An Auction Pricing Strategy for Differentiated Service Networks
Weilai Yang* , Henry Owen* , Douglas M. Blough* and Yongpei Guan†
*School of Electrical and Computer Engineering
†School of Industrial and Systems Engineering
Georgia Institute of Technology
Atlanta, GA 30332-0250 USA
Abstract
We use pricing as an effective strategy to allocate
network resources in an efficient way so as to maxi-
mize a service provider’s revenue. Among all static
and dynamic pricing strategies, an auction approach
is a widely proposed decentralized mechanism. We
propose a scenario where all clients can bid for their
required bandwidth as well as the price they are will-
ing to pay. The service provider will decide on the
admission price and differentiated service provided
for each class. These thresholds also provide a future
reference for admitting new flows later on.
1 Introduction
The Service Provider (SP) controls network resource
allocation to multi-users to provide a certain level
of Quality of Service (QoS). We study the revenue
maximization problem of a price-based resource allo-
cation scheme for Differential Service (DiffServ) data
networks. Since the network supports multiple class
services, it needs a differentiated pricing strategy in-
stead of the flat-rate pricing model used by current
Internet services. How to charge customers in the
most efficient way becomes a big issue.
Pricing has recently attracted significant attention
to achieve economic efficiency in the Internet. A num-
ber of pricing schemes have been proposed [4] [7] [10].
Despite the various strategies presented, the basic
idea is that the appropriate pricing policy will provide
incentives for users to behave in ways that improve
overall utilization and performance. An auction is a
mechanism that consists of clients submitting bids,
including the desired amount of resources and the
price they are willing to pay, and the acutioneer who
is responsible for allocating shares of the resources
based on clients’ bids. It is a decentralized mecha-
nism for efficiently and fairly sharing resources inside
a network [6].
The DiffServ domain is a class-based network. Diff-
Serv defines router behaviors expected by packets
belonging to each of these classes. Therefore, we
need a uniformed pricing model for every single class.
Paper [7] introduced the idea of the “smart mar-
ket” as an efficient pricing mechanism. It allows
users to bid on each packet. The packet gets trans-
mitted if the corresponding bid exceeds the current
marginal cost of transportation. An advantage of this
scheme is that the user usually only pay the lower
market-clearing price. The bidding process happens
on a hop-by-hop basis. However, generally customers
don’t care about anything going on inside the net-
work, but they do care about end-to-end behavior.
This causes all the hops to have to be done with
the bidding at once, which brings an overwhelming
complexity to the bidding mechanism. Another main
disadvantage is that in the real world, the computa-
tional burden of updating prices dynamically can be
quite high.
In this paper, we consider a scenario that takes cus-
tomers’ bids and gives thresholds for both price and
service offered for each class to maximize the SP’s
revenue. Flows coming and joining in later will be
subjected to those thresholds during admission con-
trol. The whole pricing threshold update will occur
within a certain time interval. From the historical
data of a SP’s charge to clients and corresponding
service offered, customers know the price range gener-
ally. They then can propose their required minimum
bandwidth as well as the price they are willing to pay.
The SP calculates a minimum bandwidth they would
provide to each class based on all the bids. Given our
newly proposed DiffServ pricing strategy, our goal is
to maximize the SP’s profit. Our contributions can
be summarized as follows:
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• We model the transaction between the customer
and the SP. We analyze the SP revenue optimiza-
tion problem based on a bidding mechanism.
The novelty is that customers have the freedom
to choose the service and price. Because of that,
the SP needs to find a way to maximize profit.
The final solution reflects not only the SP’s ben-
efit but also the customer’s real willingness to
pay for the service.
• The actual charged price will be the lowest
among those admitted flows. It assures cus-
tomers that they will never pay higher than their
bids and ususally it will be lower.
• The optimal solution also gives a guideline for
future reference as to how to admit new flows.
• We transform flow-based input information into
a class-based DiffServ domain. This makes it
possible to actually realize the auction idea in
DiffServ.
2 Related Work
Over the last several years, different methods of cre-
ating economic models for resource management have
been proposed by a number of researchers. Some of
them are based on dividing traffic into multiple pri-
ority classes, but using fixed prices for each service
class [4]. By adding congestion-dependent compo-
nents into the price gives a different dynamic pric-
ing strategy. It takes network activities into account,
improves network efficiency, and offers a more com-
petitive price. Congestion-dependent pricing charges
are determined on a per-call basis made at the time
the call is admitted. Wang [10] proposes a strategy
where the price depends on the service class’ average
demand. Specifically, the price is negotiable through
a negotiation protocol. However, it requires resource
reservation in the network, which can raise a few key
issues such as inefficient use of the network, increased
network cost, and most importantly impractical use
in real time.
