The Impact of Small-Cell Bandwidth Requirements on Strategic Operators

0.42 MB·

by  Cheng Chen

The Impact of Small-Cell Bandwidth Requirements
on Strategic Operators
Cheng Chen Randall A. Berry, Michael L. Honig Vijay G. Subramanian
NGS, Intel Corporation Dept. of EECS, Northwestern University Dept. of EECS, University of Michigan
Hillsboro, OR 97124 Evanston, IL 60208 Ann Arbor, MI 48109
[email protected] {rberry, mh}@eecs.northwestern.edu [email protected]
Abstract—Small-cell deployment in licensed and unlicensed While the deployment of small-cells will increase overall
7 spectrum is considered to be one of the key approaches to cope datacapacity,italsocomplicatesthenetworkmanagementand
with the ongoing wireless data demand explosion. Compared to
1 resource allocation for SPs. This includes how to differentiate
traditional cellular base stations with large transmission power,
0 thepricingschemesandoptimallysplittheirlimitedbandwidth
small-cellstypicallyhaverelativelylowtransmissionpower,which
2
makes them attractive for some spectrum bands that have strict resourcesbetweenmacro-andsmall-cells,takingintoaccount

n power regulations, for example, the 3.5GHz band [1]. In this the fact that users in the network are also heterogeneous
a paper we consider a heterogeneous wireless network consisting in terms of mobility patterns. Moreover, these decisions are
J ofoneormoreserviceproviders(SPs).EachSPoperatesinboth
furthercomplicatedbyregulatoryrestrictionsoncertainbands,
9 macro-cells and small-cells, and provides service to two types of
such as the designation of new spectrum in the 3.5GHz band
users: mobile and fixed. Mobile users can only associate with
] macro-cells whereas fixed users can connect to either macro- or only for small-cells.
T small-cells. The SP charges a price per unit rate for each type
I of service. Each SP is given a fixed amount of bandwidth and A. Contributions
.
s splitsitbetweenmacro-andsmall-cells.Motivatedbybandwidth Our paper analyzes the impact of regulatory requirements
c regulations, such as those for the 3.5Gz band, we assume a
that certain bands be used only for small-cells on competitive
[ minimum amount of bandwidth has to be set aside for small-
service providers that allocate bandwidth between macro- and
cells. We study the optimal pricing and bandwidth allocation
1
strategies in both monopoly and competitive scenarios. In the small-cell networks. We also analyze the associated social
v
monopoly scenario the strategy is unique. In the competitive welfare. At present new spectrum is typically apportioned
4
8 scenario there exists a unique Nash equilibrium, which depends based on an auction, and another goal is to provide insight
on the regulatory constraints. We also analyze the social welfare
3 into the social welfare achieved via winner-take-all auctions.
achieved,andcompareittothatwithoutthesmall-cellbandwidth
2 Given the policy implications of such an analysis, we briefly
constraints.Finally,wediscussimplicationsofourresultsonthe
0
effectivenessoftheminimumbandwidthconstraintoninfluencing discuss these results at a high-level; detailed results are in
.
1 small-cell deployments. Section V.
0 The scenario that we consider in the paper is the following.
7 I. INTRODUCTION
The spectrum regulator needs to allocate B units of newly
1 Current cellular networks are expected to evolve towards
available bandwidth to two competitive SPs. Each SP has an
v: heterogeneous networks (HetNets) to cope with the explo- initialendowmentoflicensedbandwidthBo andBo,andgets
i sive demand for wireless data [2], [3]. This requires service 1 2
X a proportion of the new bandwidth, denoted as Bn and Bn.
providers (SPs) to deploy small-cells in addition to traditional 1 2
Theregulatordeterminestherulesforthisassignmentusingan
r macro-cells. While typical macro-cells, such as cellular base
a appropriate auction procedure; e.g., an allocation of Bn =B
stations,typicallyhavelargetransmissionpowerandtherefore 1
and Bn =0 corresponds to the outcome of a winner-take-all
arecapableofcoveringuserswithinalargeregion,small-cells 2
auction that SP 1 wins.The initial bandwidth can be allocated
have much lower transmission power and are used to provide
byeachSPtoeithermacro-cellsorsmall-cells.Incontrast,the
servicetoalocalarea.Thisuniquecharacteristicofsmall-cells
new bandwidth can only be used for small-cells, as enforced
enablesthemtobeanattractivechoiceinsomespectrumbands
by the regulatory constraint.
