1
Spot Pricing of Secondary Spectrum Access in
Wireless Cellular Networks
Huseyin Mutlu, Murat Alanyali, and David Starobinski
Abstract—Recent deregulation initiatives enable cellular
providers to sell excess spectrum for secondary usage. In this
paper, we investigate the problem of optimal spot pricing of
spectrum by a provider in the presence of both non-elastic
primary users, with long-term commitments, and opportunistic,
elastic secondary users. We first show that optimal pricing can
be formulated as an infinite horizon average reward problem
and solved using stochastic dynamic programming. Next, we
investigate the design of efficient single pricing policies. We
provide numerical and analytical evidences that static pricing
policies do not perform well in such settings (in sharp contrast
to settings where all the users are elastic). On the other hand, we
prove that deterministic threshold pricing achieves optimal profit
amongst all single-price policies and performs close to global
optimal pricing. We characterize the profit regions of different
pricing policies, as a function of the arrival rate of primary users.
Under certain reasonable assumptions on the demand function,
we prove that the profit region of threshold pricing is optimal
and independent of the specific form of the demand function, and
that it includes the profit region of static pricing. In addition, we
show that the profit function of threshold pricing is unimodal in
price. We determine a restricted interval in which the optimal
threshold lies. These properties enable very efficient computation
of the optimal threshold policy, which is far faster than that of
the global optimal policy.
Index Terms—Management of electromagnetic spectrum, sec-
ondary markets, congestion pricing, Markov decision processes,
threshold policies.
I. INTRODUCTION
A major global effort is underway to deregulate wireless
spectrum and achieve much better utilization of this scarce
resource. The Secondary Markets Initiative [2] of the Federal
Communications Commission (FCC), is one of the major steps
towards achieving this goal. It permits leasing of spectrum
licenses subject to approval by FCC. Similar regulatory efforts
are also underway in the EU [3].
Consequences of the secondary markets initiative can al-
ready be felt with the emergence of secondary cellular
providers, commonly called Mobile Virtual Network Oper-
ators (MVNOs) [4]. MVNOs buy spectrum and (possibly
also infrastructure) from primary providers, referred to as
Mobile Network Operators (MNOs). MVNOs add the value of
better penetrating certain markets and offering differentiated
products. A notable example of successful MVNO endeavor
in the US is Virgin Mobile who has teamed up with Sprint
The authors are affiliated with the Department of Electrical and Computer
Engineering at Boston University, Boston, MA
This work was supported in part by the US National Science Foundation
under grants CNS-0132802, CNS-0721860 and ANI-0238397.
A preliminary version of this paper appeared in the proceedings of IEEE
INFOCOM 2008 [1].
Nextel as its MNO and recently reached a subscriber basis of
over 4 millions customers [5].
In this paper, we are interested in investigating how a
provider, such as an MNO, should optimally price its excess
spectrum to secondary users (SUs), such as MVNOs. On the
one hand, a provider must ensure that the quality of service
(QoS) of its primary users (PUs), who typically have long-
term contracts, is not significantly affected by the admission
of SUs. This is because the presence of SUs may increase the
blocking of PU calls and hence lead to a punishment in the
form of loss of business due to poor service. On the other
hand, the provider is interested in maximizing its profit from
the admission of SUs.
Given that the amount of excess spectrum is likely to
fluctuate over time due to the inherent randomness in the PU
traffic, spot pricing, based on real-time channel occupancy,
emerges as the solution of choice. While spot and congestion-
based pricing have been extensively studied in the literature
(Cf. Section II), the typical model assumed in previous work
differs significantly from the setting considered herein. Chiefly,
most previous work assumes that the demand functions of
all the users are elastic to price, i.e., all the arrival rates
can be regulated with price. In contrast, in our setting, only
the demand function of the SUs is elastic to price, but the
arrival rate of PUs is not. As we will show, this difference is
salient enough to result in fundamentally different structures
for optimal (or near-optimal) pricing strategies.