As Mackie-Mason and Varian mentioned, the price
for sending packets from a particular flow should be
positive [7]. From a service provider’s stand point,
the price should be differentiated when different kinds
of services are offered. When the SP faces a potential
customer, he would be able to compare his own ben-
efit by adding a new customer to the marginal cost
he imposes on other users.
Courcoubetis et al. gave a pricing model for Diff-
Serv by assigning the same amount of bandwidth
to all classes [5]. Equivalent bandwidth allocation
for DiffServ is not a reasonable strategy though it
keeps the format simple. Pricing based on equivalent
bandwidth is not fair for customers getting different
treatments. The scheme proposed in [4] used price
to reflect the resource demand and supply situation.
Pricing is worked out under a well-defined statistical
model of source traffic. However, they do not take
into account of traffic dynamic changes. This limita-
tion does not fit well with current network status.
The auction algorithm is an effective model for
solving classical assignment problems [3]. Bertsekas
also pointed out that an auction outperforms sub-
stantially its main competitors for important types of
problems, both in theory and in practice, and is also
naturally well suited for parallel computation [3].
3 Problem Formulation
The network model that we use makes the same as-
sumption as reference [9]. This assumption is that the
network can be abstracted into a single bottleneck
capacity, thus the analysis is simplified to a single
link. An absolute amount of bandwidth can also be
used to represent the capacity [9]. User experiments
reported in the literature [1] suggest that utility func-
tions typically follow a model of diminishing returns
to scale, that is, the marginal profit as a function
of bandwidth diminishes with increasing bandwidth.
Hence, [10] develops a general revenue function as a
function of bandwidth:
U
kj
= U
0j
+ W
j
log
X
kj
L
kj
where U
kj
stands for the revenue from client k, which
belongs to class j. W
j
is the sensitivity of the price
1
to
bandwidth for class j. U
0j
is the base price for class
j.
The base price for each class has been fixed by an
internet service provider already. Customers bid with
a sensitivity and a minimum bandwidth. The objec-
tive will be to maximize the SP’s revenue, subject to
the system’s available resources. The mathematical
formulation is as follows.
Decision variables:
Z
ij
=
1; if client i is admitted to class j
0; otherwise
X
ij
: bandwidth obtained by client i for class j;
L
mj
: minimum bandwidth for class j;
W
j
: price sensitivity for class j;
1
The sensitivity of the price is the amount that customers
are willing to pay if allocated bandwidth is more than the
minimum they require.
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X
j
: bandwidth assigned to each individual client in
class j;
Objective function:
max
3
j=1
i
(U
0j
+ W
j
log
X
ij
L
mj
) ∗ Z
ij
(1)
Subject to:
3
j=1
i
X
ij
<= Q
X
ij
≥ L
mj
− (1 − Z
ij
) ∗ M
W
j
≤ W
ij
+(1− Z
ij
) ∗ M
X
ij
≥ V
i
− (1 − Z
ij
) ∗ M
X
ij
≥ X
j
− (1 − Z
ij
) ∗ M
X
ij
≤ 0+Z
ij
∗ M
X
ij
≥ 0; L
mj
≥ 0; W
j
≥ 0
X
ij
≤ X
j
Parameters:
U
0j
: base price for class j
Q : total bandwidth
V
i
: minimum bandwidth required by client i
M : a very large positive number
The scenario is that all the customers propose their
values of W
ij
and L
ij
. We have to decide which flows
should be admitted for each class with the objective
of maximizing the SP’s revenue. For the flows admit-
ted to class j, we adopt the minimum W
ij
as our W
j
and the maximum L
ij
as our L
mj
.
4 Optimal Solution
First, we introduce an optimal solution within one
class, given a certain amount of bandwidth. Then,
we will expand that into multiple classes.
4.1 Solution in One Class
Within one class, i.e.j, we omit j index and simplify
the formulation. The objective function becomes max
i
(U
0
+ W log
X
i
L
i
). Suppose N is a set of all cus-
tomers who bid, M is the accepted customer set and
Q
j
is the assigned amount of bandwidth to this class
j. The bid values from customer i are L
i
and W
i
.Ac-
cording to our policy of choosing L and W values, L
should be the maximum value from the set M, while
W the minimum value. This rule also ensures that
{L∈L
x
,x∈N} and {W∈W
x
,x∈N}. Therefore, L =
max {L
i
,i∈M},andW =min{W
i
,i∈M}.Wecon-
struct another set S with all the combinations of L
x
,
W
x
values. S={(L
x
,W
x
), x∈N}. For each combina-
tion (L
x
,W
x
), we find all the flows whose L value is
less than or equal to L
x
and whose W value is greater
than or equal to W
x
. Record the number of flows as k.