that have strict power regulations. For example, in 2012, the
In Figure 1 we present a typical result that we obtain
FCC proposed to create a new Citizens Broadband Service in
for different partitions of the new bandwidths amongst the
the 3550-3650 MHz band (3.5GHz Band), previously utilized
two SPs. The blue line represents the social welfare achieved
for military and satellite operations [1]. Due to the low power
when the SPs cooperate and the new bandwidth comes with
constraint within this band, only small-cells can be deployed.
no restrictions; the red line represents the social welfare
ForSPsthatwanttousethisbandtoexpandtheirservice,this
achieved when the SPs cooperate and the new bandwidth is
type of bandwidth regulation needs to be taken into account
restricted to small cell use; and the green curve represents the
in determining optimal resource allocation strategies.
social welfare achieved when the SPs compete with the new
ThisworkwassupportedbyNSFundergrantAST-134338. bandwidthrestrictedonlytosmallcelluse.Fromtheresultsof
ourpreviouswork[13],itiseasilyverifiedthatbluelineisalso SPs. We show that in the monopoly scenario the SP simply
the social welfare achieved when the SPs compete and there increases its small-cell bandwidth to the required minimum
isnorestrictionontheusageofthenewbandwidth.Itisclear amount if its original small-cell bandwidth is less than the
fromFigure1thattheintroductionofrestrictionsontheusage constraint. This applies to both social welfare and revenue-
of new bandwidth results in a reduction in social welfare, and maximization. With two competitive SPs, there always exists
additionally,onlyspecificpartitionsofthenewbandwidthwill a unique Nash equilibrium that depends on the regulatory
lead to this reduced social welfare value being achieved with constraints. We illustrate this by considering three cases
competingSPs.Itshouldalsobenotedthattheassignmentthat corresponding to whether the original equilibrium allocation
resultsfromawinner-take-allauctionyieldsmuchlowersocial satisfies the two constraints. We characterize the equilibrium
welfare; these correspond to the two endpoints in the figure. for each case.
NumericalinvestigationsalsoshowthattheSPwiththelarger 3. Social Welfare Analysis: We also quantify the influence
amount of initial bandwidth endowment obtains the highest of the regulatory constraints on the social welfare. We con-
marginal revenue increase from any new bandwidth with the clude that if the equilibrium without constraints violates the
otherSPlosingrevenuewhenthisoccurs.ThelargerSPwould constraints, then social welfare loss is inevitable. However,
thus, bid higher to reduce the influence of the smaller SP. the social welfare loss is always bounded, and the worst
Thus one of the main contributions of our paper is to case happens when the spectrum regulator requires the SPs
highlight the possibility of such negative outcomes with the to allocate all bandwidth only to small-cells. In this extreme
specific designation chosen for small cells, and also to point case there are no macro-cells, and consequently none of the
outthenecessityofcarryingoutsuchanalysisbeforedeciding mobile users receives wireless service.
on other regulatory constraints for newly available spectrum
B. Related work
bands.
PricingandbandwidthallocationproblemsinHetNetshave
Bo=1, Bo=1.2, B=10 attracted considerable attention. In [4]–[6], small-cell service
1 2
781.5
is seen as an enhancement to macro-cell service. In contrast,
small-cellandmacro-cellserviceareconsideredtobeseparate
781
servicesin[7]–[14],[17],thesameasourmodelinthispaper.
Onlyoptimalpricingisstudiedin[4],[6],[11],[12],while[5],
780.5
e [7]–[10],[13],[14],[17]considerjointpricingandbandwidth
ra
fle allocation, as in this paper. Additionally, except for [11]–
W 780
la [14], [17] that include the competitive scenario with multiple
ic
oS SW* SPs, all the other work assumes only one SP. In this paper,
779.5 wo
SW* we investigate both monopoly and competitive scenarios, and
w adopt a model similar to that in our previous work [10], [13]
779 SWNE
w (which did not consider bandwidth regulations).
The rest of the paper is organized as follows. We present
778.5
0 1 2 3 4 5 6 7 8 9 10 the system model in Section II. We consider monopoly and
B1n competitive scenarios in Section III and Section IV, respec-
tively. Social welfare analysis is in Section V. We conclude
Fig.1. SocialwelfareversusBn withlargeB.HereSW∗ istheoptimal
1 wo in Section VI. All proofs of the main results can be found in
social welfare without regulatory constraints, SW∗ is the optimal social
welfare with regulatory constraints, and SWNE iswthe equilibrium social the appendices.
w
welfarewithregulatoryconstraints.
