A Teletrafic Theory for the Internet

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A Teletraffic Theory for the Internet

Thomas Bonald, Alexandre Proutiere

France Telecom R&D

also affiliated with Ecole Normale Superieure

thomas.bonald,[email protected]

Tutorial of Performance 2005

October 2005

T. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 1



Teletraffic theory

• Born with the developing telephone network andexemplified by the Erlang formula (1917):

B =A

C

C !

1 + A + A2

2+ . . . + AC

C !

where B = blocking rateC = number of phone linesA = traffic intensity in Erlangs




Teletraffic theory

• Born with the developing telephone network andexemplified by the Erlang formula (1917):

B =A

C

C !

1 + A + A2

2+ . . . + AC

C !

where B = blocking rateC = number of phone linesA = traffic intensity in Erlangs

• More generally, any capacity – demand – performancerelationship




The insensitivity property

• The Erlang formula does not depend on the distributionof call durations (beyond the mean)






• It only requires Poisson call arrivals






• It only requires Poisson call arrivals

• The key to simple and robust engineering rules

→1917 2005




Flow-level modeling of the Internet

• Proposed in 1998 by Massoulié & Roberts:

D =1

C −A

whereD = mean per-bit delay

C = link capacity in bit/sA = traffic intensity in bit/s






D =1

C −A



• Based on fair sharing assumption

(so-called processor-sharing model)






D =1

C −A



• Based on fair sharing assumption

(so-called processor-sharing model)• Insensitive to all traffic characteristics

(beyond the traffic intensity)




Traffic characteristics

flows think−times

• Flows are generated within sessions





flows think−times


• Sessions typically arrive as a Poisson process





flows think−times


• Sessions typically arrive as a Poisson process• Definition of traffic intensity

− flow arrival rate × mean flow size (bit/s)

− like telephone traffic




Some key results

Loss networks Bandwidth sharing(call blocking) (rate adaptation)Erlang, 1917 Telatar & Gallager, 1995

Engset, 1916 Heyman et al, 1997Roberts & Massoulié, 1998

Gimpelson, 1965 Stamatelos & Koukoulidis, 1997

Kaufman, 1981 B & Virtamo, 2005Roberts, 1981Brockmeyer et al, 1948 B & P, 2003Kelly, 1986 B, Massoulié, P & Virtamo, 2005

Ross, 1995




Outline

• Part 1: A single link− Processor-sharing model− Flow rate limits




Outline


• Part 2: Networks− Bandwidth sharing− Application to wired, cellular, ad-hoc networks




Outline

• A brief reminder− The multiclass PS queue− Kelly networks

− Whittle networks• Part 1: A single link

− Processor-sharing model− Flow rate limits



Th l i l PS



The multiclass PS queue

• 2 classes• Poisson arrivals of intensities λ1, λ2

• Exponential service requirements of parameters µ1, µ2

• PS service discipline

φ1(n1, n2) = n1n1 + n2, φ2(n1, n2) = n2n1 + n2

• A reversible Markov process

π(n1, n2) = π(0)(n1 + n2)!

n1!n2!

λn11 λn22

µn11 µ

n22

n 1

n 2

0


K ll t k



Kelly networks

• Multi-server PS (or symmetric) queues

• Poisson arrivals, exponential service requirements

•

Deterministic routes• A product-form distribution

π(n1, n2, m) = π(0)(n1 + n2)!

n1!n2!

λn1λn2

µn11 µ

n22

ν m

m!


Whi l k



Whittle networks

• PS queues with state-dependent service rates

• Poisson arrivals, exponential service requirements

• Balance property

φ1(n1, n2)φ2(n1 − 1, n2)

= φ1(n1, n2−

1)φ2(n1, n2)

Φ(n1, n2) =1

φ1(n1, n2)φ2(n1 − 1, n2) . . . φ1(1, 0)

n2

n10

π(n1, n2) = π(0)Φ(n1, n2)λn11 λn22µn11 µn22


O tli



Outline




The Erlang model



The Erlang model

• Poisson call arrivals of intensity λ

• Exponential call durations of mean τ

• An M/M/C/C queue

π(n) = π(0)

An

n! , n ≤ C

whereA = λ× τ = traffic intensity in ErlangsC = number of circuits


The Erlang model



The Erlang model

• Poisson call arrivals of intensity λ


• An M/M/C/C queue

π(n) = π(0)

An

n! , n ≤ C

whereA = λ× τ = traffic intensity in ErlangsC = number of circuits

• The Erlang formula by PASTA

B = π(C )=AC

C !

1 + A + A2

2+ . . . + AC

C !T. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 13

Insensitivity property




• Example: Erlang distribution with τ 1 + τ 2 = τ

• A Kelly queueing network

π(n1, n2) = π(0)An11

n1!

