Velocidade,  banda  ou  taxa  de  transmissão  da  Internet  no  Brasil  

Marcio  Augusto  de  Deus  Engenharia  IP  da  GlobeNet  

Professor  e  Pesquisador  do  InsAtuto  Federal  de  Brasilia  Programa  de  Pós-­‐graduação  em  Engenharia  Elétrica  da  UnB  

Capacidade  do  Canal  

•  Capacidade  =  taxa  máxima  teórica  de  transmissão  para  um  canal.  

•  Teorema  de  Nyquist:    – Largura  de  Banda  =  W  Hz  – Taxa  de  dados  <=  2W  – 2  níveis  de  codificação  – M  é  o  número  de  níveis.  

1  0  

C  =    2Wlog2  M  

bps  ≈  Baud/s  

Capacidade  do  Canal  

•  Teorema  de  Nyquist-­‐Shannon  – 1928  – define  taxa  de  transmissão  máxima  para  um  canal  de  banda  passante  limitada  

– sendo  “W”  a  largura  de  banda,  Nyquist  prova  que  a  amostragem  máxima  sobre  o  canal  é  2W  

– assim,  com  canal  de  W  Hz  transmite-­‐se  2W  Bauds      C  =  2  W  Bauds      C  =  2  W  log2  M  bps  

Capacidade  do  Canal  

•  Bit  x  Baud  –  Baud:  número  de  intervalos  de  sinalização  por  segundo  –  Bit:  0  ou  1  

•  Tipo  do  sinal:  dibit,  tribit,  etc  –  sinal  dibit:  dois  bits  codificados  em  um  intervalo  de  sinalização  

–  1  Baud  =  log2  M  bps      onde  M  é  o  número  de  níveis  sinalizáveis  

00  01  

11  

10  00  

10  

11  

01  

Capacidade  do  Canal  (Teoria  da  Informação)  

•  Shannon  –  levou  em  consideração  a  relação  sinal-­‐ruído  (S/N)  em  dB  

     C  =  W  log2  (1  +  S/N)  bps    

 •  Exemplo:  canal  de  3.000  Hz  com  relação  S/N  de  30  dB  não  transmiArá  em  

hipótese  alguma  a  mais  de  30.000  bps  •  Limite  máximo  teórico  

Relação  entre  capacidade  de  canal  e  banda  efeAva  

C  =  W  log2  (1  +  S/N)  bps  

Banda  EfeAva  (Relacionada  a  fonte  de  Informação)  

(Fonte/Receptor)  

Relação  entre  capacidade  de  canal  e  banda  efeAva  

Forma  simplificada  

C  =  W  log2  (1  +  S/N)  bps  

Banda  EfeAva  (Relacionada  a  fonte  de  Informação,    

neste  caso  na  camada  de  rede)  

(Fonte/Receptor)  

to-­‐ti  =  Intervalo  de  amostragem  

Problema  Operações  muito  simples  para  banda  efeAva  

5   10   15   20  

300Mb  

Qb  

25   tmin  

Se      to-­‐ti=5min    Beff=300Mb/(300)  =  1Mbps  

Se      to-­‐ti=10min    Beff=300Mb/(600)  =  0,5Mbps  

SNMP  ßà  MIB  (Coletas  padronizadas  em  5  min)  

Qual  é  a  banda,  taxa  ou  velocidade  neste  caso?  1Mbps  ou  0,5Mbps  ?  

Medidores  de  velocidade  na  Internet  

Internet  (Acesso+Backbone)  

Usuário  

Usuário  

Ferramenta  Lado  do  usuário  

Ferramenta  

Medidor  

upload  

download  

Metodologia  mais  usada:      

A  base  de  tempo  é  definida  no  intervalo  entre  1  e  30  segundos  

Medidores  de  velocidade  na  internet  

•  São  válidos?  Medem  taxa  instantânea,  normalmente  não  usam  modelos  matemáAcos  para  previsão  de  banda  efeAva.  

