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Modeling Max-Min Fairness for Elastic Flows in Telecommunication

Networks

Alain Dupuisa, Fabrice Guillemina and Philippe Robertb

aFrance Telecom R&D, 2, Avenue Pierre Marzin, 22300 Lannion (France)

bAlgorithms Project, INRIA, F-78150 Rocquencourt (France)

An analytical model is developed in this paper to study max-min fairness in telecommu-nication networks. This bandwidth sharing principle is very popular in packet networksand is used to allocate bandwidth to elastic ows sharing the capacity of the transmissionlinks of a network. The model considered in the present paper is a linear single networklink, on which originating points of data ows appear as a Poisson process; the length ofa data ow is exponential with unit mean and the link transmission capacity is taken asunity. While the de�nition of the max-min fair share is algorithmically clear, any analyticalcomputation rapidly proves intractable. To overcome this technical di�culty, we developsimple heuristic approximations. It turns out that the obtained formulae yield a satisfac-tory approximation of the max-min fair share. Moreover, one approximation proves fairlyaccurate when considering the amount of unused bandwidth due to max-min fairness.

1. INTRODUCTION

Max-min fairness is a very popular bandwidth sharing principle, which has been ad-vocated for a long time in the domain of data networks (see for instance the de�nitiongiven by Bertsekas and Gallager [2]). The objective of max-min fairness is to maximize,under link transmission capacity constraints, the minimum bandwidth that a data owcan obtain when traversing a network. Max-min fairness basically relies on the followingprinciple: in the domain of feasible bandwidth allocations, the rate �r allocated to thedata ow r cannot be increased without decreasing the rate �r0 allocated to some otherdata ow r0 such that �r0 < �r.Max-min fairness can be formulated in terms of an optimisation problem as follows: for

every data ow r, there is at least one link l along the route of this data ow such thatXr02Al

�r0 = Cl and �r = maxf�r0 ; r0 2 Alg; (1)

where Cl is the capacity of the link l and Al is the set of data ows crossing the link l.The problem of bandwidth sharing has recently been the subject of great interest, es-

pecially in connection with the problem of bandwidth allocation to elastic ows in the

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Internet (see for instance [6]). Those ows represent today the prevalent part of Inter-net tra�c carried over telecommunications networks and understanding how bandwidthis shared between di�erent ows crossing a link of the network is of crucial interest, forboth users and network operators. Users appreciate the bandwidth sharing principles usedwithin the network via the throughput of their applications, while network operators areinterested in optimising the use of the bandwidth of their transmission links. Fairnessis thus a design criterion for data networks. Also, fairness can be used to introduce, forinstance, tari� options.Max-min fairness is not the only fairness principle, which has been introduced so far

in the literature. Several fairness policies are discussed, for instance in [7,6]. Some ofthese new fairness criteria are intended to remedy some weak points of max-min fairness,for instance by introducing the concept of weighted fairness and minimum bandwidth per ow. However, max-min fairness is the fundamental principle, which has been used in thedesign of popular data transmission protocols (for instance TCP). More recently, max-minfairness has been advocated in the design of the available bit rate service by the ATMForum (see [1] for a survey of bandwidth sharing algorithms introduced in the context ofthe available bit rate service). Max-min fairness thus still appears as a fundamental designcriterion, even though recent studies (for instance [4]) has questioned the appropriatenessof max-min fairness for sharing bandwidth among elastic ows.In spite of the fact that max-min fairness has existed for a long time in the literature on

data networks, mathematical modeling of the behavior of max-min fairness in a networkis not very frequently investigated. In general, max-min fairness is studied via extensivesimulation in complex network con�gurations. In this paper, we study max-min fairnessfrom an analytical point of view. For this purpose, we consider a simplistic networkcon�guration, namely a single link network. Data ows appear on this link as a Poissonprocess with intensity u and have exponentially distributed lengths.While the algorithmic de�nition of the max-min fair share is clear, any analytical com-

putation rapidly turns out to be intractable. This is why we develop some heuristic ap-proximations, which are quite fair when considering the max-min fair share, but whichare less accurate for estimating the amount of unused bandwidth. This latter quantitycorresponds to the proportion of the link bandwidth, which is not used because of thebandwidth sharing principle.The organization of this paper is as follows: In Section 2, the linear model network model

is presented and some basic results for the M=M=1 system are recalled. In Section 3,di�erent approximations of the max-min fair share are developed. These approximationsare then used to estimate the amount of lost bandwidth. Some concluding remarks arepresented in Section 4.