An22

n2!n1 + n2 ≤ C A1 + A2 = A

n1+n2=n

π(n1, n2) = π(0)n!

n1+n2=n

n!n1!n2!

An11

An22

= π(0) An

n!


Processor sharing model



Processor-sharing model

• Poisson flow arrivals of intensity λ

• Exponential flow sizes of mean σ

•

An M/M/1 queueπ(n) = π(0)ρn

ρ < 1

whereA = λ× σ = traffic intensity in bit/sC = capacity in bit/s

ρ = A/C = link load


Mean per bit delay



Mean per-bit delay

• D, the ratio of the mean flow duration tothe mean flow size


Mean per-bit delay



Mean per-bit delay


• By Little’s law,

n = λ× σD =⇒ D =n

A

n = ρ1− ρ

, ρ = AC

=⇒ D = 1C −A


Mean per-bit delay



Mean per-bit delay



n = λ× σD =⇒ D =n

A

n = ρ1− ρ

, ρ = AC

=⇒ D = 1C −A

• By insensitivity, the mean transfer delay of x bits is x×D


Insensitivity to the flow size distribution



Insensitivity to the flow size distribution

• Example 1: Erlang distribution with σ1 + σ2 = σ

• A Kelly network

π(n1, n2) = π(0) (n1 + n2)!n1!n2!

ρn11 ρn22

ρ1 + ρ2 = ρ

n1+n2=n

π(n1, n2) = π(0)

n1+n2=n

n!n1!n2!

ρn11 ρn22 = π(0)ρn

n1 = ρ11− ρ

n2 = ρ21− ρ

D1 = D2 =1

C −A


Insensitivity to the flow arrival process



Insensitivity to the flow arrival process

• Example 2: Two-flow sessions with σ1 + σ2 = σ

• A Kelly network

π(n1, n2, m) = π(0)(n1 + n2)!

n1!n2!ρn11 ρn22

ν m

m!ρ1 + ρ2 = ρ

n1+n2=n

π(n1, n2, m) = π(0)eν n1+n2=n

n!

n1!n2!ρn11

ρn22

= π(0)eν ρn

n1 =ρ1

1− ρn2 =

ρ21− ρ

, D1 = D2 =1

C −A


Flow throughput



Flow throughput

• γ , the inverse of the mean per-bit delay

γ = C −A

CA

γ


Flow throughput



Flow throughput


• The mean transfer delay of x bits is x/γ

γ = C −A

CA

γ


Flow throughput



Flow throughput


• The mean transfer delay of x bits is x/γ

γ = C −A

(C = 1)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

F

l o w t h

r o u g h p u t

Link load


The Engset model



The Engset model

• K permanent sessions, jump-over blocking


•

Exponential think-time durations of mean ν

−1

• A closed Jackson network

π(n) = π(0)K

nan, n ≤ C

wherea = ν × τ = per source virtual traffic intensity

C = number of circuitsT. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 21

Traffic intensity



a c te s ty

interarrival time

calls, think times,τ ν−1

• Effective per source traffic intensityτ

τ + ν −1=

a

a + 1


Traffic intensity



y

interarrival time

calls, think times,τ ν−1

• Effective per source traffic intensityτ

τ + ν −1=

a

a + 1

• Overall traffic intensity in Erlangs

A = K ×a

a + 1T. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 22




y p p y

• Example: Erlang distribution with τ 1 + τ 2 = τ

• A closed Kelly network

π(n1, n2) = π(0) K !(K − n)!

a

n1

1n1!

a

n2

2n2!

n = n1 + n2 ≤ C a1 + a2 = a

n1+n2=n

π(n1, n2) = π(0)

K

n

n1+n2=n

n!

n1!n2!an11 an22 = π(0)

K

n

an


The Engset formula



g

• Number of ongoing calls seen by a new call

π0(n) ∝ π(n)× (K − n)ν

∝

K − 1n

an, n ≤ C


The Engset formula



g

• Number of ongoing calls seen by a new call

π0(n) ∝ π(n)× (K − n)ν

∝

K − 1n

an, n ≤ C

• Call blocking

B = π0(C ) =(K − 1) . . . (K − C )a

C

C !

1 + (K − 1)a + . . . + (K − 1) . . . (K − C )aC

C !


Engset vs. Erlang



• C = 20• K = 25, 50, 250, ∞

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

B l o c k i n g p r o b a b i l i t y

Link load


The PS model with finite source



• K permanent sessions• Exponential flow sizes of mean σ

• Exponential think-time durations of mean ν −1

• A closed Jackson network

π(n) = π(0)K !

