A mathematical model of the Paris Metro Pricing scheme for charging packet networks

Computer Networks 46 (2004) 73–85

www.elsevier.com/locate/comnet

A mathematical model of the Paris Metro Pricing schemefor charging packet networks

David Ros a,*, Bruno Tuffin b

a GET/ENST Bretagne, Rue de la Chataigneraie, CS 17607, 35576 Cesson S�evign�e Cedex, Franceb IRISA/INRIA Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France

Available online 9 April 2004

Abstract

Pricing has become one of the main challenges of the networking community and is receiving a great deal of interest

in the literature. In this paper, we analyze the so-called Paris Metro Pricing scheme which separates the network into

different and independent subnetworks, each behaving equivalently, except that they charge their customers at different

rates. In our model, each subnetwork is represented by a single bottleneck queue, and the ‘‘customers’’ (data packets)

choose their subnetwork taking into account not only the prices, but also the expected delay, which is supposed to have

an economic impact. We obtain some necessary and sufficient conditions for the stability of the system; we analyze the

problem of maximizing the network revenue and compare it with the case of a single network, and present a multi-

application extension of the model. Numerical results illustrating some key aspects of the system are provided

throughout the paper.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Internet; Packet networks; Pricing; Queueing models

1. Introduction

Pricing in IP networks is becoming a mainconcern of Internet Service Providers (ISPs).

Classical pricing schemes, based on a subscription

fee and/or a (possibly unlimited) number of hours

of network access, depending on the country and

the type of usage, are becoming obsolete due to

three major reasons:

* Corresponding author. Tel.: +33-2-99-12-70-46; fax: +33-2-

99-12-70-30.

E-mail addresses: [email protected] (D. Ros),

[email protected] (B. Tuffin).

1389-1286/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.comnet.2004.03.019

1. Such simple schemes were an incentive to stim-

ulate utilization. Even if they probably helped

to develop the network and allowed ISPs togain market share, nowadays they may prove

sub-optimal. Indeed, demand in access net-

works may easily exceed the installed capacity,

leading to congestion.

2. Some users might wish to pay more than others

to avoid congestion and get an improved ser-

vice. Why not serve them first and (possibly) in-

crease the ISP revenue?3. Unlike the telephone network, the current

Internet––and, more so, the next-generation In-

ternet––has to deal with different kinds of appli-

cations with diverse quality of service (QoS)

ed.

mail to: [email protected]

74 D. Ros, B. Tuffin / Computer Networks 46 (2004) 73–85

requirements. Including these requirements in a

pricing scheme may, in some way, improve

users’ satisfaction.

A wide range of charging methods for packetnetworks has been devised and discussed in the

literature; see for instance the survey papers [1–6].

The main classes include resource reservation,

priorities between packets, auctions for bandwidth

or pricing based on transfer rates.

In this paper, we study another method, called

Paris Metro Pricing (PMP) and introduced by

Odlyzko in [7,8]. Under the PMP scheme, thenetwork is split into independent subnetworks;

more precisely, a fixed fraction of the capacity of

each link is totally allocated to each subnetwork.

The tariff will be different for each subnetwork, so

that (hopefully) congestion will be alleviated in the

most expensive ones. This method does not offer

any QoS guarantees, so that it is somehow weak

with respect to the abovementioned techniques;however, it is very attractive because of its imple-

mentation and management simplicity. In this

paper, we develop a mathematical model which

may be helpful in setting prices in a PMP-based

network. Particularly, our model will use the delay

introduced by the network as a cost.

Class-based service differentiation has already

been proposed in the literature. In [9], packets aremarked according to customer’s QoS require-

ments, but no pricing scheme is developed––the

model is only performance-based. In [10], a pricing

scheme is presented where time is divided into slots

and only a given number of packets can be served

in each slot. Users compete by choosing in which

priority class they send their packets (higher pri-

ority classes being more expensive). Note that theperformance metric in [10] is throughput, whereas

ours is delay. Besides, a continuous-time model

like the one we introduce in this paper is more

representative in many cases.

It has been argued by Gibbens et al. [11] that

network segmentation might not work under

competition (i.e., no Nash equilibrium exists);

nonetheless, their result applies to a model quitedifferent from ours: they use the traffic load as the

measure of congestion. However, delay seems to be

more relevant since it is a QoS parameter users

actually experience. Moreover, note that even if the

optimization problem in [11] looks similar to that

presented in this paper, Gibbens et al. deal with the

competition between two subsets of subnetworks.

Nevertheless, it is worth mentioning that we arrive

at the same conclusion as Gibbens et al.––i.e., theISP is better off by not splitting its network.

Note also that in [12], PMP has been adapted in

such a way that, instead of logically separating the

subnetworks, a round-robin service discipline is

used to share the bandwidth among them. This

‘‘improved’’ version of PMP, even though it de-

serves attention, is not considered here.