Our first contribution in this paper is to formalize the profit
maximization problem of a cellular provider in the presence
of both PUs and SUs. Based on certain reasonable statistical
assumptions, we show that optimal pricing can be formulated
as an infinite horizon average reward problem and solved using
stochastic dynamic programming.
Our second contribution is to investigate the design of
efficient single pricing policies, i.e., policies where a provider
can either admit a SU and charge a fixed price or reject a
SU. These policies have the major advantage of making the
cost of spectrum much more predictable to SUs. We first show
that static pricing, which always applies the same admission
price to SUs independently of the channel occupancy, may
perform very poorly. This result stands in sharp contrast to
the case where all the users are elastic to price. On the
other hand, we provide numerical evidence that threshold
pricing, which applies a fixed admission price to SUs when the
channel occupancy is below some threshold T and rejects them
otherwise, performs very close to optimal. Further, we prove
that among all the possible single-price admission policies
(including randomized), threshold pricing is the optimal one.
2
Our third contribution is to characterize the profit regions of
static pricing and threshold pricing. Our goal is to determine
the maximum arrival rate of PUs, at which it is still possible to
achieve profit from the admission of SUs. We characterize the
profit regions of different pricing policies. We prove that the
profit region of threshold pricing is optimal, i.e., it is identical
to that of optimal pricing and larger than that of static pricing.
Through numerical example, we show that the difference
between the profit regions of threshold and static pricing can
sometimes be very large. An interesting observation is that
the profit regions of all the pricing policies depend only on
the support of the demand function of the SUs, but not on
its specific form. This result applies to quite general demand
functions.
Our last contribution is to devise an efficient computational
procedure to calculate the optimal threshold and price for
threshold pricing. In particular, we prove that, for any given
threshold T , the profit function is unimodal in price. This en-
ables us to resort to well-known logarithmic search procedures
to compute the optimal price. Moreover, we show that the
optimal threshold is a non-decreasing function of price. By
using this property, we are able to reduce the search interval
for the optimal threshold, thus speeding up calculation of
the optimal threshold policy. We provide numerical results
showing that the optimal threshold policy can be computed
considerably faster than the global optimal policy.
The rest of the paper is organized as follows. In Section II,
we survey related work. Our model and notation are introduced
in Section III. In Section IV, we show how to derive the
optimal pricing policy and characterize the optimal prices.
In Section V, we investigate single-price policies, prove the
optimality of threshold pricing, and characterize the profit
regions of static, threshold and optimal pricing. In Section
V, we also prove unimodality of profit function of threshold
pricing. Then, in Section VI, we develop an efficient method to
compute the optimal price and threshold for threshold pricing.
We conclude the paper in Section VII.
II. RELATED WORK
The problem we consider in this paper is related to two
well studied areas in communication networks, namely, pricing
and call admission control. As such, we restrict our literature
review to those papers that are the most relevant. A survey
of other work related to pricing in cellular networks can be
found in [6].
In [7], Paschalidis and Tsitsiklis investigate dynamic,
congestion-based pricing of network resources. Their model
assumes that all the users are elastic to price. They show
that static pricing achieve good performance in general and
can even be optimal in some asymptotic traffic regimes.
This result was extended in [8] and [9], in the context of
large network asymptotics. In [10], Ziya et. al. show that the
optimal static price is unique. In [11], static spectrum pricing
strategies capturing the effects of network-wide interferences
are developed.
Threshold admission control policies have been extensively
studied. Refs. [12, 13] provide useful insights into the proper-
ties of such policies. The optimality of threshold pricing for
certain optimization problem is proved in [14, 15]. None of
these papers integrate pricing into their formulations.
Refs. [16–18] integrate pricing with admission control in
cellular networks. Ref. [16] considers time-of-day pricing
methods. In our work, we consider pricing strategies that
operate at much shorter time-scales, based on real-time in-
formation. Ref. [17] develops and evaluates “charge-by-time”
pricing algorithms, while in our work we consider charg-
ing per admission. Ref. [18] develops a stochastic dynamic
programming formulation that incorporates retrials. Our main
contribution with respect to this previous body of work is to
go beyond numerical optimizations and attempt to prove gen-
eral structural properties, applicable to very general demand
functions.