2. NETWORK MODEL AND PRELIMINARY RESULTS

We consider a linear network (for instance a bus with in�nite length), which is repre-sented by the real half line [0;1). The transmission capacity of this single link network istaken as unity. We assume that data ows randomly appear on the link. A data ow is

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then characterized by an originating point and a length. We assume that the originatingpoints of data ows are distributed according to a Poisson process with intensity u andthat the length of data ows is exponentially distributed with unit mean.In this model, several data ows may share the bandwidth of the transmission link,

giving rise to clumps (see Figure 1). By de�nition, a clump is a set of data ows, whichshare a common portion of the transmission link.

clump of data flows

number ofsharing a portion

of the transmission linkin the clump

network link

data flows

Figure 1. Linear network model.

We assume that in a clump, the bandwidth of the transmission link is shared among thedi�erent data ows according to max-min fairness. Moreover, we suppose that the rateadaptation of the data ows is immediate. There are thus no time aspects in the modelconsidered. In other words, we consider a con�guration at a given time and we study theallocation of bandwidth to each data ow. Temporal aspects will be addressed in furtherstudies.The di�culty in max-min fairness consists of computing the bandwidth allocated to a

data ow. While the algorithmic de�nition of max-min fairness is simple and quite intuitive,the analytical derivation of the exact max-min fair share requires painful computations,even in extremely simplistic con�gurations as in the above linear network. The goal of thispaper is to develop approximations for the max-min fair share as well as for the amountof unused bandwidth in a clump. This last quantity may be interpreted as a performancemeasure by the network operator because it corresponds to the amount of the networkbandwidth, which is not used because of the bandwidth allocation principle adopted forsharing the bandwidth among the di�erent active connections.To derive heuristic approximations of the two above quantities, we note that clumps are

distributed as busy periods in an M=M=1 queueing system with input rate u and unitservice rate. Before proceeding with some analytical results on the above linear network

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model, let us recall some basic elements about the M=M=1 queue.Charlier polynomials form the fundamental orthogonal polynomial system associated

with theM=M=1 with input rate u and unit service rate. Charlier polynomials fQm(x; u)gare de�ned by the recursion: Q�1(x; u) � 0, Q0(x; u) � 1, and for m � 0,

uQm+1(x; u) + (x�m� u)Qm(x; u) +mQm�1(x; u) = 0: (2)

The generating function of Charlier polynomials is given by

1Xj=0

uj

j!Qj(x; u)z

j = euz(1� z)x: (3)

An easy consequence of the above relation is that Charlier polynomials satisfy

kXj=0

uj

j!Qj(x; u) =

uk

k!Qk(x� 1; u): (4)

For numerical purposes, we note that Q0(x; u) � 1 and the above relation implies

Qk(�1; u) =k!

uk

kXj=0

uj

j!: (5)

From the general relation

Q0n(0; u) = �

n�1Xk=0

k!

uk+1

kXj=0

uj

j!;

we deduce that

Q0n(0; u)

def=

@

@xQn(x; u)

����x=0

= �1

u

n�1Xk=0

Qk(�1; u);

and hence, we have the recursion

Q0n(0; u) = Q0

n�1(0; u)�1

uQn�1(�1; u): (6)

From relation (4), we have

Q0k(�1; u)�

k

uQ0

k�1(�1; u) = Q0k(0; u); Q0

k(�2; u)�k

uQ0

k�1(�2; u) = Q0k(�1; u):

(7)

The above relations allow us to recursively compute the quantities Q0k(0; u), Q

0k(�1; u) and

Q0k(�2; u), which will be of important use in the following.For a given initial state a 2 N, let Tb denote the �rst time the occupation process f�tg

of the M=M=1 queue hits the level b 2 N:

Tb = infft > 0 : �t = b j �0 = ag: (8)

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When a � b, the Laplace transform of the random variable Tb is given by Tak�acs formula[5] (see also [3]):

eTb(s) def= Ea

�e�sTb

�=

Qa(�s; u)

Qb(�s; u); (9)

where Qn(x; u) is the nth Charlier polynomial.