The layout of the paper is as follows. The PMPscheme is introduced in Section 2. In Section 3

the mathematical model is described. Necessary

and sufficient conditions for stability are then

provided, and an optimization problem (maxi-

mizing the ISP revenue) is introduced. Section 4

provides some numerical illustrations of the re-

sults of Section 3. In Section 5, we compare the

revenues of a PMP network with those of anequivalent, non-PMP network. Section 6 is de-

voted to the case of multiple applications with

different delay sensitivities. Finally, conclusions

are given in Section 7.

2. PMP: a brief description

The original PMP proposal in [8] consists of

partitioning a network into a few logically separate

networks (or classes), each having a fixed fraction

of the capacity of the entire network. Every sub-

network would route and handle packets accord-

ing to the current Internet protocols. There is no

formal guarantee of QoS, but by charging different

rates for different classes (served in the same way),it is supposed that the most expensive classes will

be less congested by way of self-regulation, hence

delivering a better QoS. The name given to this

model, Paris Metro Pricing, stems from the rules

of the Paris Metro about 20 years ago, where

trains were composed of cars of two classes,

offering exactly the same quality of seats. As

tickets prices were different, the cars for the mostexpensive class were less congested, leading to a

better perceived QoS.

D. Ros, B. Tuffin / Computer Networks 46 (2004) 73–85 75

As pointed out by Odlyzko, the advantage of

PMP is that, even if we do not have any strict QoS

guarantee using this scheme, but rather a statistical

guarantee, it would permit dispensing with com-

plex, non-scalable mechanisms (like, say, resource

reservation protocols) and keep the simpler andcheaper current model of the Internet, while

improving the QoS experience. Indeed, no signal-

ling protocols would have to be used, and the tariff

would be fixed, which is preferred by most users

[13].

From the preceding discussion, it is intuitively

clear that prices play in PMP an important role in

controlling congestion and, notably, in distribut-ing the load among subnetworks: prices are the

key to quality-of-service differentiation. The

problem of finding the ‘‘right’’ set of prices for a

given bandwidth partition may be informally sta-

ted as follows: for a PMP network to work in an

efficient manner, 1 the charge for using the most

expensive subnetwork should be high enough, so

that this subnetwork is lightly loaded, but not ‘‘toohigh’’––otherwise, the subnetwork would remain

empty, because the high quality offered would not

compensate for the (very) high monetary cost in-

curred by users.

In the next section, we present a mathematical

model of PMP that allows us to analyze its sta-

bility properties with respect to prices, as well as to

solve the revenue maximization problem.

3. Mathematical model

For the sake of clarity, we begin our study of

PMP by presenting a simplified model in which all

packets are generated by the same kind of appli-

cation and all users have the same valuation ofQoS (represented in our model by the mean packet

delay). Later, in Section 6, we will describe a more

realistic model corresponding to a multi-applica-

tion scenario where a different QoS valuation may

be associated to each application.

1 Efficiency may be defined, for instance, in terms of

optimizing the network operator’s revenue and providing an

adequate usage of each subnetwork’s resources.

3.1. Model presentation

Assume that there are I classes (i.e., subnet-

works) in a PMP network, and that the class-i per-packet price at the entrance of the network is pi(16 i6 I). We assume that there is a potential total

arrival rate ~k of packets at the network, corre-

sponding to the arrival rate when prices are set

to 0.

A utility measure is associated to each packet. It

is assumed to be a random variable U that follows

the same distribution for every packet, and that it

is independent of other packets’ utility.Also, a total cost function

pi þ cdi

is associated to a class i, where di is the mean delay

for a packet in the network and c is a constant

converting delay in money.

A packet enters (i.e., chooses) network i ifi ¼ argminjðpj þ cdjÞ and U P pi þ cdi, that is, it

chooses the less expensive subnetwork in terms of

total cost (which is a linear combination of delay

and price). If U < minj ðpj þ cdjÞ, the packet does

not enter at all, meaning that the network is too

expensive for it. Traffic elasticity is then a central

assumption of our model since a larger delay can

be accepted by a user, provided that the price isdecreased proportionally.

The actual total arrival rate is then

k ¼ ~kP ðU P minj

pj þ cdjÞ:

We denote by �F the complementary cumulative

distribution function of U , i.e., �F ðxÞ ¼ P ðU P xÞ:Finally, let ki (16 i6 I) denote the actual arrival

rate at subnetwork i, so that k ¼PI

i¼1 ki.