Ref. [19] analyzes a model similar to ours within the context
of a generic rental management optimization problem. This
work considers two type of customers, namely walk-in and
contract users. Walk-in users are priced according to the
congestion level of the system, similar to optimal pricing of
SUs in our model. Contract users, on the other hand, have
fixed prices and arrival rates which are analogous to our
PUs. Different than our work, [19] focuses on determining
structures of the optimal policy rather than providing a simple,
near-optimal alternative as done here.
III. NETWORK MODEL
In this section, we introduce our network model and notation
(additional notation specific to static and threshold pricing will
be provided in Section V). We consider a cellular network
where each cell provides access to C channels. In each cell,
calls from PUs arrive according to a Poisson process with fixed
rate λ
p
> 0. A punishment in the amount of K monetary
units is imposed if all the channels are busy and a PU call
is blocked. SUs call arrivals also form a Poisson process
that is independent of the PUs call arrivals process and its
rate is modulated by the price charged by the provider. We
thus assume that there is a demand function λ
s
(u) which
determines the arrival rate of SU calls, where u is the applied
price. The price is a function of the state of the system, i.e.,
a SU pays a price u
n
for its call, if there are n busy channels
in the cell, where 0 ≤ n < C.
For both PUs and SUs, call holding times are exponentially
distributed with rate µ, independently of any other events.
Without loss of generality, we will assume µ = 1, i.e., the
mean call holding time is one unit of time.
The goal of the provider is to maximize the average profit
per unit of time gained from accepting SUs. This quantity is
denoted by R. We are interested in finding a pricing policy
that satisfies this goal. A pricing policy is a rule that dictates
which price should be advertised by the provider at any given
point of time.
Under the above assumptions, the system behavior follows
the dynamics of a continuous-time birth-death Markov pro-
cess, and explicit expression for the average profit R can be
provided as follows. First, let π
n
be the steady-state probability
of finding the system in state n, i.e., there are n busy channels.
Next, let λ
n
= λ
s
(u
n
)+λ
p
denote the total call arrival rate in
3
state n and Λ = (λ
0
, λ
1
, ..., λ
C−1
) denote the vector of arrival
rates. Then, the probability of finding the system in state n,
denoted by π
n
(Λ), can be explicitly written as follows:
π
n
(Λ) =
λ
0
λ
1
λ
2
...λ
n−1
n!
1 +
λ
0
1!
+
λ
0
λ
1
2!
+ . . . +
λ
0
λ
1
λ
2
...λ
C−1
C!
. (1)
Due to the PASTA (Poisson Arrivals See Time Averages)
property, the probability that a PU is blocked is π
C
(Λ). Thus,
the average profit is
R =
P
C−1
n=0
π
n
(Λ)λ
s
(u
n
)u
n
− (π
C
(Λ)−E(λ
p
, C))λ
p
K,
(2)
where E(λ
p
, C) is the blocking probability of PUs in the
absence of SU arrivals. This quantity corresponds to the well-
known Erlang-B formula
E(λ
p
, C) =
λ
C
p
C!
P
C
n=0
λ
n
p
n!
. (3)
The first term in Eq. (2) represents the sum of the average
revenues collected from SUs in each state. The second term is
the average punishment due to accepted SUs. The expression
π
C
(Λ)−E(λ
p
, C) represents the increase in the blocking prob-
ability of PUs due to accepted SUs. The quantity E(λ
p
, C)
acts as the normalization term to ensure that the profit is zero
when all SUs are rejected.
In the sequel, we impose the following natural assumptions
on the demand functions. These assumptions are required to
guarantee the existence of a stationary optimal pricing policy
and prove some of our structural results.
Assumption 3.1: There exists a price u
max
for which
λ
s
(u
max
) = 0. Moreover, λ
s
(u) is a strictly decreasing,
differentiable function in u over the interval [0, u
max
] and
λ
s
(0) is finite.