k is the number of flows that can be possibly admitted
if L= L
x
and W =W
x
. As defined earlier, each flow
is sharing Q/k amount of bandwidth. If Q/k <L,
that means there are too many flows and there is not
enough bandwidth to support all of them. So we have
to reduce the number k until the inequality Q/k >L is
valid. When we have number k for each combination,
the corresponding profit U
x
=
x
(U
0
+ W log
X
x
L
x
)
can also be generated. So each combination has 2
values associated: k
x
and U
x
. Now, we need to fil-
ter out some unqualified combinations by using the
value m
j
, the number of flows that can be accepted
in class j. Beginning with m=1, find all the com-
binations whose k
x
≥m and choose the one with the
highest U
x
, recorded as U
x
1
. Increment m until m is
equal to the total number of flows n, and calculate
a U
x
for each m. From the set { U
x
1
, U
x
2
,..., U
x
n
},
the U
x
i
with the highest profit is selected. We then
can get the corresponding m and (L,W ). Therefore,
we have obtained the best values of L
i
, W
i
,andm
i
for the class. The corresponding value U is the SP’s
optimal profit for class j. L and W are used as the
bid thresholds for the flows in class j.
To elaborate on this procedure, we give a simple
example as follows. Suppose in this single class, we
have the following bids:
C1:(L
1
=2M,W
1
=10); C2:(L
2
=3M,W
2
=11);
C3:(L
3
=2.5M,W
3
=9); C4:(L
4
=10M,W
4
=12);
C5:(L
5
=8M,W
5
=6).
The available bandwidth is given at 12M and the base
price is given at 20. From these bids, we can build a
matrix, including all the combinations of L
x
and W
x
.
(L
1
,W
1
)(L
1
,W
2
)(L
1
,W
3
)(L
1
,W
4
)(L
1
,W
5
)
(L
2
,W
1
)(L
2
,W
2
)
...
(L
5
,W
1
)(L
5
,W
2
) .. .. ..
Starting from (L
1
, W
1
), we look for clients whose
L
i
≤L
1
, W
i
≥W
1
. Only client 1 itself satisfies the cri-
teria. So, k=1. Test by dividing Q
j
/k = 12M and it’s
greater than L
1
=2M; therefore, it’s a valid k value for
(L
1
=2,W
1
= 10) combination. Provided L=2 and
W=10, the revenue is U =20+10log12/2=27.78.
Proceed in this way to get all the ks and Us. Then,
the next loop involves with the number of flows get-
ting admitted. When m=1, all combinations with
k≥ 1 will be considered and the highest one with U
value is U
1
. Next, m increases to 2, and a corre-
sponding U
2
is chosen. Then, we have {U
1
, U
2
, U
3
,
U
4
, U
5
}. Among them, the highest U is our final
revenue. The m value associated with that U is the
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number of clients SP should admit. The combination
(L, W) that generated the U value is SP’s threshold.
Until now, we have generated the L and W values
to maximize SP’s revenue, based on current clients’
bids. Next, we introduce some properties to show
how the SP should admit new flows to maintain its
maximum profit.
Property 1:IfW
j
and L
j
are kept the same, as long
as the inequality Q
j
/m
j
>L
j
is valid, it’s always true
that the more flows added in, the higher value U
j
is.
Proof:
The revenue function is:
U
j
= m
j
∗ U
0j
+ m
j
∗ W
j
log
Q
j
m
j
L
j
Its derivative is:
∂U
j
∂m
j
= U
0j
+ W
j
log
Q
j
m
j
L
j
− W
j
Since Q
j
/m
j
>L
j
is valid, as long as U
0j
is greater
than W
j
,
∂U
j
∂m
j
is always greater than 0. That guar-
antees that U
j
is a strictly increasing function.
Using property 1, the SP can increase the revenue
by admitting more flows with fixed W,L values. So,
after the bidding thresholds have been decided, prop-
erty 1 tells the SP how to admit new flows to maxi-
mize the profit.