3.2. Stability: existence and uniqueness

Using this model, in equilibrium the distribu-

tion of packets among classes has to be stable,

meaning that the total cost pj þ cdj is the same forall classes j. Indeed, if for a given class j the valuepj þ cdj were smaller than the total cost of the

other classes, then new packets entering the net-

work would choose class j until its total cost

reaches that of other classes. This corresponds to a


Wardrop equilibrium [14]. Let us call ptot the valuepj þ cdj (identical for all j). As we want all sub-

networks to be used, we have the following set of

I þ 1 equations with I þ 1 unknown variables ki

(16 i6 I) and ptot:PIi¼1 ki ¼ ~kP ðU P ptotÞ

pi þ cdi ¼ ptot; for 16 i6 I ;

�ð1Þ

where di is a function of ki.

The remaining of this section is separated into

two parts. In the first part, the di represent themean waiting times (i.e., service excluded). In the

second part, results are obtained as a corollary

when the di represent the response times (i.e., ser-

vice included). What users and applications care

about is total delay, so the latter case seems more

realistic; nonetheless, we will begin by treating the

waiting time case for the sake of clarity.

In order to determine ptot and ki 8i, we have thefollowing theorem.

Theorem 1. Assume that di ¼ fiðkiÞ 2 Rþ is astrictly increasing and continuous function of thearrival rate, representing the mean waiting time of aclass-i user. Without loss of generality, we supposethat p1 > p2 > � � � > pI . Assume also that the dis-tribution of U is absolutely continuous and strictlyincreasing. Then the solution of Eq. (1) exists and isunique if and only if

p1 6 �F �1 1

~k

XIi¼1

f �1i

p1 � pic

� � !: ð2Þ

Remark 1. Note that if condition (2) is not veri-

fied, then it means that the highest price p1 is too

high to obtain an equilibrium between classes.

Consequently, this class would be ignored (i.e., the

subnetwork would remain empty) and the com-

putation would have to be carried out again withthe I � 1 remaining classes.

Thus, condition (2) has to be taken as an

assumption to get the equilibrium for I classes.

Proof of Theorem 1. Denote by �F the comple-

mentary cumulative distribution function of ran-

dom variable U . The system (1) can be re-written

as

ptot ¼ �F �1 ð1=~kÞPI

i¼1 ki

� ki ¼ f �1

i ððptot � piÞ=cÞ; for 16 i6 I :

(

In particular we can get

ptot ¼ �F �1 1

~k

XIi¼1

f �1i

ptot � pic

� � !: ð3Þ

If we are able to prove that there exists a unique

ptot satisfying this equation, then the existence and

uniqueness of the ki will be straightforward toshow.

It is important to note that, by definition,

ptot P pi 8i (i.e., ptot P p1) because di P 0.

Note that the left-hand side of (3) is a contin-

uous and strictly increasing function of ptot. On the

other hand, since f �1i is strictly increasing 8i and

�F �1 is strictly decreasing (and both are continu-

ous), then the right-hand side of (3), that is,�F �1 ð1=~kÞ

PIi¼1 f

�1i ððptot � piÞ=cÞ

� is a continuous

and strictly decreasing function of ptot. Thus the

solution of (3), if it exists, is unique.

Hence, the existence of a solution––i.e., whether

the left-hand side and the right-hand side of (3)

cross each other––depends on border values of ptot.Let us first see what happens when ptot ! þ1. Let

EðSiÞ denote the mean service time for class i. Thedelay goes to infinity when the mean arrival rateapproaches the mean service rate 1=EðSiÞ, hence:limdi!1 f �1

i ðdiÞ ¼ 1=EðSiÞ. So the left-hand side of

(3) tends to infinity and the right-hand side tends

to �F �1�ð1=~kÞ

PIi¼1 1=EðSiÞ

�when ptot tends to

infinity. Therefore, the existence of a solution de-

pends on whether, at the minimal value of ptot (i.e.,p1), the right-hand side of (3) is greater than or

equal to p1: this condition is expressed by (2). Insuch a case, Eq. (3) has a unique solution. h

Corollary 1. Assume now that di ¼ f ðRÞi ðkiÞ repre-

sents the system’s response time. Then di 2½EðSiÞ;1Þ, with EðSiÞ denoting the mean servicetime.

In this case, and without loss of generality, theclasses are supposed to be ordered by their minimumtotal cost p1 þ cEðS1ÞP p2 þ cEðS2ÞP � � � P pI þcEðSIÞ.

Then the solution of (1) exists and is unique if andonly if

2 The standard numerical and optimization packages in-

cluded in, say, Matlab or Mathematica can be used to solve this

problem.3 Qualitatively similar results (not shown for space reasons)

were obtained by taking �F uniformly distributed––see [16].


p1 þ cEðS1Þ

6 �F �1 1

~k

XIi¼1

ðf ðRÞi Þ�1 p1 � pi

c

� þ EðS1Þ

�!; ð4Þ

or equivalently, using the waiting times,

p1 þ cEðS1Þ

6 �F �1 1

~k

XIi¼1

f �1i

p1 � pic

� þ EðS1Þ � EðSiÞ

�!:

Proof. Since the mean waiting time cannot be

negative, we have to use the fact that ptot Pmaxi ðpi þ cEðSiÞÞ ¼ p1 þ cEðS1Þ. By replacing p1 inEq. (2) using f ðRÞ

i instead of fi, we get Eq. (4).