•  A  metodologia  do  teste  nem  sempre  é  apresentada.  

•  De  uma  forma  geral  usam  uma  medida  de  taxa  instantânea  e  sem  consideração  do  buffer.  

•  Certamente  usam  bases  de  tempo  diferentes  daquelas  usadas  pelos  provedores.  –  Provedor:  SNPM  ßà  MIB  (1  ou  5  minutos)  

Problemas    Uso  do  cálculo  baseado  apenas  na  média  (1ª  ordem)  é  válido?  

 

Escala  de  tempo

 crescen

te  

Self-­‐similar,  MulAfractal   Poisson  

Notas  em  Banda  EfeAva  

EsAmador  de  Capacidade  (bps)  

K  X[0,t]   Servidor

[ ]∞<<= tK

KteEtKBP

tKX

,0log),(],0[

BP (bps)

Onde: n  K é o buffer n  X[0,t] processo que define a quantidade de bits que estão chegando para serem

encaminhados n  t é o tempo.

[Kelly,  1996]  

t

bits

Interface de um roteador

Base teórica

Dependência  de  longa  e  curta  duração  

Modelos  baseados  em  Poisson  

Modelos  baseados  em  2aordem  Sem  memória    Com  memória  

Definição  da  Banda  EfeAva  (Kelly)  

Buffer  muito  pequeno  

Buffer  tendendo  ao  infinito  

Definição  da  Banda  EfeAva  (Kelly)  

Buffer  muito  pequeno  

Buffer  tendendo  ao  infinito  

Definição  da  Banda  EfeAva  (Kelly)  

Para  uma  fonte  Gaussiana  com  H=0,75  

Z(t)  é  uma  dist  gaussiana  com  média  zero  

para  um  processo  fBm    

Definição  da  Banda  EfeAva  (Kelly)  

onde,                                              são  incrementos  independentes,  então:  Se  

Para  qualquer  valor  de                                          com  s  sendo  incrementado,  temos:  

Onde                                        é  o  maximo  valor  possível  

Norros  

Processos  com  dependência  de  longa-­‐duração:      

954

-0.6;

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 13, NO. 6, AUGUST 1995

We always assume that the process At has stationary increments, i.e., that for any t E % and s1 < . . . < s, the distribution of

(A(t + SI. t + SZ), . . . , A( t + ~ ~ - 1 , t + s n ) )

is independent of t . We also assume square integrability, i.e., EA: < CO. It follows from the stationarity of the increments that EAt = mt for some constant m, the mean rate, and that, denoting v(t) = VarA+

1 2

CovA,,At = - ( v ( t ) + t f ( ~ ) - v ( t - s ) )

for s < t . Thus, the correlation structure (all so called second- order properties) of At is determined by the variance function v(t) alone.

The process At is called short-range dependent if for any s < t 5 U < v the correlation coefficient between A(as,at) and A ( a u , a v ) converges to zero when the time scale a approaches infinity. Otherwise it is called long- range dependent. If At is short-range dependent, v ( t ) is asymptotically linear. In particular, if At is a process with independent increments, say a Poisson process or a compound Poisson process, then w ( t ) is a linear function.

All traffic models traditionally used in teletraffic theory are short-range dependent. From this point of view it was rather shocking that the accurate and extensive LAN traffic measurements conducted in Bellcore [ 151 gave variance curves where v ( t ) grew rather accurately as a fractional power t p , with p taking values strictly between 1 and 2, through half a dozen of orders of magnitude. It became obvious that at least some traffic phenomena had to be studied with long-range dependent models.

The power form v ( t ) = tP is closely related to the fascinat- ing fractal nature of the traffic traces recorded at Bellcore. Indeed, the time-scaled process A,t then has the variance function

V,arAmt = (at)P = aPVarAt

which implies that Aat and apI2At have the same correlation structure, i.e., the centered process At - mt is second-order self-similar. Note that a second-order self-similar process is long-range dependent unless it has uncorrelated increments, and that the (centered) Poisson process is second-order self- similar (with p = 1).