IV. DERIVATION OF THE OPTIMAL PRICING POLICY
In this section, we derive the optimal pricing policy and
present properties characterizing the optimal prices.
A. Stochastic Dynamic Programming Formulation
The maximization of the profit function in Eq. (2) is a com-
plex multi-dimensional optimization problem and becomes
quickly intractable as C grows. One approach to alleviate this
problem is to formulate it as an average reward stochastic
dynamic programming (DP) problem [20, 21]. Specifically,
the optimal prices u
∗
n
and optimal profit R
∗
corresponding
to the optimal policy can be computed using the so-called
Bellman’s equations since all the states in the Markov chain
are recurrent (see Proposition 7.4.1 in [21]).
Bellman’s equations are usually formulated for discrete-time
Markov chains. In our case, the Markov chain is continuous,
but it can be discretized using a procedure called uniformiza-
tion, where the transition rates out of each state are normalized
by the maximum possible transition rate v, which in our case
is given by the following expression:
v = λ
s
(0) + λ
p
+ C. (4)
Fig. 1. Uniformized Markov Chain
The uniformized Markov chain with corresponding transition
rates is shown in Fig. (1).
Bellman’s equations are generally guaranteed to return the
optimal solution only for a finite action (control) space U,
where U represents the set of all possible prices advertised by
the provider. Hence, prices must be discretized. We denote the
discretization step with ∆u. The cardinality of the action space
is thus |U| = du
max
/∆ue. On the one hand, consideration
of a limited range of prices leads to a potential reduction
in the profit. On the other hand, if the demand function
λ
s
(u) is continuously differentiable in u, this reduction is
at most linear in discretization step ∆u since the profit in
Eq. (2) is a smooth function of u
0
, u
1
, · · · , u
C−1
. Hence the
alluded profit loss can be made arbitrarily small at the expense
of higher computational complexity by selecting a smaller
∆u. In Section VI, we describe an efficient computational
procedure, applicable to threshold pricing, that scales to very
large cardinality |U|.
Equipped with the above formulation, we can now compute
the optimal pricing policy using the Bellman equations:
J
∗
+ h(n) = max
u
n
∈U
[λ
s
(u
n
)u
n
+ h(n + 1)
λ(u
n
)
v
+h(n − 1)
n
v
+ h(n)(1 −
λ(u
n
)
v
−
n
v
)]
(5)
for n = 0, 1, 2...C − 1 and
J
∗
= −λ
p
K + h(C − 1)
C
v
, (6)
whereas the optimal profit is:
R
∗
= J
∗
+ E(λ
p
, C)λ
p
K. (7)
The first term in the right-hand side (RHS) of Eq. (5) repre-
sents the profit gained at state n from the acceptance of a SU.
The second and third terms are contributions to the revenue if
the next transition is an arrival or departure, respectively. The
last term is a consequence of the uniformization procedure.
The effect of punishment due to blocked PU calls is captured
by the first term in the RHS of Eq. (6). The prices maximizing
the RHS of Eq. (5) represent the optimal prices.
The unknowns in the above equations are h(n) and J
∗
.
The quantities h(n) denote the relative reward in state n with
respect to state C. When the optimal policy is applied, h(n)/v
represents the difference between the total revenue gained over
an infinite time horizon when starting the process from state n
and that gained when starting from state C. The quantities R
∗
and J
∗
differ only by a normalization constant used to ensure
the non-negativity of the profit.
4
Fig. 2. Optimal prices for various PU arrival rates (λ
p
) . C = 20, K = 100,
λ
s
(u) = (10 − u)
+
and ∆u = 10
−6
.
The solution of Bellman equation can be obtained by using
various techniques described in the literature, such as policy
iteration or relative value iteration [20, 21]. Policy iteration
theoretically requires on the order of O (|U|
C
) iterations to
converge while value iteration is not guaranteed to converge
within a finite number steps. However, value iteration has a
lower computational complexity at each iteration. In practice,
as in other infinite horizon average reward problems [22],
policy iteration appears to converge faster.
