Auction-based Congestion Pricing for Wireless Data Services

Abstract

Due to the surprising growth of mobile subscribers, GSM networks have been experiencing congestion, especially in their data services. Pricing of network resources could help resolve congestion. However, due to the absence of an accurate user demand estimation, determining suitable congestion prices for the mobile networks can be problematic. In this paper, we propose an auction-based pricing approach for resource allocation in mobile data networks, specifically GPRS networks. In addition, the impact of the reserve prices of the auction mechanism on the mean system delay of the networks is investigated. The numerical results show that, by increasing the auction reserve price, the level of network congestion is significantly reduced, as indicated by the reduction of the mean system delay of the network. The tradeoff between performance improvement, due to the proposed model, and revenue optimization is also investigated.

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Auction-based Congestion Pricing for Wireless Data
Services
Saravut Yaipairoj
Telecommunications Management
Stevens Institute of Technology
Hoboken, New Jersey 07030
Email: syaipair@stevens.edu
Fotios C. Harmantzis
Telecommunications Management
Stevens Institute of Technology
Hoboken, New Jersey 07030
Email: fharmant@stevens.edu
Abstract—Due to the surprising growth of mobile subscribers,
GSM networks have been experiencing congestion, especially in
their data services. Pricing of network resources could help re-
solve congestion. However, due to the absence of an accurate user
demand estimation, determining suitable congestion prices for the
mobile networks can be problematic. In this paper, we propose
an auction-based pricing approach for resource allocation in
mobile data networks, specifically GPRS networks. In addition,
the impact of the reserve prices of the auction mechanism on the
mean system delay of the networks is investigated. The numerical
results show that, by increasing the auction reserve price, the level
of network congestion is significantly reduced, as indicated by the
reduction of the mean system delay of the network. The tradeoff
between performance improvement, due to the proposed model,
and revenue optimization is also investigated.
I. INTRODUCTION
Network traffic in mobile networks has grown exponentially
in recent years. This trend keeps continuing, due to the
surprising growth of wireless subscribers. Penetration within
mature markets and growth of emerging markets continue to
exceed the expectation of network operators, which contributes
to increasing network congestion. Sophisticated resource allo-
cation schemes based on certain technical approaches have
been proposed to alleviate and prevent congestion. However,
these schemes could respond to only certain dynamic range of
incoming traffic. Mechanisms that provide incentives, which
influence the usage behavior of subscribers, are required to
achieve more efficient allocation of scarce wireless network
resources. Pricing could be the means to resolve congestion
challenges in wireless networks.
Using prices, the network signals to users, providing in-
centives which influence their usage behavior [2]. In fact,
pricing has become a means for both traffic management
and congestion control. A number of pricing schemes have
been recently proposed (mostly for internet services), such
as Flat Pricing, Priority Pricing, Paris-Metro Pricing, Smart-
market Pricing and so on [3]. In the wireless arena, since
certain wireless services have been enabled by the use of
IP in their data applications, there have been a lot of effort
to apply such pricing schemes to emerging wireless services
(after appropriate modifications).
The price setup in each pricing scheme can be determined
based on certain criteria, mainly, the cost of services, the
revenue expected by network operators and, most importantly,
the demand for services by the subscribers. As for the sub-
scribers demand, it can be a function of quality of services,
price structure, and application, which varies significantly
among different markets. The methodologies for estimating
user demand in particular markets require an extensive survey,
whose accuracy is affected by the forecasted group size,
customer characteristics, customer classification system and
the extent of demand survey [4]. The methodologies could be
too complex and time-consuming to achieve. In the absence
of accurate demand estimation, auction mechanism can be
used to solve certain pricing issues. The fact that an auction
mechanism draws out the demand information directly from
users allows the network to set prices for the network services
quickly and effectively [17].
Allocating network resources through economic mecha-
nisms has been studied in many literatures [2], [13], [14].
Regarding the mechanisms based on auction theory, most of
the research works aim for internet applications. Smart-Market
pricing scheme [2] is a well-known auction-based mechanism,
which is closely related to our work. Lazar et al. [5] have done
tremendous work on the auction for internet bandwidth. The
impact of auction reserve price on the network revenue is also
studied in [1], [6]. In this paper, we propose an auction-based
pricing model that is used to allocate network resources for
GPRS data services. The impact on the performance analysis
due to the change of an auction reserve price is investigated.
This paper has been much improved from our previous work
[15] as it includes the case of multiple winners in one auction
round and its effect on the network performance. In addition,
the auction reserve prices that yield optimal tradeoff between
the network performance and revenue is also investigated.