A process Yt is called (strictly) selfsimilar with Hurst parameter (or self-similarity parameter) H if, for any a > 0, the processes Yet and aH& have the same finite-dimensional distributions. Obviously, self-similarity and second-order self- similarity are equivalent for Gaussian processes since their finite dimensional distributions are by definition Gaussian and thus fully characterized by their first and second-order moments. By an important theorem of Lamperti [14], self- similarity is a generic property of wide classes of limit processes. As a general reference to self-similar processes, see, e.g., articles in the collection [5] .

When a model is built on second-order properties alone, a Gaussian process is often the simplest choice. In this paper we shall model the variation of connectionless traffic with a

0 . 2 0 . 4 0 . 6 0 .8 1.0 -0.1

v Fig. 1. A realization of Zt , t E [0,1] with H = 0.8

Gaussian self-similar process, a fractional Brownian motion (FBM). A normalized FBM with Hurst parameter H E [i, 1) is a stochastic process Zt, t E (-m, CO) , characterized by the following properties:

1) 2, has stationary increments; 2) 2, = 0, and EZt = 0 for all t ; 3) EZ? = Jt12H for all t ; 4) Zt has continuous paths; and 5 ) 2, is Gaussian, i.e., all its finite-dimensional marginal

distributions are Gaussian. This process was found by Kolmogorov [12], but relatively little attention was paid to it before the pioneering paper by Mandelbrot and Van Ness [16] (where the FBM also got its present name). In the special case H = l / 2 , Zt is the standard Brownian motion. We have ruled out the other limiting case H = 1 since the respective 2, is a deterministic process with linear paths.

The covariance of the increments in two nonoverlapping intervals is always positive and has the expression

covzt, - zt, , zt, - zt, 1 2

= - ((t4 - t p - (t3 - t p

+(t3 - t 2 y - (t4 - t p )

for tl < t2 5 t 3 < t4. Many features of FBM’s with H > 1/2 are different

from those of most stochastic processes usually appearing in traffic models. They are not Markov processes and not even semimartingales, having nondifferentiable paths with zero quadratic variation. Fig. 1 presents a simulated realization of Zt with H = 0.8, produced with a simple but somewhat inaccurate bisection method given in [17]. Note that the path looks considerably smoother than that of an ordinary Brownian motion.

For the use of the FBM as a traffic model element it is pleasant to note that in spite of the strong correlations, it is ergodic in the sense that the stationary sequence of increments Zn+l - 2, is ergodic (e.g., [2], Theorem 14.2.1).

B. Fractional Brownian TrafJic

model defined as follows. The rest of this paper is devoted to the -study of a traffic

954

-0.6;

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 13, NO. 6, AUGUST 1995

We always assume that the process At has stationary increments, i.e., that for any t E % and s1 < . . . < s, the distribution of

(A(t + SI. t + SZ), . . . , A( t + ~ ~ - 1 , t + s n ) )

is independent of t . We also assume square integrability, i.e., EA: < CO. It follows from the stationarity of the increments that EAt = mt for some constant m, the mean rate, and that, denoting v(t) = VarA+

1 2

CovA,,At = - ( v ( t ) + t f ( ~ ) - v ( t - s ) )

for s < t . Thus, the correlation structure (all so called second- order properties) of At is determined by the variance function v(t) alone.

The process At is called short-range dependent if for any s < t 5 U < v the correlation coefficient between A(as,at) and A ( a u , a v ) converges to zero when the time scale a approaches infinity. Otherwise it is called long- range dependent. If At is short-range dependent, v ( t ) is asymptotically linear. In particular, if At is a process with independent increments, say a Poisson process or a compound Poisson process, then w ( t ) is a linear function.