4.2 Solution in Multi-Classes
When we put the scenario in three classes, the objec-
tive function and corresponding constraints formu-
lated as a Lagrangian are
max[ m
1
∗ (U
01
+ W
1
log
Q
1
m
1
L
1
)+
m
2
∗ (U
02
+ W
2
log
Q
2
m
2
L
2
)+
m
3
∗ (U
03
+ W
3
log
Q
3
m
3
L
3
)];
Q
1
+ Q
2
+ Q
3
= Q;
(2)
Therefore, we have the following solution
⇒
Q
1
=(m
1
W
1
)/(m
1
W
1
+ m
2
W
2
+ m
3
W
3
) ∗ Q
Q
2
=(m
2
W
2
)/(m
1
W
1
+ m
2
W
2
+ m
3
W
3
) ∗ Q
Q
3
=(m
3
W
3
)/(m
1
W
1
+ m
2
W
2
+ m
3
W
3
) ∗ Q
We notice that the W value does not affect oth-
ers like Q, m, and L. While Q, m and L are closely
related. When value m and L are fixed, it’s always
the best to choose the highest W to get a high U. In
the last section, we introduced how to work with all
(L
x
, W
x
) combinations in one class to get the value
k. Now we use that information and work in another
way. First, start from m=1 and L
i
(i=1) and check
all the combinations (W
x
) with L
1
. From these, the
effective ones are those with k≥m; then, choose the
one with the highest W. Now, we have m, L, and
W for class j. The same procedure applies to other
classes and we have m
1
, L
1
, W
1
, m
2
, L
2
, W
2
, m
3
,
L
3
, W
3
.NowQ
1
, Q
2
, Q
3
can be solved according to
equation 2. Lastly, we check the feasibility of each
solution by calculating Q
1
/m
1
,Q
2
/m
2
, Q
3
/m
3
. Only
if all of them are greater than L
1
,L
2
,L
3
respectively,
then we consider this a feasible solution and record
the corresponding U value. Otherwise, it is aban-
doned. Following the same steps by changing the
value of m and L
i
, we can get all the possible feasible
solutions. Finally, among all the feasible solutions,
the highest U is the optimal solution.
5 Simulation and Analysis
We randomly generate a set of clients who partici-
pate in the auction. The bids include the required
service class, desired minimum bandwidth, and the
price they are willing to pay. We use their bids and
the network’s capacity as inputs into our multiclass
pricing model. The SP’s revenue is produced as out-
puts. Meanwhile, we manually assign a fixed network
capacity to each class. Then, our single-class solution
is used to solve the case individually within a class.
The final revenue will be the sum of three individual
ones. The motivation of doing so is to show that the
multiclass solution takes into account not only the
single class, but also the competition among those
classes. The process is actually a two-step auction.
In the first step, clients inside each class compete with
each other to get the price as their offered price. In
the second, each class tries to get as much bandwidth
as it can. Finally, it comes to a converged balance
point where it reaches the highest point of the SP’s
final revenue.
We use three simple techniques get the bandwdith
allocation for each class, run the single class optimiza-
tion algorithm inside the class and get the final total
revenue by adding them together. We compare those
results with the ones generated by our multi-class op-
timization algorithm to show that the multi-class op-
timization outperforms the three simple ways.
The ways to decide on each class’ bandwidth al-
location are: 1) Get the ratio of each class band-
width assignment by selecting the highest bid from
the desired minimum bandwidths for that class. For
instance, the highest bid from class 1 is 20M, 12M
from class 2, and 0.2M from class 3. The ratio is
then, 20:12:0.2. So, class 1 gets 20/(20+12+0.2) of
total capacity. The same formula for class 2 and
3. 2) Use the number of customers who bid as
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the reference for calculating the ratio. For exam-
ple, 2 customers bid for class 1, 4 for class 2, and
10 for class 3. The ratio is 2:4:10. The amount of
bandwidth is divided proportionally. 3) Calculate
the total bandwidth bidded for each class if all the
customers get admitted. The bandwidth allocation
for classes would be proportional to the total band-
width. For example, 2 customers in class 1 bidding
for 2M and 3M; 3 customers in class 2 bidding for 1M
and 1.5M; 6 customers in class 3 bidding for .09M,
.1M, .15M, .2M, .21M, .08M. The ratio comes at:
(2+3):(1+1.5):(.09+.1+.15+.2+.21+.08).
We set the revenue results from multi-class opti-
mization algorithm to a value of one on the y-axis.
The cases shown in Figure 1 are for six different sets
of customer demands and bids. These customer de-
mands and bids are varied to allow our algorithm
to be compared to the other three simple algorithms
over a variety of traffic inputs and bids. The rev-
enue generated by our multi-class algorithm (set to
units) exceeds the revenue generated by the simplier
algorithms.
Figure 1: Comparison
6 Conclusion
This paper was motivated by the problem of provid-
ing differentiated QoS to clients to maximize service
provider’s profit. We presented a novel pricing strat-
egy of maximizing service provider’s revenue based
on clients’ bids of price as well as desired service.
This scheme allows customers to express their will-
ingness to pay along with their required service. The
service provider calculates the thresholds for each ser-
vice class according to network resource availability.
The thresholds can also be used as a future refer-
ence for admitting new clients. Our simulation re-
sults show that the algorithm works well in assigning
proper bandwidth to each class in terms of maximiz-
ing the service provider’s profit.
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