II. SYSTEMMODEL
We now summarize our other contributions in this paper: Weadoptthemathematicalmodelinourpreviouswork[13]
1. Incorporating Bandwidth Regulations into the HetNet for the analysis. We now describe the different aspects of it
Model: Prior related work that considers bandwidth allocation while pointing out the additional elements considered here.
assumes SPs are free to split spectrum between macro- and
A. SPs
small-cells in any way. Here, we add additional small-cell
bandwidth constraints that impose a minimum amount of We consider a HetNet with N SPs providing separate
small-cell bandwidth to the SPs. This is primarily motivated macro- and small-cell service to all users. Denote the set
by the 3.5GHz Band released by the FCC that can only be of SPs as N. Each SP is assumed to operate a two-tier
usedtodeploysmall-cells.Theintroductionofsuchbandwidth cellular network consisting of macro- and small-cells that are
constraints has a direct influence on the optimal pricing and uniformly deployed over a given area. We further assume
bandwidth allocation strategies of SPs. all SPs have the same density of infrastructure deployment.
2. Characterizing the impact of the regulatory constraints We normalize the density of macro-cells to one. The density
for SPs in both monopoly and competitive scenarios: We of small-cells is denoted as N . In our setting, macro-cells
S
analyze scenarios with both a monopoly SP and competitive havehightransmissionpower,andthereforecanprovidelarge
coverage range. In contrast, small-cells have low transmission C. User and SP Optimization
power, and consequently local coverage range.
Figure 2 illustrates the network and market model. We
Each SP i has a total amount of bandwidth B exclusively
i now introduce the optimization problems corresponding to
licensed.1 Since we assume all macro- and small-cells use
both users and SPs. We assume each user is endowed with a
separate bands, each SP i needs to decide how to split its
utility function, u(r), which only depends on the service rate
bandwidthintoB ,bandwidthallocatedtomacro-cells,and
i,M it gets. For simplicity of analysis, in this paper we assume
B , bandwidth allocated to small-cells. When determining
i,S that all users have the same α-fair utility functions [15] with
this partition, every SP is required to conform to (possible)
α∈(0,1):
bandwidth regulations enforced by the spectrum regulator.
Specifically,SPiisrequestedtoguaranteeaminimumamount r1−α
u(r)= , α∈(0,1). (1)
of bandwidth allocated to small-cells, and this lower bound is 1−α
denoted as B0 .
i,S Thisrestrictionenablesustoexplicitlycalculatemanyequilib-
For a fixed bandwidth allocation, the total achievable data
riumquantities,whichappearstobedifficultformoregeneral
rate provided by the macro-cells of SP i is C =B R ,
i,M i,M 0 classesofutility.Furthermore,thisclassiswidelyusedinboth
where R is the (average) spectral efficiency of the macro-
0 networking and economics, where it is a subset of the class
cells. The total available rate in small-cells of SP i is given
of iso-elastic utility functions.2
by C = λ B R , where λ > 1 reflects the increase in
i,S S i,S 0 S
Each user chooses the service by maximizing its net payoff
spectralefficiencyduetosmallercellsize,andpossiblygreater
W, defined as its utility less the service cost. For a service
deployment density. Each SP i provides separate macro- and
with price p, this is equivalent to:
small-cell services and charges the users a price per unit rate
for associating with its macro-cells or small-cells, namely,
W =max u(r)−pr. (2)
pi,M and pi,S. r≥0
B. Users For α-fair utility functions, (2) has the unique solution:
Weassumetheusersinthenetworksarealsoheterogeneous
and categorize them into two types based on their mobility r∗ =D(p)=(u(cid:48))−1(p)=(1/p)1/α, (3)
patterns. Mobile users can only be served by macro-cells. In
contrast, fixed users are relatively stationary, and can connect where D(p) here can be seen as the user’s rate demand
to either macro- or small-cells (but not both). Denote the function. The maximum net payoff for a user is thus:
rdeesnpseictiteivseloyf. Nmootbeiltehautstehreshaentderofigxeendeituyseorfsthase uNsemrsacnadn aNlsfo, W∗(p)=u(D(p))−pD(p)= α p1−α1. (4)
1−α
arise from an equivalent model that assumes (N +N ) as
m f
the total density of users, who are mobile with probability Recall that fixed users can choose between any macro- or
N /(N +N ) and stationary with probability N /(N + small-cell service offered by any SP, while mobile users can
m m f f m
N ). After user association, let K and K denote the onlychoosethemacro-cellserviceprovidedbyaSP.However,
f i,M i,S
mass of users connected to the macro- and small-cells of SP here, we assume mobile users have priority connecting to
i, respectively. (Note that K consists of fixed users only, macro-cells, which means macro-cells will only admit fixed
i,S
whereas K can consist of both mobile and fixed users.) users after the service requests of all mobile users have been
i,M
addressed.