II. PROBLEM STATEMENT
General Packet Radio Service (GPRS) has been deployed
as a data extension of GSM networks. It supports best-effort
data service and provides a major improvement in spectrum
efficiency for both voice and data services in cellular networks.
Even though theoretical speed of GPRS data service is up to
171.2 kbps, in reality, the achievable transmission rate is much
lower than expected (approximately 50kbps), due to GPRS
best-effort architecture. In addition, GPRS data service fails

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to support most real-time applications due to the resource
sharing between voice and data traffic [7], [8]. Since the
priority of voice services is higher than that of data services,
few traffic channels are available for data traffic. Performance
could be even worse when the system experiences high volume
of aggregated traffic. Therefore, resource allocation techniques
for the data services in GPRS networks can be a real challenge.
There has been enormous amount of work that investigates
the integration of mixed services in cellular networks, aiming
to improve the quality of the data services [9], [10], [7].
However, most technical schemes can improve the system
performance of GPRS networks within a certain dynamic
range of incoming traffic. When the traffic arrival rate is
temporarily very high, such as during the busy hours, the
difference between peak and off-peak demand for certain
mobile services can be of a magnitude of 20 to 1 [11]. Due to
the best-effort architecture of GPRS, these technical schemes
cannot sufficiently improve quality of service (regardless of
how well the parameters are adjusted [12]). Besides the
engineering approaches used to improve GPRS performance,
certain approaches on resource allocation based on economics
theory can be effectively used to alleviate the congestion
in the networks. In this paper, we propose an economics
approach based on auction theory that effectively enhances
GPRS services and alleviates network congestion.
III. MODEL DESCRIPTION
GSM voice services and GPRS data services would share
traffic channels in a single cellular cell. There are C channels
in total, where Cg channels are reserved as guard channels
for GPRS traffic; Cv channels are shared by voice and data
services. However, voice calls have the right of priority and
preemption over data calls. Both voice call arrival and GPRS
data packet arrival are modeled according to two mutually
independent Poisson processes with rates λv and λg, respec-
tively. The channel holding time of a voice call and the service
time of a GPRS data packet are exponentially distributed with
mean 1/µvand 1/µg, respectively. The incoming GPRS traffic
is treated as FCFS transmission and queued in a buffer of size
B whenever all traffic channels are occupied. The delay of
GPRS services is monitored by the base station to determine
the system congestion. The handoff calls or guard channels for
voice calls are not considered here. During network congestion
indicated by the increasing delay in the GPRS system, we
employ an auction-based pricing mechanism to admit data
users into the system. A user has to associate her bid with
her service request. The user’s bid is transmitted as part of
the QoS profile to the Serving GPRS Support Node (SGSN)
via the base station. The base station identifies the number of
winners who could gain access to the system, based on channel
availability. By considering the QoS profile and the submitted
bid, the SGSN determines the radio priority level used for
user data. Subsequently, the SGSN informs the winning MS
for its radio priority level during the PDP Context Activation
and the PDP Context Modification procedure. The winning
MS will be granted access by receiving an IP address and a
QoS profile based on their winning bids. The winners of the
auction are allowed to access the network resource as long as
they do not violate the QoS profile. Regarding the handoff
calls for the data services, to eliminate the call dropping
in the situation when auction winners lose in the auction
as they move to a new base station, we assume that each
base station reserves certain number of traffic channels to
accommodate these coming handoff calls. This scheme allows
the users to access the traffic channels without participating in
an auction. Since handoff calls utilize separate traffic channels
that are not allocated through auctions, we do not consider the
handoff calls in our model. The auction rounds are established
periodically. Mobile terminals monitor auction information
through the GPRS attached procedure. Users, who are willing
to participate in an auction, are required to bid greater or
equal to a reserve price assigned by the network (The reserve
price is discussed later in section V). Users, who lose in the
auction, are allowed to resubmit a new bid in a later round,
or leave the system. Since the auction technique is simply
used for admitting users to the system, the data services are
still based on best-effort services. However, users can expect
better services during the congestion period due to the fact
that only the data calls from winning users would be able to
access the traffic channels at that time [15].
IV. GPRS DELAY ANALYSIS
Due to the prioritization of voice calls over data calls, the
voice blocking probability of GSM/GPRS networks can be
approximately determined by the erlang-B formula [16]. The
GPRS data services with a certain number of traffic channels,
m, can be modeled as M/M/m/m+B queuing model which
can be used to obtain performanceparameters such as blocking
probability and mean system delay. According to Liu et al.