(K − n)!n

wherea = ν × σ = per source virtual traffic intensityC = capacity in bit/s

= a/C = per source virtual link loadT. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 26

Traffic intensity



interarrival time

think times, ν−1

σflows,

• Effective per source traffic intensityσ

σ

C + ν −1

=a

a

C + 1


Traffic intensity



interarrival time

think times, ν−1

σflows,

• Effective per source traffic intensityσ

σ

C + ν −1

=a

a

C + 1

• Overall traffic intensity in bit/s

A = K ×a

a

C + 1T. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 27

Mean per-bit delay



• D, the ratio of the mean flow duration to themean flow size


Mean per-bit delay





n = λ× σD

λ =K

n=0

π(n)ν (K − n) = ν (K − n) =⇒ D =n

K − n

1

a


Mean per-bit delay





n = λ× σD

λ =K

n=0

π(n)ν (K − n) = ν (K − n) =⇒ D =n

K − n

1

a

• By insensitivity, the mean transfer delay of x bits is x×D





• Example: Erlang distribution with σ1 + σ2 = σ• A closed Kelly network

π(n1, n2) = π(0) K !(K − n)! n!n1!n2!n11 n22

n1 + n2 = n1 + 2 =

n1+n2=n

π(n1, n2) = π(0)K !

(K − n)!n, n1 =

1

n n2 =2

n


Finite vs. infinite source



• C = 1• K = 10, 100, 1000, ∞

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

F l o w t h r o

u g h p u t

Link load


Outline






A common flow rate limit



• Maximum flow bit rate c• Poisson flow arrivals of intensity λ

• Exponential flow sizes of mean σ

• If C/c is an integer m, an M/M/m queue

π(n) = π(0)

(ρm)n

n!if n ≤ m

π(m)ρn−m if n > mρ < 1

whereA = λ× σ = traffic intensity in bit/sC = capacity in bit/s

ρ = A/C = link loadT. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 32

Mean per-bit delay



• Exact expression:

D =1

c+

B

C − (1−B)A×

A

C −A

where B is the blocking probability in thecorresponding Erlang model:

B =Am

m!

1 + A + A2

2+ . . . + Am

m!


Mean per-bit delay



• Exact expression:

D =1

c+

B

C − (1−B)A×

A

C −A

where B is the blocking probability in thecorresponding Erlang model:

B =Am

m!

1 + A + A2

2+ . . . + Am

m!

• Bound:

D ≤1

c+

1

C ×

A

C −A


Flow throughput



• C = 1• c = 0.2, 0.4, 0.6, 0.8, 1

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

F l o w t h r o

u g h p u t

Link load

Bound

Exact


PS models with a common rate limit



• Infinite source (cf. Erlang model)− Poisson flow/session arrivals− a multi-server PS queue


PS models with a common rate limit



• Infinite source (cf. Erlang model)− Poisson flow/session arrivals− a multi-server PS queue

• Finite source (cf. Engset model)− non-Poisson flow arrivals− a closed network with one multi-server PS queue andone infinite-server queue


Multirate loss systems



• K classes, bit rates c1, . . . , cK • Poisson call arrivals of intensities λ1, . . . , λK

• Exponential call durations of means τ 1, . . . , τ K • A reversible Markov process

π(n1, n2) = π(0)An11

n1!

An22

n2!n1c1 + n2c2 ≤ C

where

A1 = λ1 × τ 1 = class-1 traffic intensity in ErlangsA2 = λ2 × τ 2 = class-2 traffic intensity in ErlangsC = capacity in bit/s


Blocking probability



• Class-1 blocking probability

B1 =

C −c1<n1c1+n2c2≤C

An11

n1!

An22

n2!

n1c1+n2c2≤C

An11

n1!

An22

n2!

• Class-2 blocking probability

B2 =

C −c2<n1c1+n2c2≤C

An11

n1!

An22

n2!

n1c1+n2c2≤C

An11

n1!

An22

n2!


The Kaufman-Roberts formula



• Assume C and c1, c2 are integers





• Assume C and c1, c2 are integers• Define:

P (n) = n1c1+n2c2=n

An11

n1!

An22

n2!

2

1

c1=1, c2=2






P (n) = n1c1+n2c2=n

An11

n1!

An22

n2!