3. ANALYSIS OF MAX-MIN FAIRNESS FOR A LINEAR NETWORK

3.1. Approximation of the max-min fair share

The determination of the max-min fair share of one data ow requires the computationof the number of ows simultaneously sharing a portion of the link with the date owconsidered. Such a computation relies on the transient analysis of the number of activedata ows at a given point of the link. Once a clump is clearly identi�ed, a completerecursive procedure is required to determine the max-min fair share of a data ow. Thisnested procedure makes any mathematical analysis extremely di�cult. Nevertheless, asimple lower bound for the max-min fair share can easily be obtained by computing thenumber � of data ows, which have a portion of the network in common with the data owunder consideration. Once � is computed, 1=(� +1) is a lower bound for the max-min fairshare. The random variable � is characterized as follows.

Proposition 1 The number � of data ows, which share a portion of the transmission

link with a data ow with an arbitrary originating point on the half line [0;1), has the

generating function �(z) given by

jzj � 1; �(z) =e�u(1�z)

1 + u(1� z): (10)

and mean value E[�] = 2u.

Proof. Consider a data ow with an arbitrary originating point. The number �1 of data ows with originating points before the point considered and sharing a portion of the linkwith the data ow considered is equal to the stationary distribution in the correspondingM=M=1 system and is hence distributed as a Poisson random variable with parameter u.Thus,

Pf�1 = kg =uk

k!e�u: (11)

Knowing the length d of the data ow considered, say, d = � for � > 0, the number �2 ofdata ows, which appear on the link after the originating point of the data ow consideredand which share a common portion of the link with that ow, is distributed as Poissonrandom variable with parameter u�

Pf�2 = k j d = �g =(u�)k

k!e�u�: (12)

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and hence,

Pf�2 = kg =

Z 1

0

(u�)k

k!e�u�e�� d� =

uk

(u+ 1)k+1: (13)

Since the random variables �1 and �2 are independent of each other, the result follows.This completes the proof. 2

Corollary 1 The mean value of the rate � obtained by an arbitrary data ow under max-

min fairness is such that

� � E[bmin] = E

�1

� + 1

�=

e

u[E1(1) �E1(1 + u)] ; (14)

where E1(z) is the exponential integral de�ned by

E1(z) =

Z 1

z

e�t

tdt; j arg zj < �: (15)

Proof. We have

E

�1

� + 1

�=

Z 1

0

E[z� ] dz =

Z 1

0

e�ut

1 + utdt =

e

u[E1(1)� E1(1 + u)] :

This completes the proof. 2

As it will be shown by simulation results presented in the following, the lower boundgiven by 1=(� +1) may be too drastic. Another lower bound for the max-min fair share ofa data ow is obtained when computing the maximum number � of data ows sharing aportion of the transmission link with the data ow considered. The lower bound 1=(�+1)is exact, when the data ow belongs to the group of data ows realizing the maximumnumber of simultaneous data ows in a clump. This lower bound is not exact, when in thegroup of data ows simultaneously active with the data ow considered, there is a data ow sharing a portion with another group of data ows such that the maximum number ofsimultaneous data ows is greater than the maximum number of simultaneous data owsin the former group. This phenomenon is illustrated in Figure 2.

Proposition 2 The maximum number of data ows simultaneously sharing a portion of

the transmission link with an arbitrary data ow is � characterized by be generating function

jzj � 1; �(z) = e�u(1�z) � (1 � z)Q(z; u); (16)

where Q(z; u) is de�ned by

Q(z; u) = e�u1Xh=0

zhuh

h!

Qh(�2; u)

Qh+1(�1; u): (17)

The mean value of the random variable � is

E[�] = u� e�uQ(1; u) = u� e�u1Xh=0

uh

h!

Qh(�2; u)

Qh+1(�1; u): (18)

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max-min fair share = 1=(�+ 1) = 1=4

max-min fair share = 1=2 > 1�+1 = 1=3

network link

Figure 2. Comparison between the max-min fair share and 1=(� + 1).

Proof. Consider a given data ow and let n0 be the number of data ows sharing thesame portion of the transmission link with the data ow considered at its originating point.Then, by denoting by � the length of the data ow considered,

Pf� < h j n0 = kg = PkfTh > �g =

Z 1

0

PkfTh > sge�s ds = 1�Ek�e�Th

�= 1�

Qk(�1; u)

Qh(�1; u):

Since n0 has a Poisson distribution with parameter u, we have

Pf� < hg = e�uh�1Xk=0

uk

k!