[18], the blocking probability of GPRS can be obtained by
Pbg=
C
?
m=Cg
A(m)gm,m+B
(1)
where A(m) is the probability that there are m channels
available for a GPRS data packet and gm,m+B is the steady-
state probability of n GPRS packets in the system, which are
shown as follows:
gm,n=
?
gm,0
ρn
m!mn−m
ρn
g
n!
:
:
1 ≤ n ≤ m
m ≤ n ≤ m + B
gm,0
g
(2)
gm,0= [1 +
m−1
?
n=1
ρn
n!+
g
m+B
?
n=m
ρn
g
m!mn−m]−1
(3)
where pg= λg/µg is the GPRS traffic load in Erlang. To
calculate mean system delay, let us consider (2) and (3). Upon
a GPRS data packet arrival, it will find the system ready to
serve, having n users waiting, or being blocked with respective
probabilities, denoted by Pm,S, Pm,n, and Pm,b, which are
given by

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Pm,s=
m−1
?
i=0
gm,i
(4)
Pm,n= gm,m+n,0 ≤ n < B
Pm,b= gm,m+B
(5)
(6)
where S = λg(1 − Pbg) is the throughput of GPRS.
Assuming that there are m channels for GPRS packets, the
mean system delay,¯tm, according to the phase-type renewal
processes [19], [18] is given as
¯tm= Pm,S
1
µg
+
B−1
?
n=0
Pm,n¯tm,n
(7)
where¯tm,n=
the rate of a packet departure from the queue (mµg) and the
rate of serving a packet (µg). Finally, the mean system delay
for GPRS is [18]
1
µg+n+1
mµgis the mean system delay given
¯t =
1
(1 − Pbg)
C
?
m=Cg
¯tmA(m)
(8)
V. AUCTION-BASED PRICES FOR CONGESTION CONTROL
Our auction-based pricing model is implemented at the
admission phase of the GPRS networks. The auction used in
the model is a multi-unit second-price sealed bid auction or
multi-unit Vickery auction [20]. To maximize revenue gained
from the auction, we employ a revenue-maximizing auction
or a so-called optimal auction [1]. The optimal auction in this
case is basically a Vickery auction with an introduction of a
reservation price.
Let us assume that the network admits a certain number of
auction winners, K. The bidders’ private value (the value of
network access through the auction) is independently and iden-
tically distributed (IID) drawn from the common distribution
F(v) where; F(v) = 0, F(v) = 1, and F(v) strictly increasing
and differentiable over the interval (v,v). We assume further
that, all bidders are risk neutral. From the revenue equivalent
theorem based on the work by Riley and Samuelson [1],
the common equilibrium bidding strategy for Vickery auction
yields expected revenue to the seller as follows
E = n
?v
v∗
(vF′(v) + F(v) − 1)p(v)dv
(9)
where v∗ is the reserve price, n is the number of bidders
and v is the maximum bidders’ value. p(v) is the probability
of a bidder who can be one of the auction winners. p(v) is
simply the sum of binomial pdf with parameter n − 1 and
1 − F(v) shown as
p(v) =
K−1
?
i=0
?
n − 1
i
?
(1 − F(v))iF(v)n−1
(10)
Even though the revenue equivalent theorem is used to
analyze the auction of a single item, the theorem can be
generalized for our auction model. The auction in our case
is a multi-unit auction in which each winner requires only
one unit, i.e. one-time access to the networks. Hence, the
revenue-equivalence theorem still holds for these auctions
[17]. Based on the generalization of Vickery auctions, the K
highest bidders are admitted into the system and pay the value
of the highest losing bid.