2

1

c1=1, c2=2

• Then:

P (n) =1

n(A1c1P (n− c1) + A2c2P (n− c2))

with P (n) = 0 if n < 0


Example



• C = 100• ck = 1, 5, 10, 30

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4


Link load


Example (cont’d)



• C = 100• ck = 1, 30

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4


Link load


Multirate PS systems



• K classes, bit rates c1, . . . , cK • Poisson flow arrivals of intensities λ1, . . . , λK

• Exponential flow sizes of means σ1, . . . , σK • A Whittle network

π(n1, n2) = π(0)Φ(n1, n2)An11 An2

2

A = A1 + A2 < C

whereA1 = λ1 × σ1 = class-1 traffic intensity in bit/s

A2 = λ2 × σ2 = class-2 traffic intensity in bit/sC = capacity in bit/s

provided

the allocation is balanced(so-called balanced fairness)T. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 41

Balance property



• The product of service rates φ1, φ2 (allocated bit rates)does not depend on the considered path

Φ(n1, n2) =

1

φ1(n1, n2)φ2(n1 − 1, n2) . . . φ1(1, 0)

n2

n10


Balance property



• The product of service rates φ1, φ2 (allocated bit rates)does not depend on the considered path

Φ(n1, n2) =

1

φ1(n1, n2)φ2(n1 − 1, n2) . . . φ1(1, 0)

n2

n10

• A necessary and sufficient condition for insensitivity!


Max-min fairness



• Allocation by water-filling

φ1(n1, n2) = n1c1

φ2(n1, n2) = n2c2

φ1(n1, n2) = C − n1c1

φ2(n1, n2) = n2c2

φ1(n1, n2) = n1n1+n2

C

φ2(n1, n2) = n2n1+n2 C


Max-min fairness



• Allocation by water-filling

φ1(n1, n2) = n1c1

φ2(n1, n2) = n2c2

φ1(n1, n2) = C − n1c1

φ2(n1, n2) = n2c2

φ1(n1, n2) = n1n1+n2

C

φ2(n1, n2) = n2n1+n2 C • The balance property is violated!


Balanced fairness



• Allocation by balancing the service rates

φ1(n1, n2) =Φ(n1−1, n2)

Φ(n1, n2), φ2(n1, n2) =

Φ(n1, n2−1)

Φ(n1, n2)


Balanced fairness



• Allocation by balancing the service rates

φ1(n1, n2) =Φ(n1−1, n2)

Φ(n1, n2), φ2(n1, n2) =

Φ(n1, n2−1)

Φ(n1, n2)

• A unique balance function

Φ(n1, n2) =1

n1!cn11×

1

n2!cn22 if n1c1 + n2c2 < C

Φ(n1, n2) =1

C (Φ(n1−1, n2) + Φ(n1, n2−1)) otherwise




A recursive formula




P (n) =

n1c1+n2c2=nΦ(n1, n2)A

n11 A

n22

2

1

c1=1, c2=2

• Then: P (n) = 1n

(A1c1P (n− c1) + A2c2P (n− c2))

P ≡n>C

P (n) =A1

C −A

C −c1<n≤C

P (n)+A2

C −A

C −c2<n≤C

P (n)


Example



• C = 100• ck = 1, 5, 10, 30

0

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1

F l o w t h

r o u g h p u t

Link load

BoundExact




Outline



• Part 1: A single link− The processor-sharing model− Flow rate limits

•

Part 2: Networks− Bandwidth sharing− Application to wired, cellular, ad-hoc networks

We consider:− data networks only− no flow rate limit


The linear network



C C

1

2 3

• Network state: n = (n1, n2, n3) numbers of active flows on eachroute

• Bandwidth allocation: φ(n) = (φ1(n), φ2(n), φ3(n)) ∈ C

φ1(n) + φ2(n) ≤ C

φ1(n) + φ3(n) ≤ C

φ1

φ2

φ3


The linear network



• A network of PS queues with state-dependent service rates- Each class-i flow served at rate φi(n)/ni

(TCP fairly shares the bandwidth among connections with the

same characteristics)

- A PS node per flow class

1

2 3

C C

ρ2

ρ1

ρ3

φ1(n)

φ2(n)

φ3(n)




Outline




•

Part 2:Networks− Bandwidth sharing− Application to wired, cellular, ad-hoc networks


The model



• N flow classes: flows of the same class require the samenetwork resources (e.g., a set of links in wired network)


The model




• Network state: n = (n1, . . . , nN ), ni number of class-i flows


The model





• Bandwidth allocation: φ(n) = (φ1(n), . . . , φN (n)), class-i flows

served at rate φi(n). C, a convex, compact, and monotone

capacity set


The model







capacity set

• Class-i flows generated in sessions according to a Poisson

process, traffic intensity ρi = λi/µi. ρ = (ρ1, . . . , ρN )


The model







capacity set

• Class-i flows generated in sessions according to a Poisson

process, traffic intensity ρi = λi/µi. ρ = (ρ1, . . . , ρN )

•Issues:- Is the network stable?

- What is the mean time to transfer a class-i flow?