Besides, using the fact that f ðRÞi is given by

f ðRÞi ¼ fi þ EðSiÞ, we can easily obtain the equation

with the waiting times. h

3.3. Optimization problem

The idea, from a network provider’s perspec-

tive, is to find the pi maximizing the revenue of the

network, i.e.,

R ¼ maxpi ;8i

XIi¼1

kipi;

subject to pi P 0 8i.Based on the existence and uniqueness condi-

tion of the arrival rates and total cost (defined in

terms of pi), and using the previous notations, the

problem can be reformulated as

maxpi;8i

XIi¼1

f �1i

ptot � pic

� �pi;

subject to

ptot P pi 8i;

ptot ¼ �F �1 1

~k

XIi¼1

f �1i

ptot � pic

� � !;

pi P 0 8ior its equivalent form, if we rather talk about the

response time instead of the waiting time.

As this problem looks analytically intractable in

the general case, in the next section we are going to

present numerical results illustrating the stability

domain, as well as the impact of prices and band-

width partition on delay, total cost and revenue.

Note that, in some sense, our approach can berelated to that of Honig and Stieglitz [15] (albeit

applied to a different model), where revenue, prices

and performance were studied.

4. Numerical results

In this section we will present some numericalresults obtained with the above model. 2 More

detailed results can be found in [16]. In what fol-

lows we will use the response time of the system as

a measure of delay, that is, di ¼ f ðRÞðkiÞ. We will

also relax the restriction p1 þ cEðS1ÞP p2 þ cEðS2Þimposed in the previous section.

For the sake of simplicity, let us consider the

case in which the ISP partitions its network inI ¼ 2 subnetworks. Indeed, even if the case I > 2

is, in principle, not harder to solve numerically

than the I ¼ 2 case, we will only consider the latter

in order to being able of graphically presenting the

results. Each subnetwork i is viewed as a single

bottleneck, modeled as a M/G/1 FIFO queue with

service rate li ¼ 1=EðSiÞ, so

f ðRÞi ðkiÞ ¼

1

liþ

1þ CV2� �

qi

2li 1� qið Þ ;

where qi ¼ ki=li and CV2 is the squared coefficient

of variation of the service law.

Unless stated otherwise, the following parame-

ters will be used throughout this section:

• Total capacity of the network: c ¼ 2.

• Potential total arrival rate: ~k ¼ 3.

• Utility distribution function �F : exponential 3

with mean �U ¼ 1.

0.5 1 1.5 2 2.5 3p1

0.5

1

1.5

2

2.5

3p2

p2 = p1γ γ

−1 µµ 2

+

2.5

3p2

p2 = p1γ γ

−1 µµ 2

+

(a)


• Cost per unit of delay: c ¼ 1.

• Service time distribution: exponential (i.e., each

queue is a M/M/1 queue and so CV2 ¼ 1). Note

that the M/D/1 queue gives similar results.

Note that, in the case of TCP flows, bandwidth

sharing among flows has been successfully mod-

eled as a M/G/1 processor sharing (PS) queue (see

for instance [17]). However, such a queueing sys-

tem has exactly the same response time as a M/M/

1/PS queue with identical mean service time, and

the latter has in turn the same response time as a

M/M/1/FIFO queue with the same characteristics.Therefore, our analysis of a M/G/1/FIFO queue

actually encompasses the M/G/1/PS case.

We assume, without loss of generality, that the

total bandwidth c of the network is shared among

subnetworks according to:

l1 ¼ ac; l2 ¼ ð1� aÞc with 0:56 a < 1:

ð5Þ

0.5 1 1.5 2 2.5 3p1

0.5

1

1.5

2

(b)

Fig. 1. Equilibrium region, �F exponential. (a) a ¼ 0:5, (b)

a ¼ 0:7.

4.1. Equilibrium region

It is interesting to look at the shape of the

equilibrium region, defined as the set E of all pairs

ðp1; p2Þ such that the system of equations (1) has aunique solution, that is: E ¼ E0 [ E1 [ E2, with E0

defined by the set of prices such that

p1 þ c=l1 ¼ p2 þ c=l2 that also verify the stability

condition and, for i 2 f1; 2g

Ei ¼ ðp1; p2Þ: p�i

�þ c

l�i

< pi þcli6 �F �1 1

~kðf ðRÞ

�i Þ�1 pi � p�i

c

��þ 1

li

�� ;

where �i denotes the element of f1; 2g which is not

i. Ei, for i 2 f1; 2g, can be regarded as the price

area such that the charge plus the cost of service,

i.e., the total cost minus the waiting cost, is more

expensive for class i (and such that the stability

condition is verified).