All traffic models traditionally used in teletraffic theory are short-range dependent. From this point of view it was rather shocking that the accurate and extensive LAN traffic measurements conducted in Bellcore [ 151 gave variance curves where v ( t ) grew rather accurately as a fractional power t p , with p taking values strictly between 1 and 2, through half a dozen of orders of magnitude. It became obvious that at least some traffic phenomena had to be studied with long-range dependent models.

The power form v ( t ) = tP is closely related to the fascinat- ing fractal nature of the traffic traces recorded at Bellcore. Indeed, the time-scaled process A,t then has the variance function

V,arAmt = (at)P = aPVarAt

which implies that Aat and apI2At have the same correlation structure, i.e., the centered process At - mt is second-order self-similar. Note that a second-order self-similar process is long-range dependent unless it has uncorrelated increments, and that the (centered) Poisson process is second-order self- similar (with p = 1).

A process Yt is called (strictly) selfsimilar with Hurst parameter (or self-similarity parameter) H if, for any a > 0, the processes Yet and aH& have the same finite-dimensional distributions. Obviously, self-similarity and second-order self- similarity are equivalent for Gaussian processes since their finite dimensional distributions are by definition Gaussian and thus fully characterized by their first and second-order moments. By an important theorem of Lamperti [14], self- similarity is a generic property of wide classes of limit processes. As a general reference to self-similar processes, see, e.g., articles in the collection [5] .

When a model is built on second-order properties alone, a Gaussian process is often the simplest choice. In this paper we shall model the variation of connectionless traffic with a

0 . 2 0 . 4 0 . 6 0 .8 1.0 -0.1

v Fig. 1. A realization of Zt , t E [0,1] with H = 0.8

Gaussian self-similar process, a fractional Brownian motion (FBM). A normalized FBM with Hurst parameter H E [i, 1) is a stochastic process Zt, t E (-m, CO) , characterized by the following properties:

1) 2, has stationary increments; 2) 2, = 0, and EZt = 0 for all t ; 3) EZ? = Jt12H for all t ; 4) Zt has continuous paths; and 5 ) 2, is Gaussian, i.e., all its finite-dimensional marginal

distributions are Gaussian. This process was found by Kolmogorov [12], but relatively little attention was paid to it before the pioneering paper by Mandelbrot and Van Ness [16] (where the FBM also got its present name). In the special case H = l / 2 , Zt is the standard Brownian motion. We have ruled out the other limiting case H = 1 since the respective 2, is a deterministic process with linear paths.

The covariance of the increments in two nonoverlapping intervals is always positive and has the expression

covzt, - zt, , zt, - zt, 1 2

= - ((t4 - t p - (t3 - t p

+(t3 - t 2 y - (t4 - t p )

for tl < t2 5 t 3 < t4. Many features of FBM’s with H > 1/2 are different

from those of most stochastic processes usually appearing in traffic models. They are not Markov processes and not even semimartingales, having nondifferentiable paths with zero quadratic variation. Fig. 1 presents a simulated realization of Zt with H = 0.8, produced with a simple but somewhat inaccurate bisection method given in [17]. Note that the path looks considerably smoother than that of an ordinary Brownian motion.

For the use of the FBM as a traffic model element it is pleasant to note that in spite of the strong correlations, it is ergodic in the sense that the stationary sequence of increments Zn+l - 2, is ergodic (e.g., [2], Theorem 14.2.1).

B. Fractional Brownian TrafJic

model defined as follows. The rest of this paper is devoted to the -study of a traffic

Variância  é  uma  potência  de  t  

Processo  de  tempo-­‐escala  

954

-0.6;

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 13, NO. 6, AUGUST 1995

We always assume that the process At has stationary increments, i.e., that for any t E % and s1 < . . . < s, the distribution of

(A(t + SI. t + SZ), . . . , A( t + ~ ~ - 1 , t + s n ) )

is independent of t . We also assume square integrability, i.e., EA: < CO. It follows from the stationarity of the increments that EAt = mt for some constant m, the mean rate, and that, denoting v(t) = VarA+

1 2

CovA,,At = - ( v ( t ) + t f ( ~ ) - v ( t - s ) )

for s < t . Thus, the correlation structure (all so called second- order properties) of At is determined by the variance function v(t) alone.