Utility-based allocations



• Usual allocation are based on the notion of utility

max

i niU (φi(n)/ni)

φ(n) ∈ C- Max throughput: U (r) = r

- Proportional fairness (Kelly’97): U (r) = ln r

- Minimal potential delay (Massoulie-Roberts’99): U (r) = 1/r

- α-bandwidth sharing (Mo-Walrand’00): U (r) = r1−α/1 − α

α = 0 max throughput

α → 1 proportional fairness

α = 2 minimal potential delay

α → +∞ max-min fairness (Rawls’71, Bertsekas-Gallager’87)


Utility-based allocations in practice



• Decentralized algorithms- e.g., A model of TCP: proportional fairness in wired networks can

be arbitrarily closely approximated by the following decentralized

algorithm, Kelly-Maullo-Tan’98: λi = φi/ni,

∂λi

∂t= wi − λi(t)

l∈i

pl(j:l∈j

λj)


Utility-based allocations in practice



• Decentralized algorithms- e.g., A model of TCP: proportional fairness in wired networks can

be arbitrarily closely approximated by the following decentralized

algorithm, Kelly-Maullo-Tan’98: λi = φi/ni,

∂λi

∂t= wi − λi(t)

l∈i

pl(j:l∈j

λj)

• Centralized algorithms- e.g. the gradient-based algorithm (for any capacity set, any utility

function), Stolyar’05: at time t choose φ∗ ∈ C such that

φ∗ = argmaxi

U (λi(t))φ∗i , λi(t + 1) = (1 − β )λi(t) + βφ∗i (t)

The proportional fair algo (Tse) in CDMA/HDR system is built that

wayT. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 56

Utility-based allocations: flow-level stability

If ll i b d ili f i f h f



• If ρ ∈ C, allocations based on utility functions of the formλ1−α/(1 − α), with α > 0, stabilize the network

Proof. Using classical fluid limit and the following Lyapounov function

f (λ) =i

ρ−αiλα+1i

α + 11/µi


Utility-based allocations: flow-level stability

If C ll ti b d tilit f ti f th f



• If ρ ∈ C, allocations based on utility functions of the formλ1−α/(1 − α), with α > 0, stabilize the network

Proof. Using classical fluid limit and the following Lyapounov function

f (λ) =i

ρ−αiλα+1i

α + 11/µi

• The linear network paradox with the max throughput allocation,B.-Massoulie’01

- Stability condition: ρ ∈ K = ρ : ρ1 < (1− ρ2)(1 − ρ3) C -

Maximizing the throughput in all static scenarios can minimize thethroughput in a dynamic scenario


Utility-based allocations: performance

P ti l f i h li id t k



• Proportional fairness on homogeneous linear, grid networks(B.-Massoulie’01)

- On these networks, PF is

insensitive- The stationary distribution is

explicit

• For a general non trivial capacity set C, almost all utility-based

allocations are sensitive, e.g., maxmin is always sensitive


Utility-based allocations: performance

Proportional fairness on homogeneo s linear grid net orks



• Proportional fairness on homogeneous linear, grid networks(B.-Massoulie’01)

- On these networks, PF is

insensitive- The stationary distribution is

explicit

• For a general non trivial capacity set C, almost all utility-based

allocations are sensitive, e.g., maxmin is always sensitive• How can we predict the performance of these usual

allocations?


Balanced fairness

Introduced by B P ’03 as the most efficient insensitive



• Introduced by B.-P. 03 as the most efficient insensitivebandwidth allocation


Balanced fairness

•Introduced by B P ’03 as the most efficient insensitive



Introduced by B.-P. 03 as the most efficient insensitivebandwidth allocation

• Insensitivity implies the existence of a balance function Φ such

that∀i, φi(n) =

Φ(n − ei)

Φ(n)


Balanced fairness






that∀i, φi(n) =

Φ(n − ei)

Φ(n)

• Balanced fairness satisfies the capacity constraintsΦ(n − e1)

Φ(n), . . . ,

Φ(n − eN )

Φ(n)

∈ C


Balanced fairness






that∀i, φi(n) =

Φ(n − ei)

Φ(n)

• Balanced fairness satisfies the capacity constraintsΦ(n − e1)

Φ(n), . . . ,

Φ(n − eN )

Φ(n)

∈ C

• Efficiency means that φ(n) belongs to the border of C

Φ(0) = 1, Φ(n) = minα : (Φ(n − e1)

α, . . . ,

Φ(n − eN )

α) ∈ C


Balanced fairness (cont’d)

•Invariant measure with balanced fairness



Invariant measure with balanced fairness

π(n) = π(0)Φ(n)ρn11 . . . ρnN N







π(n) = π(0)Φ(n)ρn11 . . . ρnN N

• Network stability under balanced fairness if and only ifn

Φ(n)ρn11 . . . ρnN N < +∞







π(n) = π(0)Φ(n)ρn11 . . . ρnN N


Φ(n)ρn11 . . . ρnN N < +∞

• Balanced fairness maximizes the probability the system is

empty







π(n) = π(0)Φ(n)ρn11 . . . ρnN N


Φ(n)ρn11 . . . ρnN N < +∞

• Balanced fairness maximizes the probability the system is

empty

•The only possible Pareto-efficient and insensitive allocation isbalanced fairness