Fig. 1 shows the equilibrium region for twodistinct bandwidth allocations, in the case where

the utility is exponentially distributed. The equi-

librium region is shown in gray; the straight line

pointed to by an arrow corresponds to the ‘‘fron-

tier’’ E0 where the total costs (minus the waiting

cost) are the same for both classes.Note how the equilibrium region above the

frontier gets narrower as the parameter a in-

creases. Note also that, as prices get higher and

higher, the equilibrium region degenerates into the

straight line given by the frontier condition, no

matter the value of a. Since there is no upper

bound to the utility (meaning that some users are

willing to tolerate a very high cost), and the fixedcost pi þ c=li increases for both subnetworks,

queueing delay tends to zero (together with the

arrival rate), so equilibrium can only be reached if


the fixed cost is about the same in each subnet-

work; otherwise, traffic would switch entirely to

the less-expensive one.

4.2. Total delay

Delay in each subnetwork is shown in Fig. 2,

for two different bandwidth allocations. Observe

that, for subnetwork i, delay is always minimal

along the frontier of the equilibrium region Ei.

The tradeoff between prices and delay is fairly

evident in these figures. When one of the prices

(say, p1) is kept fixed, delay in the corresponding

subnetwork (i.e., d1) decreases as the other sub-network’s price (p2, in this case) decreases, while

delay d2 increases. This is due to the fact that a

00.5

11.5

22.5 0

0.5

1

1.5

2

2.5

0

0.5

1

1.5

2

p2

p1

d 1

0.51

1.52

2.5 0

0.5

11.5

22.5

0

0.25

0.5

0.75

1

p2

p1

d 1

(a)

((c)

Fig. 2. Delay in each subnetwork (shown as a dark-gray surface), �F ex

region; the set of prices E0 is shown as a solid line over the equilibrium

all figures. (a) a ¼ 0:5, subnetwork 1. (b) a ¼ 0:5, subnetwork 2. (c) a

lower p2 will tend to attract more customers to

subnetwork 2, increasing queueing delay in it––

and so decreasing queueing delay in subnetwork 1.

4.3. Equilibrium cost, revenue and optimality

Fig. 3 shows that the total cost ptot is strictly

increasing with pi. This, indeed, can easily be

proved analytically.

Fig. 4 shows the revenue of the network, R, as afunction of prices, for two different bandwidth

allocations. It is interesting to remark that there

appears to be a single maximum (i.e., there is a

single pair of prices which is optimal, in the sensethat revenue is maximized). As a increases, the

position of the maximum value of R moves closer

00.5

11.5

22.5

00.5

11.5

2

0

0.5

1

1.5

2

p2

p1

d 2

0.51

1.52

2.5

00.5

1

1.5

2

2.5

0

0.5

1

1.5

2

2.5

p2

p1

d 2

(b)

d)

ponential. The light-gray shadow corresponds to the equilibrium

region. Note that the orientation of the axes is not the same in

¼ 0:7, subnetwork 1. (d) a ¼ 0:7, subnetwork 2.

0

0.5

1

1.5

2

p1

0

0.51

1.52

p2

1

2

3

4

ptot

0

0.5

1

1.5

2

p1

2

0

1

2

3

p1

0

1

2

3

p2

1

2

3

4

ptot

1

2

3

p

1

(a)

(b)

Fig. 3. Equilibrium cost, �F exponential. (a) a ¼ 0:5, (b) a ¼ 0:7.

0

1

21 0

1

2

p2

0

0.1

0.2

0.3

R

0

1

2p1

0.51

1.5

2

2.51 0

1

2

p2

0

0.1

0.2

0.3

0.4

R

0.51

1.5

2p

(a)

(b)

Fig. 4. Revenue, �F exponential. (a) a ¼ 0:5, (b) a ¼ 0:7.


to the rear ‘‘edge’’ of the surface, which corre-

sponds to the upper edge of the equilibrium regionE2 in Fig. 1.

4.4. Sensitivity of the optimal revenue

4.4.1. Sensitivity to the bandwidth partition

Let us now examine how the optimal revenue, as

well as the corresponding prices, delay and total

cost, depend on the bandwidth allocation param-eter a (see Fig. 5).