�1�

Qk(�1; u)

Qh(�1; u)

�;

and by using relation (4),

Pf� < hg = e�u

h�1Xk=0

uk

k!�

uh�1

(h� 1)!

Qh�1(�2; u)

Qh(�1; u)

!and hence,

Pf�= hg = e�u�uh

h!�

uh

h!

Qh(�2; u)

Qh+1(�1; u)+

uh�1

(h� 1)!

Qh�1(�2; u)

Qh(�1; u)

�and equation (16) follows. The mean value is obtained by taking the derivative at pointz = 1. This completes the proof. 2

Corollary 2 The mean value of the rate � obtained by an arbitrary data ow under max-

min fairness is such that

� � E

�1

�+ 1

�=

1� e�u

u� e�u

1Xh=0

uh

(h+ 2)!

Qh(�2; u)

Qh+1(�1; u); (19)

where Qh(x; u) is the hth Charlier polynomial.

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Remark. When the load u of the system is very small (u� 1), we have

E

�1

� + 1

�� E

�1

�+ 1

�� 1 � u

and thus, approximations (14) and (19) lead to the same asymptotic behaviour.

The two approximations (14) and (19) are illustrated in Figure 3 for di�erent values ofthe load u. The exact value of max-min fair share has been computed by simulation. Itturns out that for small values of the load u (namely u � 1), these two approximationsare quite accurate. The accuracy rapidly degrades when the load u increases but they stillgive a good idea for the value of the max-min fair share. Moreover, approximation (19)is a little bit better than approximation (14). For u � 5, the relative error is about 93%for approximation (14) and 22% for approximation (19). It thus appears that the bestapproximation is the latter one, whose computation is however much more complicatedthan that of approximation (14).

approximation E[1=(� + 1)]approximation E[1=(� + 1)]

Max-Min fair share

Load u

Meanallocatedbandwith

6543210

1

0.8

0.6

0.4

0.2

0

Figure 3. Mean value of the max-min fair share for the linear network model.

3.2. Amount of lost bandwidth

We de�ne the lost bandwidth as the long range ratio of bandwidth of the link, which isnot used in clumps due to the max-min bandwidth sharing principle. Let on and fn denotethe originating and the terminating points of the nth data ow, respectively. Then, theproportion of bandwidth, which is not used up to point t because of max-min fairness, is

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mathematically de�ned by

Bt =

R t

01Ifns>0g ds�

R t

0

P1n=1 �n1Ifon�s�fng dsR t

01Ifns>0g ds

; (20)

where ns is the number of active data ows at point s and �n is the rate allocated to thenth data ow. Of course, the random variables dn = fn � on and �n are correlated in anintricate way and any analysis of their interdependence seems to be out reach, when �n iscomputed by using the exact man-min fairness allocation.When considering the long range average, it is easily checked that

Bdef= lim

t!1Bt = 1 �

u

1� e�uE [d�] (21)

where d and � are the length and the rate allocated to an arbitrary data ow, respectively.B is clearly upper bounded by the value obtained when taking � = 1=(� + 1) in de�ni-

tion (21), where � is de�ned by Proposition 1. We speci�cally have the following result.

Proposition 3 The proportion of lost bandwidth B due to max-min fairness is upper

bounded as

B � B1 = �u

(1 + u)(eu � 1)+

e

1 � e�u[E1(1)� E1(1 + u)] ; (22)

where E1(z) is the exponential integral de�ned by equation (15).

Proof. With the notation of the proof of Proposition 1, we have

E[d�] = E

�E

�d

� + 1

���� d��It is easily checked that

E [z� j d = �] = e�u(1+�)(1�z);

and then,

E

�1

� + 1

���� d = �

�=

Z 1

0

e�u(1+�)(1�z) dz:

It follows that

E[d�] =

Z 1

0

e�u(1�z)

[1 + u(1� z)]2dz =

1

u

�1�

e�u

1 + u

��

e

u[E1(u)�E1(1 + u)] :

Equation (22) is obtained by using relation (21). This completes the proof. 2

Now, we consider the second approximation of the max-min fair share, which is givenby equation (19).