Balanced fairness: stability

• If ρ ∈ C, balanced fairness stabilizes the network˜



If ρ ∈ C, balanced fairness stabilizes the networkProof. Let Φ be a balance function satisfying the network capacity

constraints, i.e.,

(Φ(n − e1)

Φ(n), . . . ,

Φ(n − eN )

Φ(n)) ∈ C,

then Φ(n) ≤

˜Φ(n)

.

Since (1 + )ρ ∈ C, the balance function Φ corresponding to the static

allocation φ(n) = (1 + )ρ satisfy the network constraints and is stable.

Finally, n

Φ(n)ρn ≤n

Φ(n)ρn < +∞


Balanced fairness: performance

•Under the stability condition, the performance can be evaluated

li itl i Littl ’ f l



U de t e stab ty co d t o , t e pe o a ce ca be e a uatedexplicitly using Little’s formula

γ i =ρi

E i[ni]and the network stationary distribution

π(n) = π(0)Φ(n)ρ

n1

1 . . . ρ

nN

N


Outline

•Part 1: A single linkThe processor sharing model



Part 1: A single link− The processor-sharing model− Flow rate limits



Static routing

•A wired network is a set of L links and K routes where eachroute k is a subset of links Capacity of link l C



route k is a subset of links. Capacity of link l, C l


Static routing

•A wired network is a set of L links and K routes where eachroute k is a subset of links Capacity of link l C




• Static routing (N = K - the class of a flow is defined by a

route): capacity set

C = φ : φA ≤ C = (C 1, . . . , C L)

A is a N × L matrix, Akl = 1 if l ∈ k, Akl = 0 otherwise


Static routing

•A wired network is a set of L links and K routes where eachroute k is a subset of links Capacity of link l Cl




• Static routing (N = K - the class of a flow is defined by a

route): capacity set

C = φ : φA ≤ C = (C 1, . . . , C L)

A is a N × L matrix, Akl = 1 if l ∈ k, Akl = 0 otherwise

• Stability condition: ∀l,

k:l∈k ρk < C l


Static routing: balanced fairness

•An insensitive allocation is defined by a balance function Φ





•An insensitive allocation is defined by a balance function ΦC it t i t



• Capacity constraints

∀l,k:l∈k

Φ(n − ek)

Φ(n) ≤ C l

equivalent to

∀l, Φ(n) ≥ 1C l

k:l∈k

Φ(n − ek)



• An insensitive allocation is defined by a balance functionΦ

• C cit constr ints



• Capacity constraints

∀l,k:l∈k

Φ(n − ek)

Φ(n) ≤ C l

equivalent to

∀l, Φ(n) ≥ 1C l

k:l∈k

Φ(n − ek)

• Balanced fairness recursively defined byΦ(0) = 1

and

Φ(n) = maxl

1

C l

k:l∈k

Φ(n − ek)


The linear network

• A homogeneous 2-link line2 3



C C

1

n3

n1

n2

n+

+


The linear network

• A homogeneous 2-link line

• Both links are saturated:

2 3




e.g. if n1 > 0 and n2 > 0,

Φ(n) = Φ(n − e1) + Φ(n − e2)

C C

1

n3

n1

n2

n+

+


The linear network

• A homogeneous 2-link line

• Both links are saturated: 1

2 3




e.g. if n1 > 0 and n2 > 0,

Φ(n) = Φ(n − e1) + Φ(n − e2)

• Φ(n) is the number of direct

paths from n to 0

Φ(n) =

n1 + n2 + n3

n1 This is proportional fairness

C C

1

n3

n1

n2n

+

+


A symmetric tree network

• A trunk and several branches with identical capacities

φ3



2

1

1

1

φ1 φ2

φ3

C

C = (2, 1, 1, 1) A =

1 1 0 0

1 0 1 0

1 0 0 1


A symmetric tree network (cont’d)

• Flow throughputs: γ 1 = γ 2 = γ 3 =(2−)(3−)(6+)

(4−)(9+) , where = ρ1 + ρ2 + ρ3



ρ1 + ρ2 + ρ3

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2

F l o w t h r o u g h p u t

Traffic intensity

Max-min fairProportional fair

Balanced fair

• For max-min and proportional fairness, simulations with

Poisson flow arrivals and exponentially distributed flow sizes.T. Bonald, A. Proutiere, A Teletraffic Theory for the Internet – 68