For the association rules, we adopt the same process de-
Bandwidth Allocation :
scribed in [13]. That is, users always choose the service with
Pricing Decision :
lowestpriceandfillthecorrespondingcapacity.Ifmultipleser-
vices have the same price, then the users are allocated across
them in proportion to the capacities. Once a particular service
Macro-cells Small-cells
capacity is exhausted, then the leftover demand continues to
fill the remaining service in the same fashion.
Each SP determines the bandwidth split and service prices
SP 1 SP 2
to maximize its revenue, which is the aggregate amount paid
by all users associating with their macro- and small-cells.
Mobile Fixed
Service Competition Users Users Meanwhile they also need to conform to the constraints on
small-cell bandwidth allocation. Specifically, SP i solves the
2Ingeneralα-fairutilitiesrequirethatα≥0toensureconcavity;requiring
Fig.2. SystemModel.
α>0 ensures strict concavity but allows us to approach the linear case as
α→0.Therestrictionofα<1ensuresthatutilityisnon-negativesothata
usercanalways“optout”andreceivezeroutility.Notealsothatasα→1,
1ForthemonopolySPscenario,wewillignorethesubscript. weapproachthelog(·)(proportionalfair)utilityfunction.
following optimization problem: this case it is given by:
N λ1/α−1B
maximize Si =pi,MKi,MD(pi,M)+pi,SKi,SD(pi,S), BSW =Brev = f S , (7a)
(5a) S S N λ1/α−1+N
f S m
subject to Bi,M +Bi,S ≤Bi,Bi,M ≥0,Bi,S ≥Bi0,S, BSW =Brev = NmB . (7b)
(5b) M M N λ1/α−1+N
f S m
0<p ,p <∞. (5c)
i,M i,S 2.IfB0 > Nfλ1S/α−1B ,theoptimalbandwidthallocation
Alternatively,asocialplanner,suchastheFCC,mayseekto is changeSd to:Nfλ1S/α−1+Nm
allocate bandwidth and set prices to maximize social welfare,
BSW =Brev =B0, BSW =Brev =B−B0. (8)
which is the sum utility of all users, subject to the same S S S M M S
constraints (5b) and (5c). This is given by: Consequently there will be both a welfare and revenue loss if
this case applies.
N
(cid:88) In both cases the optimal macro- and small-service prices
maximize SW= [K u(D(p ))+K u(D(p ))].
i,M i,M i,S i,S
are market-clearing prices, i.e., the prices that equalize the
i=1
(6) total rate demand and the total rate supply in both cells.
Theorem 1 states that if the original optimal bandwidth
D. Sequential Game and Backward Analysis allocation without the bandwidth regulations already satisfies
the imposed constraint, then the SP just keeps the same
We model the bandwidth and price adjustments of SPs in
bandwidth allocation. If the original bandwidth allocation
the network as a two-stage process:
violates the regulatory constraint, then the SP increases the
1) Each SP i first determines its bandwidth allocation small-cell bandwidth to the required level. This is because
B ,B betweenmacro-cellsandsmall-cells.Denote the added regulatory constraint does not change the concavity
i,M i,S
the aggregate bandwidth allocation profile as B. of the revenue or social welfare function with respect to the
2) GivenB(assumedknowntoallSPs),theSPsannounce small-cell bandwidth, and further increasing the bandwidth
prices for both macro-cells and small-cells. The users allocation to small-cells will only lead to more revenue or
then associate with SPs according to the previous user social welfare loss.
association rule.