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Proposition 4 The mean value of the amount of lost bandwidth B is upper bounded as

B � B2 = �u

eu � 1

@

@s

"1

s

1Xh=0

uh

(h+ 2)!

Qh(�s� 1; u)

Qh+1(�s; u)

#�����s=1

=u

eu � 1b2; (23)

where b2 can be written as

b2 =1Xh=0

uh

(h+ 2)!

Qh(�2; u)

Qh+1(�1; u)+

1Xh=0

uh

(h+ 2)!

Q0h(�2; u)

Qh+1(�1; u)

�

1Xh=0

uh

(h+ 2)!

Qh(�2; u)Q0h+1(�1; u)

Q2h+1(�1; u)

; (24)

with Qh(s; u) being the hth Charlier polynomials and

Q0n(x; u) =

@

@xQn(x; u):

Proof. By using the same arguments as in the proof of Proposition 2, we have by condi-tioning on the length of the data ow considered,

E

�1

�+ 1

���� d = �

�= e�u

1Xh=0

1

h+ 1

hXk=0

uk

k!PkfTh+1 > �g �

1Xh=0

1

h+ 1

h�1Xk=0

uk

k!PkfTh > �g

!:

It follows that

E [�d] = e�u1Xh=0

1

h+ 1(Qh �Qh�1) ;

where

Qh =hX

k=0

uk

k!

Z 1

0

�PkfTh+1 > �ge�� d�:

Noting that Z 1

0

PkfTh > �ge�s� d� =1

s

�1�

Qk(�s; u)

Qh(�s; u)

�;

we have Z 1

0

PkfTh > �g�e�� d� = �@

@s

1

s

�1�

Qk(�s; u)

Qh(�s; u)

�����s=1

and then, by using relation (4),

E[�d] =1

u(1� e�u) + e�u

@

@s

"1

s

1Xh=0

uh

(h+ 2)!

Qh(�s� 1; u)

Qh+1(�s; u)

#�����s=1

:

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Relation (23) is obtained by using de�nition (21). This completes the proof. 2

Remark. When the load of the system is very small (u� 1), then

B1 � u:

Moreover, when u� 1,

1

s

1Xh=0

uh

(h+ 2)!

Qh(�s� 1; u)

Qh+1(�s; u)�

u

2s(s+ u)

and then, B2 � u. Hence, the two approximations lead to the same asymptotic behaviorwhen u� 1.

When u!1, we see that B1 ! e�E1(1) � 60% while B2 decreases. When consideringthe exact max-min fairness allocation, we may reasonably conjecture that the amount oflost bandwidth should tend to 0 as u ! 1. Indeed, when u ! 1, the system is heavilyloaded and the superposition of all data ows exhibits a smooth behavior (central limittheorem). Hence, the amount of lost bandwidth should be small. The di�erence betweenthe two approximations is illustrated in Figure 4.

e:Ei(1; 1) = 0:59635

approximation (B1)

approximation (B2)

Max-min alloc.

Load u

Meanvalueofthelostbandwidth(%)

6543210

60

50

40

30

20

10

0

Figure 4. Mean value of the bandwidth lost in a clump due to max-min fairness.

It follows that approximation B1 does not follow the general behavior of the exactamount of lost bandwidth under max-min fairness. Approximation B2 exhibits the correct

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behavior and yields a quite accurate estimate for the amount of lost bandwidth. Forinstance, when u is in the neighborhood of 6, the relative error is about 25% as illustratedin Figure 4. ApproximationB2 is numerically computed by using the recursions (7) betweenthe derivatives of Charlier polynomials.

4. Conclusion

Two approximations have been proposed in this paper to estimate the fair share as wellas the amount of lost bandwidth when resources are allocated to elastic ows accordingto max-min fairness in a single link linear network. It turns out that the approximationbased upon the maximum number of ows sharing the transmission capacity of the link(equations (19) and (23)) yields the best results for the two above performance parameters.This opens the door to analytically modeling max-min fairness in more complex situations,for instance when the network is a graph. This will be investigated in further studies.Finally, more accurate estimates for the max-min fair share and the amount of lost band-

width could certainly be obtained when considering heavy tra�c approximations (speci�-cally when u becomes very large). The development of approximations under heavy tra�cconditions is however made di�cult by the complexity of max-min fairness, in particular bythe algorithmic de�nition of the fair share. This point will be investigated in forthcomingpapers.

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