The reservation price should correspond to the cost of net-
work resource realized by network operators (or the network’s
private valuation) and the users’bid distribution. Based on
[1], by making the IID assumption and consider risk-neutral
bidders, the reservation price, v∗, that maximizes the expected
revenue satisfies
v∗= v0+ 1 −F(v∗)
F′(v∗)
(11)
where v∗is the optimal reservation price, and v0is the net-
work’s private valuation. As discussed earlier, we focus on the
scenario when reserve prices can be used to prevent congestion
in the system. We first assume that the winners of an auction
generate a certain amount of traffic load into the network. Let
vcbe an increased reserve price (a congestion reserve price),
set by the network operators during the congestion period. The
new reserve price reflects the congestion cost attributed to the
network’s private valuation (v0). We further define the term
“eligible bidder” as the bidder whose bid is higher than the
reserve price. Hence, the number of eligible bidders during
congestion time (Nc) is the number of bidders whose bid
is higher than the congestion reserve price (vc). We assume
that, when there is no congestion, the reserve price is set at
an optimal value (v∗); however, it increases to vc when the
GPRS service experiences congestion. By assuming a certain
bid distribution and certain value of vc, we can prove that the
ratio of the area under the tail distribution, i.e. (1 − F(v)),
of the bid between vcand v∗would identify the ratio of the
number of eligible bidders between the state of congestion
and the normal state of operation, i.e. (Nc
we further assume that all eligible bidders, Nc, can be winners
(i.e., K ≥ Nc), then the ratio can be used as a reduction factor
of the offered load to the GPRS system. Hence, the reduced
offered load due to the auction mechanism can be described
as
N∗)(see equation 12). If
ρgc= ρg·Nc
N∗
= ρg·1 − F(vc)
1 − F(v∗)
(12)
where N∗is the number of eligible bidders when the reserve
price is at optimal value (v∗). ρg= λg/µgis the GPRS offered
load as described in the delay model. ρgcis the reduced offered
load based on the congestion reserve price, vc. Since GPRS
offered load is reduced, the GPRS system yields performance
improvement in the form of a smaller delay in the system.
In the case of K being small (K < Nc), meaning that only a
fraction of eligible bidders are admitted (according to K), the

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ratio (Nc
load. However, the impact on the traffic load can be described
by the probability of Nc being larger than K. Based on the
above assumptions, the reduced offered load can be described
as
N∗) cannot be used as the reduction factor of the traffic
ρgc(K) = ρg·(
K/N
1 − F(v∗)·P(vc)+
K−1
?
i=0
(K − i − 1)/N
1 − F(v∗)
·Pi)
(13)
where
P(vc) = 1 −
K−1
?
i=0
Pi= 1 −
K−1
?
i=0
?
n
i
?
(1 − F(vc))iF(vc)n−i
where P(vc) is the probability of Ncbeing larger or equal
to K. The underlined assumption is that the offered load
applied to GPRS network depends heavily on the number of
winners of the auction. Since the auction reserve prices limit
the number of eligible bidders who are potential winners in
each auction round, we can effectively reduce GPRS traffic
load by adjusting auction reserve prices, which consequently
yield reduction in the mean system delay of the GPRS.
Even though increasing auction reserve prices can reduce
traffic load resulting in improvement in terms of mean system
delay, the improvement is achieved through the cost of revenue
loss. It is important for network operators to identify the
performance improvement of the network according to the
revenue changed by the congestion reserve price vc. The
optimization problem in this context is simply how much the
system delay and network revenue will be reduced by the use
of the auction reserve price. We first define the normalization
of the mean system delay improvement of GPRS and the
revenue reduction as shown in (14). Then, we maximize the
difference of those terms in order to identify the auction
reserve price that achieves maximum delay improvement with
minimum loss of revenue. The objective function for the
optimization problem can be shown as
f(t,E) = max
vc
(¯t(v∗) −¯t(vc)
¯t(v∗) −¯t(v)−E(v∗) − E(vc)
VI. NUMERICAL ANALYSIS
In our experiment, we consider a single GPRS cell, which
could accommodate up to 30 traffic channels. The guard
channel Cg is equal to 1 and buffer size B is equal to 10.
The voice call blocking probability is held below the typical
value of 2 percent. We assume that the users bidding valuation
is distributed between 0 and 1. The average number of bidders
in each round is 10. The private valuation of the network v0is
zero if there is no network congestion. However, it increases
and contributes to the congestion reserve price (vc) during the
time of congestion. Fig. 1 illustrates the mean system delay
of GPRS versus GPRS traffic load when the reserve price of
the auction is increased from 0.6 to 0.7. We assume normal
distribution of the bid valuation by simply setting the mean
value and standard deviation at 0.5 and 0.2 respectively, which
E(v∗) − E(v))
(14)
0510 15
2
3
4
5
6
7
GPRS Load (Erlang)
Mean System Delay of GPRS (sec)
Vres is at optimal (0.6)
Vc is at 0.7 (Normal distributed)
Fig. 1. Means System Delay of GPRS when reserve price is 0.7, K=4, mean
service time is 0.2 sec.
yields the pdf of the normal distribution to be sufficiently
within 0 and 1. Based on our assumptions of bidder valuation
and delay analysis, it is clear that the mean system delay drops
significantly when the reserve price is increased. This is due to
the pricing incentive enforced by the congestion reserve prices
which discourage bidders from participating in the auction.
TABLE I
RESERVE PRICE OF THE AUCTION ROUND VERSUS MEAN
SYSTEM DELAY OF GPRS, (GPRS LOAD = 10 ERLANG).