IV. COMPETITIVESCENARIOWITHTWOSPS
We then do backward induction. That is, we first derive the
In this section we turn to the competitive scenario with two
priceequilibriumunderafixedbandwidthallocation.Wethen
SPs,eachofwhichmaximizesitsindividualrevenue.Applying
characterizethebandwidthallocationequilibriumbasedonthe
the results from [13], the price equilibrium given any fixed
price equilibrium obtained.
bandwidth allocation is always the market-clearing price. We
thereforefocusonthebandwidthallocationNashequilibrium.
III. MONOPOLYSCENARIOWITHASINGLESP Considering the case without the additional regulatory con-
straint, using the results from [13], there exists a unique
We first study the bandwidth allocation when a single SP
Nash equilibrium and the bandwidth allocations of two SPs
is operating in the network. This is similar to the analysis in
at equilibrium are given by:
our previous work [10], except here we have an additional
regulatory constraint that imposes a minimum bandwidth N λ1/α−1B N B
BNE = f S 1 ,BNE = m 1 , (9a)
allocation to small-cells. This added constraint will change 1,S N λ1/α−1+N 1,M N λ1/α−1+N
the optimal bandwidth allocation strategy for the monopoly f S m f S m
SP. In [10] it is concluded that for the set of α-fair utility BNE = Nfλ1S/α−1B2 ,BNE = NmB2 . (9b)
functions we use in this paper, the revenue-maximizing and 2,S N λ1/α−1+N 2,M N λ1/α−1+N
f S m f S m
social welfare-maximizing bandwidth allocation turn out to
With the additional regulatory constraints, we have the fol-
be the same. The following theorem states that the optimal
lowing theorem characterizing the corresponding Nash equi-
bandwidthallocationsunderbothobjectivesarestillthesame,
librium between two SPs.
butaddingalargevalueforthebandwidthsetasideforsmall-
Theorem 2: With two SPs, with a constraint on minimum
cells changes the optimal bandwidth allocation.
small-cell bandwidth, the Nash equilibrium exists and is
Theorem 1 (Optimal Monopoly Bandwidth Allocation):
unique. Moreover, the total bandwidth allocated to small-cells
For a monopoly SP, the optimal revenue-maximizing and
by the two SPs is no less than that without the regulatory
social welfare-maximizing bandwidth allocation strategies are
constraints.
the same and can be determined by the following cases:
Theorem 2 states that the existence and uniqueness of the
rem1.aiInfsBtS0he≤saNmfNeλfS1aλ/sS1α/−tαh1−a+1tNBwmi,ththoeutotphteimreaglublaantodrwyidctohnastlrloaicnatt.ioInn Nstraasihntesq.uTihliibsricuamnbisepprreosveerdveudsianfgtesrimadidlainrgmtehtehoredgsualsatporroyvcidoend-
in our previous work [13], with some modifications. The last Case C: Only one constraint is violated. Without loss
part of the theorem may be more subtle than it appears. One of generality, we assume at the Nash equilibrium without
may try to argue that if any of the constraints is violated, that regulations, only SP 2’s small-cell bandwidth allocation falls
SP then needs to increase its bandwidth allocation to small- below the required threshold. In this case, the new Nash
cells. It would then hold that the total bandwidth allocated equilibrium is characterized by the following proposition:
to small-cells surely increases. However, the logic does not Proposition 2: In case C, the Nash equilibrium with regu-
carry through if only one constraint is violated at the Nash latory constraints is one of the following two types:
equilibrium omitting the constraint. In that case, the SP with Type I. Both SPs increase their small-cell bandwidth alloca-
violated constraint must increase the bandwidth allocation to tionstoexactlytherequiredamount,i.e,B =B0 ,B =
1,S 1,S 2,S
small-cells. However, the other SP, whose equilibrium small- B0 .
2,S
cell bandwidth allocation without regulations satisfies the Type II. Only SP 2 allocates exactly the required minimum
constraint, may potentially decrease its bandwidth in small- amountofbandwidthto small-cells,i.e,B >B0 ,B =
1,S 1,S 2,S
cells in response to the increase in bandwidth allocation of B0 .
2,S
its competitor. In that case, determining the change in total Notethatequation(10)alsoappliestogiveconditionswhen
bandwidth requires a more detailed analysis. Nonetheless, a type I equilibrium arises.