Recursive algorithm

• State space decomposition

Ω N3 0 Ω Ω Ω Ω Ω Ω Ω



Ω = N3 = 0 + Ω1 + Ω2 + Ω3 + Ω12 + Ω13 + Ω23 + Ω123

where Ω I = n : ni > 0 iff i ∈ I


Recursive algorithm (cont’d)

• Normalization constant G(ρ) = n

Φ(n)ρn

G( ) 1 G ( ) G ( ) G ( ) G ( ) G ( ) G ( ) G ( )



G(ρ) = 1+G1(ρ)+G2(ρ)+G3(ρ)+G12(ρ)+G13(ρ)+G23(ρ)+G123(ρ)

where G I =

n∈ΩI π(n)




Φ(n)ρn

G( ) 1+G ( )+G ( )+G ( )+G ( )+G ( )+G ( )+G ( )




where G I =

n∈ΩI π(n)• We have Gi(ρ) = ρi

1−ρi, Gij(ρ) =

ρiρj(1−ρi)(1−ρj)






Φ(n)ρn

G(ρ) 1+G (ρ)+G (ρ)+G (ρ)+G (ρ)+G (ρ)+G (ρ)+G (ρ)




where G I =

n∈ΩI π(n)• We have Gi(ρ) = ρi

1−ρi, Gij(ρ) =

ρiρj(1−ρi)(1−ρj)

• Recursion

G123(ρ) =ρ1G23(ρ) + ρ2G13(ρ) + ρ3G12(ρ)

2 −

•Flow throughtput

γ i =

∂

∂ρiln G(ρ)

−1


The store-and-forward bound

• Store-and-forward policy: the flows on a given route are

transmitted sequentially on each link of this route



Each link fairly shares its capacity among active flows.


The store-and-forward bound

• Store-and-forward policy: the flows on a given route are

transmitted sequentially on each link of this route



Each link fairly shares its capacity among active flows.

1

2 3

C C

ρ1

ρ2 ρ3

C C

PS PS




Fast routing

• Each flow class is assigned a set of routes




Fast routing


• All flows of class i chooses one route in the subset si at any



time.

A set R of N × K matrices such that if R ∈ R, Rik = 1 if class-iflows take route k and Rik = 0 otherwise


Fast routing





time.


• At any time a routing scheme R ∈ R is chosen


Fast routing





time.


• At any time a routing scheme R ∈ R is chosen

• Capacity set

C = convex hull of φ : ∃R ∈ R,φRA ≤ C


Fast routing (cont’d)

• Example

1 0

1 0

1 0



C = (1, 1), A =

1 0

0 1

, R =

1 0

0 1

or R =

0 1

0 1

1

1

or

class 1

class 3

class 2

φ1 φ2

φ3

C


Traffic splitting

• Each class i can use all routes in the set si at the same time




Traffic splitting


• A set S of N × K stochastic matrices such that if S ∈ S ,

S f ll k if k



S ik = 0 for all k except if k ∈ si

- S ik is the proportion of the total bandwidth φi (offered to class-itraffic) class-i flows get on route k


Traffic splitting


• A set S of N × K stochastic matrices such that if S ∈ S ,

S 0 f ll k t if k



S ik = 0 for all k except if k ∈ si

-S ik

is the proportion of the total bandwidthφi

(offered to class-i

traffic) class-i flows get on route k

• Capacity set

C = φ : ∃S ∈ S ,φSA ≤ C


Traffic splitting (cont’d)

• Example

C = (1, 1), A =

1 0

0 1

,

φ2

2



0 1

S =

1 0

α 1− α

0 1

1

1

class 1

class 3

andclass 2

φ1 φ3

C

1

2

1 1


Traffic splitting (cont’d)

• Performance of balanced fairness

γ 1 = γ 3 =2(2 − )(3 − )

12 5, γ 2 = 2 −



12 − 5

0

0.5

1

1.5

2

0 0.5 1 1.5 2

Traffic intensity

Classes 1,3


Balanced fair

0

0.5

1

1.5

2

0 0.5 1 1.5 2

F l o w t h

r o u g

h p u t

Traffic intensity

Class 2


Balanced fair


Outline




• Part 2: Networks

− Bandwidth sharing− Application to wired, cellular, ad-hoc networks


Downlink of cellular networks

• Formalism: a set of M transmission profiles, each

corresponding to a particular allocation of downlink radio

resources







resources



• At any time, a transmission profile is chosen





resources




• C is the M × N capacity matrix such that C mi is the rate

allocated to class-i flows in transmission profile m





resources




• C is the M × N capacity matrix such that C mi is the rate

allocated to class-i flows in transmission profile m

• T the set of M -dimensional non-negative row vector summingto 1. τ ∈ T corresponds to a particular schedule