The process At is called short-range dependent if for any s < t 5 U < v the correlation coefficient between A(as,at) and A ( a u , a v ) converges to zero when the time scale a approaches infinity. Otherwise it is called long- range dependent. If At is short-range dependent, v ( t ) is asymptotically linear. In particular, if At is a process with independent increments, say a Poisson process or a compound Poisson process, then w ( t ) is a linear function.

All traffic models traditionally used in teletraffic theory are short-range dependent. From this point of view it was rather shocking that the accurate and extensive LAN traffic measurements conducted in Bellcore [ 151 gave variance curves where v ( t ) grew rather accurately as a fractional power t p , with p taking values strictly between 1 and 2, through half a dozen of orders of magnitude. It became obvious that at least some traffic phenomena had to be studied with long-range dependent models.

The power form v ( t ) = tP is closely related to the fascinat- ing fractal nature of the traffic traces recorded at Bellcore. Indeed, the time-scaled process A,t then has the variance function

V,arAmt = (at)P = aPVarAt

which implies that Aat and apI2At have the same correlation structure, i.e., the centered process At - mt is second-order self-similar. Note that a second-order self-similar process is long-range dependent unless it has uncorrelated increments, and that the (centered) Poisson process is second-order self- similar (with p = 1).

A process Yt is called (strictly) selfsimilar with Hurst parameter (or self-similarity parameter) H if, for any a > 0, the processes Yet and aH& have the same finite-dimensional distributions. Obviously, self-similarity and second-order self- similarity are equivalent for Gaussian processes since their finite dimensional distributions are by definition Gaussian and thus fully characterized by their first and second-order moments. By an important theorem of Lamperti [14], self- similarity is a generic property of wide classes of limit processes. As a general reference to self-similar processes, see, e.g., articles in the collection [5] .

When a model is built on second-order properties alone, a Gaussian process is often the simplest choice. In this paper we shall model the variation of connectionless traffic with a

0 . 2 0 . 4 0 . 6 0 .8 1.0 -0.1

v Fig. 1. A realization of Zt , t E [0,1] with H = 0.8

Gaussian self-similar process, a fractional Brownian motion (FBM). A normalized FBM with Hurst parameter H E [i, 1) is a stochastic process Zt, t E (-m, CO) , characterized by the following properties:

1) 2, has stationary increments; 2) 2, = 0, and EZt = 0 for all t ; 3) EZ? = Jt12H for all t ; 4) Zt has continuous paths; and 5 ) 2, is Gaussian, i.e., all its finite-dimensional marginal

distributions are Gaussian. This process was found by Kolmogorov [12], but relatively little attention was paid to it before the pioneering paper by Mandelbrot and Van Ness [16] (where the FBM also got its present name). In the special case H = l / 2 , Zt is the standard Brownian motion. We have ruled out the other limiting case H = 1 since the respective 2, is a deterministic process with linear paths.

The covariance of the increments in two nonoverlapping intervals is always positive and has the expression

covzt, - zt, , zt, - zt, 1 2

= - ((t4 - t p - (t3 - t p

+(t3 - t 2 y - (t4 - t p )

for tl < t2 5 t 3 < t4. Many features of FBM’s with H > 1/2 are different

from those of most stochastic processes usually appearing in traffic models. They are not Markov processes and not even semimartingales, having nondifferentiable paths with zero quadratic variation. Fig. 1 presents a simulated realization of Zt with H = 0.8, produced with a simple but somewhat inaccurate bisection method given in [17]. Note that the path looks considerably smoother than that of an ordinary Brownian motion.