First, note that the optimal revenue

Ropt ¼ maxp1;p2P 0 R is highly sensitive to the value

of the bandwidth allocation parameter a. Fig. 5(a)suggests that the ISP may find interesting, from an

economic point of view, to operate at the maxi-

mum-revenue allocation aopt. However, since Ropt

(and, hence, the revenue for any non-optimal set ofprices) rapidly decreases for values of a above aopt,

a too-unequal bandwidth allocation policy may

result in lower income. On the other hand, a

‘‘safe’’ bandwidth allocation (say, a ¼ 0:5) may

result in a maximum revenue as low as � 70% ofthe highest Ropt.

Note that, interestingly enough, the optimal

revenue is maximized when the price p2 of the

lowest-capacity subnetwork drops to zero. For

higher values of a, since p2 is at its lowest-possiblevalue, the ISP cannot compensate for the poorer

performance of subnetwork 2 by lowering its price,

so more and more traffic tends to flow throughsubnetwork 1 and the price p1 must be raised in

order to have a stable network. For a < aopt, a

falling p2 tends to attract more traffic to subnet-

work 2 in spite of the degradation in delay d2 whena increases, resulting in an increasing revenue.

4.4.2. Sensitivity to the users’ valuation of delay

The c parameter, which expresses the cost perunit of delay, can be regarded as how much users

value having a good quality of service (in terms of

delay): the higher the value of c, the higher the

impact of delay on the total cost pi þ cdi, and

the lower the probability that a given packet enters

the network.

0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

α

Ropt

0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7

8

9

10

α

price

p1

p2

0.5 0.6 0.7 0.8 0.92

3

4

5

6

7

8

9

10

11

α

ptot

0.5 0.6 0.7 0.8 0.90

2

4

6

8

10

12

α

delay

d1

d2

(a)

(c)

(b)

(d)

Fig. 5. Optimal revenue and prices (and corresponding total cost and delay) as a function of a, with an exponentially distributed

utility. (a) Optimal revenue Ropt; (b) optimal prices pi; (c) total cost ptot; (d) delay di.

0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α

Ropt

γ = 0.25

γ = 0.5

γ = 1

γ = 2

Fig. 6. Effect of the c parameter on the optimal revenue, with

an exponentially distributed utility.


Fig. 6 illustrates the sensitivity of the optimal

revenue to the value of c, for a in the ½0:5; 0:95�range. For c ¼ 0:25, Ropt is fairly stable for a

wide range of bandwidth allocations: Roptð0:5Þ=RoptðaoptÞ � 0:87, with aopt � 0:91.

On the other hand, when c ¼ 2 (meaning that

users’ valuation of delay is eight times higher than

in the previous case) we have that Roptð0:5Þ=RoptðaoptÞ � 0:66, i.e., the ‘‘loss’’ incurred by

operating the network at a safe bandwidth allo-

cation is higher than in the c ¼ 0:25 case. In other

words, the model predicts that revenues are higherand fairly insensitive to bandwidth allocation

when users are more tolerant of delay (which is an

intuitively appealing result).


Notice also how the optimal allocation aopt is

dependent on c, going from aopt � 0:91 to 0.67. In

practical terms, this means that an ISP would have

to operate its network on the ‘‘safe’’ side (i.e., with

a close to 0.5) if the value of c cannot be accuratelyestimated.

5. Comparison with the case of a single network

An interesting question is how well a PMP

network performs, in terms of revenue, with re-

spect to a one-tiered (I ¼ 1), non-PMP network.

We will focus on the two-tiered (I ¼ 2) case ofPMP. We will suppose that both the one-tiered

network and the two-tiered PMP network have the

same link capacity c; in the latter case, this capacity

is shared among subnetworks according to Eq. (5).

This would correspond to the scenario in which an

ISP wants to evaluate the consequences of intro-

ducing PMP in its network by deploying the

appropriate mechanisms in routers (e.g., packetclassification and scheduling mechanisms), but

without increasing the installed capacity.

We observed in our preliminary numerical

experiments 4 that the revenue generated by a

single network (I ¼ 1) is greater than the revenue

generated by two subnetworks (I ¼ 2) when the

total network capacity––i.e., the service rate––is

unchanged, regardless of the partition parametera. Let us formally prove this result for the M/M/1

queue in the following theorem.

Theorem 2. Consider a M/M/1 queue (representingthe network) with service rate l and the responsetime as the delay cost. The revenue generated by thissingle queue is greater than the revenue generated bytwo separate queues with service rates al andð1� aÞl, respectively.

