Server, a

Obs. 1: the comparison is

performed between Erlang and Engset fixing the same

ffered load Ao

SLIDE 47

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

SLIDE 48

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

µ λ σ = =

2

A

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

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

, 1 , , 1 , 1 , 1 , , 1 , 1 , , 1 , 1 ,

= + = + − = + + + = + +    − = − = + + + + = + +

∑ ∑

= = − − − + − + + − i j m i n j m n m n m n i n i n i n i n i j i j i j i j

p p p p m n m i p p i p p i n m i n j p i p j p p i j λ λ µ λ µ λ µ λ µ µ λ µ λ

48

SLIDE 49

Traffic theory

Theoretical analysis

Overflow servers – Finite case

 Distributions

( ) ( ) ( ) ( ) ( )

) ,..., 1 ( 1 1 1 1 ) ,..., 1 ( 1 1 1 r) (Brockmeye 1 S S ! ! S S N.B. 1 !

1 1 , n 1 n , 1 n 1 j

m r S r v S S a S K m k a k r K S i x i K p A E i A j A v r v v m A A S

v n m r v n r m n r m n r k r m k r k x j x i x i x i m x i j

n

n i i

j
v

m

m

v

m

r

=         − − = = =         − − − =         + − =               = =         − + − =

+ = + + − = − + + − = = − =

∑ ∑ ∑ ∑ ∑

n j S S p P S i x i K p Q

n j i j m i j x n x i x i x i m x i j n j i

= = =         + − = =

∑ ∑ ∑

= − + + + − = =

per B

Erlang

servers primary

)

1 ( servers

verflow
1

, 1 ,

49

SLIDE 50

Traffic theory

Theoretical analysis

Overflow servers – Finite case

 Grade of service

( ) ( ) ( ) ( )

[ ]

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

n
m

n w p p

m

n n m n m p

m

n p p

n

p p m i m n m n n n

m

n

n
p

w i s

m

n

p
n
w

A E A E A A m A E S S m S A E m n m n S A E n n S S S S S A A E A E A A A iQ A A E A A A E A A

, 1 , 1 , 1 1 , 1 , 1 1 1 , 1 , 1 , 1 , 1 + + + + = + + + +

= = Π = = + Π = + = Π =         − = − = − = = = =

∑

50

SLIDE 51

Traffic theory

 Overflow servers: m = ∞

Theoretical analysis

Overflow servers – Infinite case

51

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

! ,... 2 , 1 ! 1 ! 1 Solution

1

,... 1 , 1 ,... 1 , 1 ,..., 1 1 ) ( chain Markov d 2

1

, , 1 , 1 , , 1 , 1 , , 1 , 1 ,

h A e C h s h A s s v e C C C C v A i v C p p i p p i p p i n i n j p i p j p p i j

h

A

h s h

h

s A h v n v n v j v v

v

n i i j i j i n j i n i n i n i n i j i j i j i j

−

− = − + ∞ = ∞ = = − + − + + −

= = −         − + = −         − = = = + + + = + +    = − = + + + + = + + ∞ = −

∑ ∑ ∑ ∑

m λ µ λ µ λ µ µ λ µ λ

SLIDE 52

Traffic theory

Theoretical analysis

Overflow servers – Infinite case

( )

[ ]

( )

1956)

(Wilkinson

1 1 (Riordan) ! servers

verflow

for give

f

moments Factorial

group
verflow

in traffic

ffered

= traffic carried

2

, 1 1 , 1 1

        − + + + − = ⇒ = = ⇒ = =         = → ∞ =

∑

∞ =

w
w

w w

n
w
n
n

k n k

i

i k i

A A n A A A A E A M A A E A M C C A Q k i k M Q m σ

52

SLIDE 53

Traffic theory

Analysis of channel group

BCC (CPS)

 Channel group

m channels
Ao Poisson
BCC

( )

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

1 1 1 1 1 1 1 1 1 1 1 RVM

1

1

1

1 RVM

1
2

2 2 , 1 , 1 2 , 1 2 2 , 1

> − + + = − + − − + = − + − − − + = = − + + + − − + = − + + + − = =         − + + + − = = < − − = = − − = − + − = − =

s s s s p s s s p p

s
s

p p p

p
p

p p

p
p

p p

m
p

s s

m
s

s s

m
s

s s p s

s
m
s

A m A m A m A A A m A m A A A A m A A A m A A A A m A A A A A m A A A A E A A A A m A E A A A m A E A A A A mA A A A E A A σ σ σ σ σ

53

OFFERED CARRIED LOST

Wilkinson’s Formula

SLIDE 54

Traffic theory

Dimensioning of overflow channels

Wilkinson approach

 Channel group loaded by Ao, σo

2 with RVM >1

 Dimensioning/analysis

( ) ( )

e

m m

e

p

p

p p

e
e
e
e

m

e
e
e

A E A A A m m A A m A A A A E A A A m A

e e

+

= Π = Π Π         − + + + − = =

, 1 2 , 1 2

r

knowing

r

Compute 2 1 1 e knowing and Compute 1 σ σ

54

SLIDE 55

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

( ) ( )

m

i i

m
m

A m E i A m A A E , ! !