For the use of the FBM as a traffic model element it is pleasant to note that in spite of the strong correlations, it is ergodic in the sense that the stationary sequence of increments Zn+l - 2, is ergodic (e.g., [2], Theorem 14.2.1).

B. Fractional Brownian TrafJic

model defined as follows. The rest of this paper is devoted to the -study of a traffic

Variância  

Aαt  bαp/2 .  At                        isto  é,  ambos  os  processos  possuem  a  mesma  correlação.  

Yαt  bαH .  Yt                        Y  é  auto-­‐similar  para  ½  <  H  <  1  ou  de  outra  forma,  se  as  distribuições  finitesimais  de  Yαt  e  αH  .  Yt  forem  iguais.  

Norros  

Generalizando:  

NORROS: ON THE USE OF FRACTIONAL BROWNIAN MOTION 957

In the Brownian case H = 1/2, (8) reduces to

l-p . x = const (9)

with p = 7n/C. Here we may roughly say that reducing the relative free capacity 1 - p by half costs doubling the storage size. Note that the link capacity C does not appear in the equation except within p.

With H > l / Z the situation is different. Let us first write

Pa

(8) in the form of a buffer dimensioning formula - VPC p m - H ) )

(1 - o)H/(l--H) = f - 1 ( t )a1 / (2 (1-H) )C(2H--1) l (2 (1~H))

Fig. 4. connections.

Reduction of VPC's by the use of a CLS instead of pairwise ATM

for the virtual waiting time in a queueing system ([20], cited according to [l]).

Dejinition 3. I : The (stationary) fractional Brownian stor- age with input parameters m, a and H and output capacity C > m is the stochastic process X t defined as

X t SUP (At - A , - C( t - s ) ) , t E (-CO, CO) (6) s s t

where At is the fractional Brownian traffic process with parameters m, n and H . In the special case H = l / Z we call X t the Brownian storage.

The stationarity of X t follows from the stationarity of the increments of A t . That X t is almost surely finite is a consequence of Birkhoff's ergodic theorem. Indeed, the ergodicity of Zt implies that limt,, Zt / t = E21 = 0 a.s., which together with the assumption rn < C yields

lim ( A t - A , - C( t - s ) ) = --x as. Y ' - m

Note that the storage process X t is always nonnegative, although the arrival process has also negative increments.

B. A Scaling Law

The fractional Brownian storage X t obeys an interesting scaling law which is easily deduced from the self-similarity of the FBM. Consider the typical requirement that the probability that the amount of work in system exceeds a certain level x must be small. (The value x is the substitute for the buffer size in our infinite storage model.) If the largest allowed buffer saturation probability (or time congestion probability) is t, then the equation

t = P(X > x) (7)

holds at the maximal allowed load. Now, the self-similarity of Zt allows for deriving from (7) a more explicit relation between the design parameters x (buffer space, or requirement) and C (link capacity) and the traffic parameters m, a and H at the critical boundary.

Theorem 3.2: [ 181 Assuming (7), the following equation holds

where the function

\ V I

(10) It is seen that when H is high, a substantial increase in utiliza- tion, say again halving the free capacity, requires a tremendous amount more storage space. Thus we have a new argument for the widely accepted view that for connectionless packet traffic the utilization factor cannot be practically improved by enlarging the buffers.

The scaling relation (8) can also be written as the bandwidth allocation rule

C = + ~ ~ l ( ~ ) a l / ( 2 H ) x - ( l - H ) / H ~ ~ ~ l / ( z H ) (11)

showing that, for H > l / Z , the link requirement C increases slower than linearly in m so that a multiplexing gain is obtained by using links with higher capacity.

As a practical example where the multiplexing gain plays a central role, compare the use of painvise ATM virtual path connections (VPC's) between n + 1 LAN/ATM intenvorking units (IWU's) with the use of a centralized routing function (connectionless server (CLS)). The essential difference be- tween an ATM switch and a CLS is that the former performs switching purely at the ATM layer whereas the latter looks at the network layer address, found in the payload of the first cell of each datagram. The two alternatives, called in [lo] indirect and direct connectionless service provision over an ATM network, respectively, are depicted in Fig. 4 in the case ri = 3.