For different PU arrival rates λ
p
, Figure (2) shows the
values of the optimal prices (computed using policy itera-
tion), for the demand function λ
s
(u) = (10 − u)
+
(where
(·)
+
= max(·, 0)), and parameters C = 20, K = 100, and
∆u = 10
−6
. The figure indicates that, as λ
p
increases, the
prices become higher in each state, and that SUs should not be
accepted when the number of busy channels exceeds a certain
threshold. More insight into this behavior will be provided in
the sequel.
B. Properties of the Optimal Policy
In this section, we provide some results characterizing the
optimal prices. First, we consider the ideal case of unlimited
capacity.
Lemma 4.1: In the infinite capacity case (i.e., C → ∞),
the optimal prices for all states are equal to
u
∞
= arg max
u∈U
(λ
s
(u)u),
and the corresponding profit is
R
∞
= λ
s
(u
∞
)u
∞
.
Note that R
∞
is an upper bound on the profit achievable in
any finite capacity system.
The following lemma states that in a finite capacity system,
the optimal price in each state is larger than the optimal price
in the infinite capacity case.
Lemma 4.2: For any 0 ≤ n ≤ C − 1, u
∗
n
≥ u
∞
.
The next result states that the optimal prices are monoton-
ically increasing in n.
Lemma 4.3: For any 0 ≤ n ≤ C − 1, u
∗
n+1
≥ u
∗
n
.
Proofs of these properties follow similar methods to those
used in [7]. The main difference lies in taking into considera-
tion the effects of PU arrivals and punishments. These proofs
can be found in [23].
A consequence of the above properties is that the optimal
price for any state lies between u
∞
and u
max
. This fact can
be exploited to reduce the size of the action space U when
computing the optimal prices using Eq. (5).
V. SINGLE-PRICE POLICIES
In this section, we investigate the design of single-price
policies. In each state, these policies can either admit a SU and
charge a fixed price u or reject a SU (which is equivalent to ask
for a price u
max
or higher). For such policies the objective is
to optimize the value of u as well as the admission policy
i.e., the decision of whether or not to admit a SU that is
willing to pay the price. These policies are attractive because
they allow a provider to advertise a single-price. They are
also computationally easier to derive. Moreover, compared to
optimal pricing, they provide more insight into the structure
of good pricing policies.
A simple single-price policy is the so-called static pricing
where SU calls are always applied the same admission price,
unless all the channels are busy. For the cases where the
demand functions of all the users are elastic to price and
punishments are not imposed, static pricing is known to
perform well and to be even asymptotically optimal in several
regimes [7–9]. However, in this section, we show that, in
the presence of inelastic users (PU) and punishments for
blocked PU calls, the performance of static pricing degrades
significantly.
Instead, we show next that among all single-price poli-
cies (including randomized), a deterministic threshold pricing
policy performs optimally. In threshold pricing, SU calls are
admitted and charged a price u when the channel occupancy is
smaller than some threshold T and rejected otherwise. We also
provide numerical evidence showing that threshold pricing
performs very close to the optimal.
A. Optimality of Threshold Pricing
Theorem 5.1: For any price u (including the optimal one),
a threshold admission policy is optimal among all single-price
policies.
Proof: Let us redefine the system such that punishment
in the amount of u units are imposed for each rejected SU call
instead of rewarding u for an accepted one. Optimizing such
a system follows same methods and results the same optimal
policy. This new formulation of the problem is identical to the
well-known MINOBJ problem analyzed in [14] where SU and
PU calls are analogous to new and handover calls, respectively.
It is shown in [14] that a threshold admission policy is the
optimal solution for the MINOBJ problem and, thus, the same
result applies to our setting. Note that the analogy is valid
when K > u. If this is not the case, then the admission policy
5
Fig. 3. Average profit vs primary load (λ
p
) for different pricing policies.