,

1

= =

∑

=

( )

[ ]

( )

[ ]

integer 1

1

1 , 1 1 , 1

m A E A m A E

m
m

+ =

− − −

( )

[ ]

( )

real 1

1

, 1

m dy y e A A E

m y A

m
∫

∞ − −

+ =

55

SLIDE 56

Traffic theory

Dimensioning of overflow channels

Overflow traffic - Average

2 4 6 8 10 12 14 10-2 10-1 100 101 m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 m=11 m=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, A

(Erlang)

Valore medio del traffico di trabocco, A

p (Erlang)

56 Average offered traffic, AO (Erlang) Calculation based on Erlang-B formula Average lost traffic, Ap (Erlang)

SLIDE 57

Traffic theory

Dimensioning of overflow channels

Overflow traffic - Average

10 15 20 25 30 35 40 45 50 10-2 10-1 100 101 102

Formula di Erlang-B

Intensita media del traffico offerto, A

(Erlang)

Valore medio del traffico di trabocco, A

p (Erlang)

m=10-50

57 Average offered traffic, AO (Erlang) Average lost traffic, Ap (Erlang) Calculation based on Erlang-B formula

SLIDE 58

Traffic theory

Dimensioning of overflow channels

Overflow traffic - Variance

2 4 6 8 10 12 14 10-2 10-1 100 101 m=1 m=2 m=3 m=4 m=5 m=6 m=7 m=8 m=9 m=10 m=11 m=12 m=13 m=14 m=15 m=16 m=17 m=18 m=19 m=20

Coda M/M/m/0

Intensita media del traffico offerto, A

(Erlang)

Varianza del traffico di trabocco (Erlang

2)

58 Average offered traffic, AO (Erlang) Variance of lost traffic, Ap (Erlang)

SLIDE 59

Traffic theory

Dimensioning of overflow channels

Overflow traffic - Variance

10 15 20 25 30 35 40 45 50 10-2 10-1 100 101 102

Coda M/M/m/0

Intensita media del traffico offerto, A

(Erlang)

Varianza del traffico di trabocco (Erlang

2)

m=10-50

59 Average offered traffic, AO (Erlang) Variance of lost traffic, Ap (Erlang)

SLIDE 60

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

SLIDE 61

Traffic theory

Dimensioning of overflow channels

Sum of multiple flows

 Numerical solution for two Wilkinson equations ⇒ Rapp approximation

∑ ∑

= =

n i pi

pi

n i

A

1 2 2 1

A

σ σ

( ) ( ) ( )

e

m m

e

p

p
e

m

e
e

A E A A A A E A A A z z z A

e e

+

= Π = = = − + =

, 1 , 1 2 2

1 3 σ σ

61

SLIDE 62

Traffic theory

Dimensioning of channel groups

Fredericks model

 Applicable for a traffic Ao, σo

2 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

SLIDE 63

Traffic theory

Dimensioning of channel groups

Fredericks model

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

group

63

SLIDE 64

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 z A E

z

m p , 1

[ ] [ ] [ ] [ ] [ ] [ ]

2 2 2

g
g
g

zA z A z Y Var z X Var A z A z Y zE X E zY X z A Y Var Y E σ = = = = = = = = = =

64

( )

s

A A ≅

SLIDE 65

Traffic theory

Fredericks model

Analysis

        − − =       − − =                 −         − = =                 − =                 − = =         − + + + − =             − + + + − = =         =         = =

p s s s s pz s sz

z

m

z

sz sz s

z

m

z

m

sz

s

p
p

p

z

pz

z

pz pz pz p

z

m

z

m

pz

p

A zA A m zA z A m A z A z A z m z A E A A z z z A E A z A E z A z zA A A A m z A z A zA A A z m A A A z z z A E A z A E z A z zA A 1 1 1 1 1 1

2 , 1 2 2 2 2 , 1 , 1 2 2 2 2 , 1 , 1

σ σ σ σ

65

SLIDE 66

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

f m/zi servers

⇒ Same Fredericks equations if

Aoi, σoi

2 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

1

/z

1

A

n /z

n

A

i

/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

SLIDE 67

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)

f

indip. (stat.

1 1 2 2 2 2

n A A A A z z

i

n i n i

i
i
i
i
i

i p pi

∑ ∑

= =

= = = Π Π σ σ σ σ

( ) { } ( )

) gives ( 1 ,..., formula Erlang d generalize ! 1 ,..., Pr ,...,

1 1 1 1 1

G k k k A G k X k X k k

n i k

z

n i n n n

i i

=         = = = =

∑ ∏

=

π π

i i n i

z X X

∑

=

1

{ }

i pi

z m X − > = Π Pr

67

A

1

/z

1

A

n /z

n

A

i

/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

SLIDE 68

Traffic theory

Dimensioning/analysis of channel groups

Overall equations

( ) ( )

2 2 2 2 2 2 2 , 1

1 1 1 1 1 1 esimo Rivolo Globale

pi p s si i si p s s s s pi

i

si p

s
i
p

pi pi

p
p

p p pi

i

pi p

p

i p pi

z

m p

A A m A A z A zA A m zA A A A A z A A A m z A z A zA A A A A z z z A E i − + =         − − = Π − = Π − =         − + =         − + + + − = Π = Π = Π = Π         = Π − σ σ σ σ σ σ

p p p

i
i

i

A

z A z A z

2 2 2

σ σ σ = = =

68