A single cell

• We compare two access technologies

- The ideal broadcast channel

- TDMA access mode




A single cell



- TDMA access mode



• For the sake of clarity: mobility/fading are not modelled


A single cell



- TDMA access mode




• User positions determine their feasible rate: when all resources

are allocated to class-i flows, they receive a rate C i


A single cell



- TDMA access mode




• User positions determine their feasible rate: when all resources

are allocated to class-i flows, they receive a rate C i

C 2

class 1 class 2

C 1


TDMA

• One user scheduled at a time: N transmission profiles (one per

class), C = diag(C 1, . . . , C N )

orφ2



class 1 class 2

3 1

3

φ1

1 C


TDMA

• One user scheduled at a time: N transmission profiles (one per

class), C = diag(C 1, . . . , C N )

orφ2



class 1 class 2

3 1

3

φ1

1 C

• Scheduling

- Fair time sharing: realized by proportional or balanced fairness

φ1(n) = 3n1/(n1 + n2), φ2(n) = n2/(n1 + n2)

- Fair rate sharing: realized by max-min fairness

φ1(n) = n1/(n1 + 3n2), φ2(n) = 3n2/(n1 + 3n2)


Performance of TDMA

• Fair time sharing γ 1 = 3(1 − ), γ 2 = 1 − where = ρ1/3 + ρ2

• Fair rate sharing: A DPS queue, results by Fayolle et al’81



0

0.5

1

1.5

2

2.5

3

0 0.25 0.5 0.75 1 1.25 1.5Traffic intensity

Class 1


Balanced fair

0

0.5

1

1.5

2

2.5

3

0 0.25 0.5 0.75 1 1.25 1.5

F l o w t h

r o u g h p u t

Traffic intensity

Class 2


Balanced fair


The broadcast channel

• A Gaussian broadcast channel

• Capacity set

C φ φ W l 1P 1

φ W l 1P 2

P P P



C = φ : φ1 ≤ W log21 +P1

N 1 , φ2 ≤ W log2 1 +P2

N 2 + P 1 , P 1+P 2 ≤ P

class 1 class 2

3 1

andφ2

3

φ1

1 C


Performance of the broadcast channel

2

2.5

3

Class 1


Balanced fair

2

2.5

3

p u t

Class 2


Balanced fair



0

0.5

1

1.5

2

0 0.25 0.5 0.75 1 1.25 1.5 1.75Traffic intensity

0

0.5

1

1.5

2

0 0.25 0.5 0.75 1 1.25 1.5 1.75

F l o w t h r o

u g h

Traffic intensity


Cell coordination

• 2 interfering base stations - 3 transmission profiles

33

φ2



class 1 class 2

class 1 class 2

class 1 class 2

3

2 2

(1)

(2)

(3)

3

2

2

C

φ1

C = 3 00 3

2 2


Cell coordination: performance

2

2.5

3

h p u t

Max-min fairProportional fairBalanced fair



0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5 4

F l o w t h

r o

u g h

Traffic intensity


Outline






− Bandwidth sharing− Application to wired, cellular, ad-hoc networks


Ad-hoc networks

• A set of L node-to-node links and K routes




Ad-hoc networks


• A incidence matrix (i.e., Akl = 1l∈k)




Ad-hoc networks



• M transmission profiles used one at a time: C is the M × L



matrix, with M ml = capacity of link l in profile m


Ad-hoc networks



• M transmission profiles used one at a time: C is the M × L



matrix, with M ml = capacity of link l in profile m• N = K classes of flow


Example

3 links2 transmission profiles

2

class 3

2

(1)

class 1

class 2



2(2)

φ3

φ1

C

φ2

1

2

2

A =

1 1 10 1 0

0 0 1

C =

2 0 2

0 2 0


Example (cont’d)

1.5

2

Class 1, 3


Balanced fair1.5

2

g h p u t

Class 2


Balanced fair



0

0.5

1

0 0.5 1 1.5 2Traffic intensity

0

0.5

1

0 0.5 1 1.5 2

F l o w t h r

o u g

Traffic intensity


Summary

• Any data network can be represented by a network of

state-dependent PS queues




Summary



• Usual utility-based allocations

- can be implemented in a centralized or distributed way



can be implemented in a centralized or distributed way

- stabilize the network at flow level

- are sensitive (performance unknown)


Summary



• Usual utility-based allocations

- can be implemented in a centralized or distributed way



can be implemented in a centralized or distributed way

- stabilize the network at flow level

- are sensitive (performance unknown)

• Balanced fairness (the most efficient insensitive allocation)

- stabilizes the network at flow level

- has an explicit performance that approximates that of usual

utility-based allocations

- A distributed algorithm to implement BF is not known yet


A Teletrafic Theory for the Internet

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