Denote the bandwidth needed per IWU with centralized routing by Cdir and the total bandwidth needed per IWU with painvise VPC's by Cindir,n. Assume that a fixed amount x of output buffer space is allocated for any VPC and that all traffic streams between the IWU's are equal. It is then seen from (1 1) that

Cin&r,lL = m + (Cdir - m)711-1 ' (2H) . (12)

Note that this expression for the relative multiplexing gain does not contain the unknown factor f - ' ( t ) .

C. The Approximate Queue Length Distribution

No explicit formula for the distribution of the fractional Brownian storage seems to be known. Instead, we shall approach the distribution of X t through a lower bound.

Theorem 3.3: [ 181 Let X t be the fractional Brownian stor- age with parameters m, a , H and C. Then

depends on H but not on m, a , C or x.

Média   Parâmetros  relacionados  ao  2º  momento    

O  EsAmador  de  Banda  EfeAva  FEP  

•  O  buffer  é  aqui  representado  pela  letra  K  •  a  letra  a  representa  a  média  •  H  é  o  parâmetro  de  Hurst    •  σ  representa  o  desvio  padrão  das  amostras  •  Ploss  representa  a  probabilidade  da  perda  de  dados  por  transbordo  do  buffer.    

 para  H  pertencente  ao  intervalo:  0,5  <  H  <  1.  

( ) ( ) HH

Hloss

HH

HHPKaEN−−

−−+=111

1**)ln(*2* σ

Base teórica

[Fonseca]  [Norros]  

Estimador FEPBanda média = 108Mbpsσ = 52,34 e Ploss=2%

100120140160180200220240260280300

0,5 0,540,580,620,66 0,7 0,7

40,780,820,86 0,9 0,9

40,98

H

Ban

da E

stim

ada

(Mbp

s)

b=5minb=1minb=5sb=1sb=0,5s

Base teórica (Curvas de FEP) Capacidade  -­‐        FEP  

C=Ba

nda  Efe3

va  (M

bps)  

ForecasAng    MulAfractal  mBm  

AðtÞ ¼Z t

0

!aþ jrHðxÞxHðxÞ%1 dx;

a  =  Average  k  =  buffer  size  

σ    =  Standard  DevitaAon  H(x)  =  Holder  FuncAon  

MulAfractal  Forecasted  

h(t)=t2+2t-­‐5  

Forecast    MulAfractal              

0

1e+07

2e+07

3e+07

4e+07

5e+07

6e+07

7e+07

0 10000 20000 30000 40000 50000 60000

Acc

umul

ated

traf

fic (

in B

ytes

)

Time (in milliseconds)

TraceEP Multifractal

TraceEP Multifractal

TraceEP Multifractal

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

8e+08

9e+08

1e+09

0 20000 40000 60000 80000 100000 120000 140000

Acc

umul

ated

traf

fic (

in B

ytes

)

Time (in milliseconds)

0

1e+08

2e+08

3e+08

4e+08

5e+08

6e+08

7e+08

8e+08

0 50000 100000 150000 200000 250000 300000

Acc

umul

ated

traf

fic (

in B

ytes

)

Time (in milliseconds)

(a) (b)

(c)

Conclusões  

A  banda  efeAva  a  ser  usada  nas  seguintes  condições:    

-­‐  Bases  de  tempo  iguais  -­‐  O  cálculo  simplificado,  sem  considerar  o  buffer.  -­‐  Lembrar  que  o  roteador  não  possui  velocímetro.    

-­‐  Bases  de  tempos  disAntas:  -­‐  Deve  ser  caracterizada.  -­‐  Deve  ser  definido  um  modelo  que  possa  ser                usado  em  diferentes  bases  de  tempo.