On-line Pricing of Secondary Spectrum Access with
Unknown Demand Function and
Call Length Distribution
Huseyin Mutlu, Murat Alanyali, and David Starobinski
Department of Electrical and Computer Engineering
Boston University, Boston, MA 02215
E-mail: {hmutlu,alanyali,staro}@bu.edu
Abstract—We consider a wireless provider who caters to two
classes of customers, namely primary and secondary users.
Primary users have long term contracts while secondary users
are admitted and priced according to current availability of
excess spectrum. Secondary users accept an advertised price
with a certain probability defined by an underlying demand
function. We analyze the problem of maximizing profit gained by
admission of secondary users. Previous studies in the field usually
assume that the demand function is known and that the call
length distribution is also known and exponentially distributed.
In this paper, we analyze more realistic settings where both of
these quantities are unknown. Our main contribution is to derive
near-optimal pricing strategies under such settings. We focus on
occupancy-based pricing policies, which depend only on the total
number of ongoing calls in the system. We first show that such
policies are insensitive to call length distribution except through
the mean. Next, we introduce a new on-line, occupancy-based
pricing algorithm, called Measurement-based Threshold Pricing
(MTP) that operates by measuring the reaction of secondary
users to a specific price and does not require the demand function
to be known. MTP optimizes a profit function that depends
on price only. We prove that while the profit function can be
multimodal, MTP converges to one of the local optima as fast as if
the function were unimodal. Lastly, we provide numerical studies
demonstrating the near-optimal performance of occupancy-based
policies for diverse sets of call length distributions and demand
functions and the quick convergence of MTP to near-optimal
on-line profit.
I. INTRODUCTION
As a result of continuing efforts to deregulate wireless
spectrum management, policy agencies are granting providers
with the right to lease their spectrum [1]. This policy reform
promises more efficient use of excess spectrum, which other-
wise may be wasted. Implications of this reform can be seen in
the novel services provided by spectrum brokerage companies,
which match potential lessees and spectrum providers (licence
holders). One such service is an on-line spectrum trading and
leasing platform, called SpecEx.com, which was launched by
Spectrum Bridge Inc. in 2008 [2].
Success of the aforementioned spectrum reforms hinges on
the design of efficient pricing strategies, since a spectrum
provider strives to maximize its profit from leasing its excess
spectrum. In this paper, we aim at developing a realistic pricing
This work was supported in part by the US National Science Foundation
under grants CNS-0721860, CCF-0916892 and CNS-0238397
framework to achieve this goal. We consider a wireless spec-
trum provider who caters to two classes of customers namely
primary users (PU) and secondary users (SU). PUs have long
term contracts and are not subjected to on-line pricing. On the
other hand, SUs are admitted and priced according to current
availability of excess spectrum. The fraction of SUs that accept
the currently advertised price by the provider is dictated by
an underlying demand function. The provider must ensure
that admission of SUs does not significantly affect quality of
service of PUs. This is because presence of SUs may increase
blocking of PU calls and hence lead to a punishment in the
form of loss of business due to poor service.
The problem of pricing of shared resources has been widely
studied in the literature [3–5]. Recent works introduced pric-
ing strategies specifically tailored for secondary access of
resources [6–8]. Yet, the overwhelming majority of papers in
this area assume that the demand function of users is known
(see Sec. II for exceptions). Precise knowledge of the demand
function, which may vary over time, is, however, hard to
acquire, and raises the question of how to apply this body
of existing work in practice.
Furthermore, much of the previous work assume (for analyt-
ical tractability) that call lengths are exponentially distributed
(see again Sec. II for exceptions). A recent study based on
measurement of real traces in a cellular networks shows that
this assumption does not hold in practice [9]. In particular, [9]
observes that the variance of call length is significantly higher
than that of the exponential distribution and questions the
validity of previous work (and [6] in particular) based on the
exponential distribution assumption.
In this paper, we develop an on-line, measurement-based
pricing framework for secondary access applicable to settings
where both the demand function and the call length distri-
bution are unknown. We focus on pricing policies, which
we refer to as occupancy-based, that depend only on the
total number of ongoing (SU and PU) calls in the system.
We prove that occupancy-based policies are insensitive to
the call length distribution, except through the mean. The
proof follows through the establishment of a connection with
coordinate convex policies in call admission control that are
known to enjoy product-form equilibrium distributions and to
be insensitive to the call length distribution [10].
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