System parameters: C = 20, K = 100, λ
s
(u) = (10 − u)
+
and ∆u =
10
−6
.
is obvious, namely, always admit SU calls.
Figure (3) compares the average profits achieved by the
optimal, static, and threshold policies for a linear demand
function λ
s
(u) = (10 −u)
+
(we explain in Section VI how to
compute the optimal price and threshold). Figure (4) makes the
same comparison for the following popular non-linear demand
function [24]
λ
s
(u) = (Ae
−γu
2
− ²)
+
, (8)
where A and γ are scaling factors, and ² > 0 is a small
constant introduced to enforce Assumption 3.1. Both figures
show that threshold pricing performs close to optimal while
static pricing performs significantly worse. Furthermore, we
observe that beyond a certain value of λ
p
, static pricing stops
generating profit while threshold pricing continues doing so.
We next provide some intuition on why threshold pricing
performs so well. In Section V-D, we will show that the
maximum value of λ
p
, denoted λ
p,max
, for which threshold
pricing achieves positive profit is the same as the maximum
value of λ
p
for which optimal pricing achieves positive profit.
Furthermore, we know that when λ
p
→ 0, both static pricing
and threshold pricing perform very well. This regime is equiv-
alent to the case where all the users are elastic to price, a model
studied in [7]. There it is shown that static pricing is optimal
in certain asymptotic regimes. These results obviously extend
to threshold pricing since it is the optimal single price policy.
The arguments above explain the near-optimal performance of
threshold pricing for the cases λ
p
→ 0 and λ
p
→ λ
p,max
.
Thus, one can expect that in between these two extremes,
the profit of threshold pricing will not differ much from the
optimal profit.
B. Properties of Threshold Pricing
Having showed that threshold pricing is the optimal single-
price policy, we next derive an expression for the profit
Fig. 4. Average profit vs primary load (λ
p
) for different pricing policies.
System parameters: C = 20, K = 100, λ
s
(u) = (10e
−0.04u
2
− 0.1)
+
and
∆u = 10
−6
.
obtained with this policy, denoted by R
T
(λ
s
). Note that, the
profit function is defined as a function of λ
s
rather than u. This
considerably simplifies the notation and proofs in the rest of
the paper.
We start by computing the blocking probabilities for the
PUs and SUs:
B
P U
(λ
s
, T ) = π
C
(9)
=
(λ
s
+λ
p
)
T
λ
C−T
p
C!
P
T −1
n=0
(λ
s
+λ
p
)
n
n!
+ (λ
s
+λ
p
)
T
P
C
n=T
λ
n−T
p
n!
;
B
SU
(λ
s
, T ) =
C
X
n=T
π
n
(10)
=
(λ
s
+λ
p
)
T
P
C
n=T
λ
n−T
p
n!
P
T −1
n=0
(λ
s
+λ
p
)
n
n!
+ (λ
s
+λ
p
)
T
P
C
n=T
λ
n−T
p
n!
.
Note that, arrival rate until congestion level reaches T channels
is λ
s
+λ
p
and just λ
p
afterwards. Finally, we can provide an
explicit expression for R
T
(λ
s
) as follows:
R
T
(λ
s
)=(1−B
SU
(λ
s
, T ))λ
s
u(λ
s
) − B
P U
(λ
s
, T )λ
p
K
+E(λ
p
, C)λ
p
K.
(11)
where u(λ
s
) is the inverse function of λ
s
(u). The first term in
Eq. (11) is the revenue collected from SU calls. The second
term is a result of the punishment due to blocked PU calls.
The last term is the normalization term which is used to ensure
that profit is zero when there are no SUs (see Eq. (3)).
Next, we derive an important property of the blocking
probabilities B
P U
and B
SU
, that will be exploited in the next
section. Specifically, we show that the ratio of these blocking
probabilities depends only on the PU’s call arrival rate λ
p
and
threshold T but not on the price or the demand function of
the SU.
Lemma 5.2: The ratio
B
P U
(λ
s
,T )
B
S U
(λ
s
,T )
is independent of u and
λ
s
.