Proof. Consider first the case I ¼ 1 with a price pper packet. Since the response time d ¼ 1=ðl � kÞis also given by ðptot � pÞ=c, we can write

k ¼ l � c=ðptot � pÞ, which yields the stability

equation

4 For an example, see [16].

�F �1ðptotÞ ¼1

~kl

�� cptot � p

�:

In the case I ¼ 2 with prices p1 and p2, we have

�F �1ðptotÞ

¼ 1

~kal

�� cptot � p1

þ ð1� aÞl � cptot � p2

�

¼ 1

~kl

�� cptot � p1

� cptot � p2

�:

The proof of the theorem is in two steps: we first

show that, for a given ptot, the revenue generated is

greater in the case I ¼ 1; next, we show that, for allptot obtained in the case I ¼ 2, there exists a p in

the case I ¼ 1 giving the same ptot.Hence, for a given ptot, from the above equa-

tions in ptot we have that, necessarily,

1

ptot � p¼ 1

ptot � p1þ 1

ptot � p2;

that is,

p2 ¼ ptot �ðptot � p1Þðptot � pÞ

p � p1:

As p2 P 0 and p; p1; p2 6 ptot, we necessarily havethat pP p1. By symmetry, we also get that pP p2.Since, for a given ptot, the total arrival rate is the

same in both the I ¼ 1 and the I ¼ 2 cases, we

obtain that the revenue ðk1 þ k2Þp in the case I ¼ 1

is greater than k1p1 þ k2p2 in the case I ¼ 2.

Consider now a ptot obtained in the case I ¼ 2.Then a price

p ¼ ptot �1

1=ðptot � p1Þ þ 1=ðptot � p2Þ

gives the same value of ptot in the case I ¼ 1. The

only thing that remains to be proven is that such ap is non-negative. Since 1=ðptot � p1Þ þ 1=ðptot �p2Þ > 1=ptot, this is always the case. h

Remark 2. This result can be related to the well-

known problem of ‘‘splitting’’ a server in two.

Such separation introduces a supplementary mean

waiting time as one of the two servers may by idle

while the other has some customers waiting forservice, something that does not occur in the sin-

gle-server case (see, for instance, [18]).


Remark 3. The same result applies if we consider

the waiting time instead of the response time.

Moreover, in the former, the revenue generated in

the case I ¼ 2 tends to the revenue in the case

I ¼ 1 if a tends to 1. The reason why this does nothappen when considering the response time is that

the first class (which gets almost all the service rate

when a gets close to 1) has to match the total cost

of the second class, which tends to infinity with its

mean service time 1=ðð1� aÞlÞ.

6. A multi-application extension with per-applica-tion delay valuations

We will now consider the (more realistic) case in

which there are K types of applications/customers,

each having a different delay valuation. Let cðkÞ

denote the delay valuation for the class-k appli-

cation. Note that we will use the superscript k for

denoting application classes, which are differentfrom the PMP classes––i.e., subnetworks––noted

by the subscript i.Instead of a total potential arrival rate ~k and a

utility random variable U , we now have a collec-

tion of such inputs ð~kðkÞ;U ðkÞÞ for 16 k6K. De-

note by kðkÞ the total arrival rate of application-kpackets, kðkÞ

i the arrival of such packets in sub-

network i and, as before, ki the total arrival rate insubnetwork i. We have ki ¼

PKk¼1 kðkÞ

i and

kðkÞ ¼PI

i¼1 kðkÞi :

In this case, the set of Wardrop equilibrium

equations (1) becomes:

kðkÞ ¼PI

i¼1 kðkÞi ¼ ~kðkÞP ðU ðkÞ P pðkÞtotÞ

for 16 k6K;pi þ cðkÞdi ¼ pðkÞtot

for 16 i6 I and 16 k6K

8>>><>>>:

ð6Þ

with di ¼ fiðkiÞ:This means that the set (6) has to be verified for

all k, but with a different ptot for each k. Note

however that the equations cannot be solved

independently 8k since they are related through

the queueing delays di (16 i6 I).

Remark 4. In this case, the system behaves like a

game where each player k (i.e., each application)

tries to stabilize its traffic, that is, it tries to

equalize pi þ cðkÞdi for all i, meaning that pðkÞtot exists.

In this sense, a solution of (6), if it exists, is a Nash

equilibrium between all the classes.

Proving the existence and uniqueness of thesolution is more complicated in this case, since we

do not have a single ptot (see [16, Section 6.2.1] for

the single-valuation, multi-application case).

Therefore, finding out conditions for existence and

uniqueness is still an open question.

Nevertheless, based on the results in previous

sections we can still find a necessary, but not suf-

ficient, condition for equilibrium.