Reserve Price
0.6
0.65
0.7
0.75
0.8
Mean System Delay of GPRS (sec)
3.2413
2.8186
2.6331
2.5458
2.5019
To elaborate our analysis, we continue increasing the auc-
tion reserve price to further reduce the mean system delay.
Table I shows the mean system delay of GPRS and the
corresponding reserve prices. From the table, the optimal
reserve price obtained from equation (11) is approximately
0.6. We can see that when we increase the reserve price
from 0.6, the mean system delay reduces significantly at the
beginning. As the reserve price approaches to 1, the mean
system delay of GPRS reduces at a slower rate. This is due to
the characteristics of normal distribution of the bidders value.
In Fig. 2, we illustrate the effect of the reserve price on the
mean system delay of GPRS. From Fig. 2(a), K is set at a large
value, e.g., K = 10, meaning that all eligible bidders are the
auction winners. To verify the numerical results, we simulate
auction rounds based on the Monte Carlo methodology as
follows: First bids are randomly generated according to the
normal distribution described earlier. The number of winners
is counted at the normal and congestion state, i.e. v∗ and
vc are used, respectively. The simulation runs 1000 times,

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to represent 1000 rounds of auctions, for each congestion
reserve price (vc). Based on the average number of winners
from the simulation, the ratio of the winners that identifies the
winner reduction factor is determined by (12). The revenue
generated from the auction is also collected. We can see that
the result from simulation firmly verifies the numerical results.
In addition, Fig. 2(b) shows the effect of the reserve price when
K is small, based on (13). We can see that the mean system
delay drops significantly as we limit the number of winners
in the auction by the use of the auction reserve price.
Fig. 3 shows the expected revenue versus the reserve price
based on (9). It shows how the expected revenue in an auction
round is affected by the number of winners allowed (K) and
the reserve price (vc). To reduce the mean system delay, we can
either reduce the number of winners allowed (K) or increase
the congestionreserve price (vc). Increasing vcyields a gradual
change in the auction revenue; reducing K results in an abrupt
change in revenue, which could damage the revenue figure of
the operator by overly preventing the congestion. Therefore,
enforcing reserve price is a more suitable choice for improving
performance of the GPRS system.
According to the results in Fig. 2 and Fig. 3, it is important
to illustrate the tradeoff between the improvement of network
performance and revenue loss caused by increasing the auction
reserve price. Fig. 4 illustrates delay improvement and revenue
loss after they are normalized as described in the previous
section. As we start increasing the auction reserve price away
from the optimal value, the mean system delay improvement
increases more quickly than the revenue loss. The increse
is more significant when K is large (see Fig. 4(b)) The
results allow us to identify the auction reserve price that
yields the largest difference between the two curves, which is
simply the price at which we can achieve the maximum delay
improvement with minimum revenue loss. Fig. 5 illustrates
such tradeoff according to (14). As we can see, when K=1, the
curve shows two points (vcequals to 0.78 and 0.89) where we
can set the reserve prices that yield good tradeoff, depending
on how much delay improvement network operators would
require. For K=4, the auction reserve price should be set at
approximately 0.7.
VII. CONCLUSION
In this paper, we have proposed an auction-based congestion
pricing scheme, which can improve performance of data ser-
vices in GPRS networks. When determination of the conges-
tion price through the estimation of user demand in particular
markets is too complex and time-consuming, auctions can be
used to solve certain pricing issues. The auction mechanism
we use for our model is based on a revenue-maximization
auction (optimal auction). This type of auction introduces a
reservation price as a mean to maximize the revenue gained
from the auction. However, we employ the reservation price
for the revenue maximization as well as the congestion control.
Our analytical results show that, by increasing the auction
reserve price, we can improvethe mean system delay of GPRS.
This is due to the fact that only data traffic generated by
0.60.7 0.80.91
2.6
2.8
3
3.2
3.4
Reserve Price
Mean System Delay (sec)
Numerical
Simulation
0.60.70.80.91
2.5
2.6
2.7
2.8
2.9
3
3.1
3.2
Reserve Price
Mean System Delay (sec)
Numerical K=1
Simulation K=1
Numerical K=4
Simulation K=4
Fig. 2.
Delay of GPRS vs Reserve price with K=1 and 4
(a) Mean system delay of GPRS vs Reserve price (b) Mean system
0.60.70.80.91
0
0.5
1
1.5
2
Reserve Price
Revenue from an auction
Numerical K=1
Simulation K=1
Numerical K=4
Simulation K=4
Fig. 3.Expected Revenue of an auction round in the GPRS system.