Proposition 1. Assume that p1 P p2 P � � � P pI . Anecessary condition for solving (6) is that

XKk¼1

~kðkÞ�F ðkÞðp1ÞPXIi¼1

f �1i

p1 � picðkÞ

� �: ð7Þ

Proof. We have, 8k,

pðkÞtot ¼ ð�F ðkÞÞ�1 kðkÞ

~kðkÞ

!: ð8Þ

where �F ðkÞðxÞ ¼ P ðU ðkÞ P xÞ. If (6) is verified, thenwe also have that, 8i,

pðkÞtot ¼ pi þ cðkÞdi: ð9Þ

We will follow now an argument similar to that

we used in the proof of Theorem 1. From (8), we

can see that kðkÞ decreases when pðkÞtot increases, so

k ¼PK

k¼1 kðkÞ ¼PK

k¼1~kðkÞ�F ðkÞðpðkÞtotÞ decreases when

pðkÞtot increases. From (9), we see that ki increases

when pðkÞtot increases, so k ¼PI

i¼1 ki ¼PI

i¼1 f�1i

ðpðkÞtot � piÞ=cðkÞ�

increases as well. We want to find

a necessary condition under which there exists pðkÞtot

such thatPK

k¼1 kðkÞ ¼PI

i¼1 ki. SincePI

i¼1 f�1i

ððp � piÞ=cðkÞÞ is a strictly increasing function of p,and

PKk¼1

~kðkÞ�F ðkÞðpÞ is a strictly decreasing func-

tion of p, the inequality given in the proposition

must hold at the smallest value p ¼ p1, since

pðkÞtot P p1 8k. h


The reason why this proof does not provide a

sufficient condition for the existence and unique-

ness of equilibrium is that, in the proof, we look at

the existence of pðkÞtot for a given k, and not at the

existence of the pðkÞtot all together.

7. Conclusions

In this paper, we have introduced a mathe-

matical model of the Paris Metro Pricing (PMP)

scheme for charging packet networks. This pricing

method looks convenient for Internet Service

Providers since it is fairly easy to implement and

deploy in an ISP network using current, off-the-

shelf technologies. Even if PMP does not provide

strict QoS guarantees, users would likely prefer itto more complex pricing schemes since the total

charge is linear in the volume of data, hence pre-

dictable.

Our model has allowed us to find some neces-

sary and sufficient conditions on the prices re-

quired to obtain an equilibrium. An extension to

the model, taking into account the multi-applica-

tion case, is presented. Numerical illustrations areprovided for the basic model. Other possible

extensions include: time-of-day pricing and con-

sidering other QoS metrics than delay only. In [16],

we discuss in detail a model in which quality, as

perceived by a user, is a mixture of delay and loss

probability.

The model has pointed out a possible drawback

of the PMP scheme, namely, that for a given net-work capacity revenues may be lower in a network

implementing PMP. This limitation, coming from

the adopted bandwidth-sharing policy among

subnetworks, could be alleviated by means of

more efficient scheduling mechanisms.

As directions for future research, we wish to

investigate more deeply the multi-application case

with a per-application delay valuation. Morespecifically, whereas in the case of a single delay

valuation we have obtained a sufficient condition

(Eq. (2)), we only have a necessary condition for

the multiple-valuation case. Besides, we have not

considered yet the case where condition (2) is not

satisfied for some applications, so that, for each

k, the expressions pi þ cðkÞdi ¼ pðkÞtot are only taken

into account for a subset of subnetworks i sincethe other subnetworks are considered too

expensive. Indeed, we have forced in (6) that

each application type send packets to all sub-

networks. All the possible combinations of those

cases have to be addressed. We are also inter-ested in trying to carry out the same kind of

analysis for the round-robin scheme of [12]. Fi-

nally, the case of strict QoS requirements could

be investigated.

Acknowledgements

The authors would like to thank Peter Reichl

and the anonymous reviewers for their construc-

tive remarks.

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Research report PI-1512, IRISA (March 2003). URL

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David Ros received his B.Sc. (withhonors) and M.Sc. degrees, both inElectronics Engineering, from theSim�on Bol�ıvar University, Caracas,Venezuela in 1987 and 1994, respec-tively, and his Ph.D. in ComputerScience from the Institut National deSciences Appliqu�ees (INSA), Rennes,France in 2000. He is currently work-ing as a Research Engineer at GET/ENST Bretagne, Rennes, in the Net-working and Multimedia Department.His active research interests includenetwork modelling and simulation, as

well as pricing and quality of service issues in IP networks.

Bruno Tuffin received his Ph.D. degreein applied mathematics from Rennes 1University, France in 1997. Since, hehas been with INRIA-Rennes, France.His research interests include develop-ing Monte Carlo and quasi-MonteCarlo simulation techniques for theperformance evaluation of computerand telecommunication systems, andmore recently developing Internet ac-tive measurement techniques and newpricing schemes. On this last topic, he isthe coordinator of the INRIA’s coop-erative research action PRIXNET (see

http://www.irisa.fr/armor/Armor-Ext/RA/prixnet/ARC.htm).

http://www.irisa.fr/bibli/publi/pi/2003/1512/1512.html

http://www.irisa.fr/armor/Armor-Ext/RA/prixnet/ARC.htm

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A mathematical model of the Paris Metro Pricing scheme for charging packet networks