Theorem 2 indicates even in that case the total bandwidth While the type I Nash equilibrium in both cases B and
in small-cells would not decrease. We will present a specific C indicate both SPs allocate exactly the required minimum
example later. amount to small-cells, they are quite different. In case B both
Depending on whether the regulatory constraints are vio- SPs increase their small-cell bandwidth allocations, whereas
lated or not at the Nash equilibrium without the constraints, in case C, one SP increases its small-cell bandwidth while
there are three cases we need to cover independently. We will the other SP decreases its small-cell bandwidth. Another
seethat,ineachcase,theNashequilibriumbehavesdifferently. difference that is worth pointing out is that in case C, the
Case A: Both constraints are satisfied. The new Nash SP whose small-cell bandwidth allocation without regulations
equilibrium is the same as the Nash equilibrium without violates the constraint always operates at exactly the required
regulations. minimum point at the new Nash equilibrium, while it will
CaseB:Bothconstraintsareviolated.TheNashequilibrium furtherincreaseitssmall-cellbandwidthbeyondtheminimum
without regulations is no longer valid. The following proposi- point in a type II equilibrium for case B.
tion characterizes the properties of the new Nash equilibrium.
Next we use a specific example in Figure 3 to illustrate the
Proposition 1: In case B, the Nash equilibrium with regu- different Nash equilibrium regions as a function of the small-
latory constraints is one of the following types: cell bandwidth constraints discussed in the preceding cases.
Type I. Both SPs increase their small-cell bandwidth alloca- Thesystemparametersforthiscaseare:α=0.5,N =N =
m f
tionstoexactlytherequiredamount,i.e,B =B0 ,B = 50,R = 50,λ = 2,B = 2,B = 1. In this example the
1,S 1,S 2,S 0 S 1 2
B0 . original equilibrium small-cell bandwidth allocations without
2,S
Type II. One SP increases its small-cell bandwidth exactly the regulatory constraints are: B =1.34,B =0.67. Re-
1,S 2,S
to the required amount, while the other SP increases further gionAcorrespondstotheNashequilibriumincaseA,whichis
beyond the required amount, i.e, B = B0 ,B > alsotheequilibriumwithouttheregulatoryconstraints.Region
1,S 1,S 2,S
B0 or B >B0 ,B =B0 . B.IandRegionB.IIcorrespondtothetype-Iandtype-IINash
2,S 1,S 1,S 2,S 2,S
It is conceptually easy to characterize the necessary and equilibrium in case B where both constraints are violated at
sufficient conditions for the first type of Nash equilibrium to the original equilibrium, and the same rule applies to Region
hold since at that equilibrium the marginal revenue increase C.I and C.II.
with respect to per unit of bandwidth increase in small-cells
V. SOCIALWELFARE
should be non-positive for both SPs. This can be analytically
expressed via the two corresponding inequalities: In this section we focus on social welfare analysis. In our
previous work [10] [13], we showed that for the set of α-
αλ2B0 R
λ R0−α−R0 −α− S i,S 0R0−α−1+ fair utility functions we use here, the bandwidth allocation
S S M Nf S at equilibrium is always socially optimal in both monopoly
(B −B0 )R and competitive scenarios. With the additional regulatory
i i,S 0R0 −α−1 ≤0, for i=1,2. (10)
N M constraints on the minimum amount of small-cell bandwidth
m
allocations, this is not necessarily true. Obviously, if the
Here, R0 and R0 are defined as follows:
S M equilibriumwithoutregulationsalreadysatisfiestheregulatory
λ (B0 +B0 )R constraints, then the preceding result still holds, i.e., in case
R0 = S 1,S 2,S 0, (11a) A in the previous section. Otherwise, a social welfare loss is
S N
f incurred compared to the case without regulatory constraints.
R0 = (B1−B10,S +B2−B20,S)R0. (11b) DenoteSW∗wo,SWNwE astheequilibriumsocialwelfarewithout
M Nm and with regulatory constraints, respectively. The following
1 2) The optimal social welfare with the regulatory con-
straints, which we denote as SW∗.
0.9 Region Region w
Region 3) The equilibrium social welfare with regulatory con-
0.8 C.II B.II B.I straints, SWNE.
w/
0.7 The next theorem compares the three scenarios depending
Region C.I
0.6 on the total amount of newly available bandwidth B.
0BS,20.5 Theorem 4: Depending on the amount of new bandwidth
B, there exists a bandwidth threshold T
0.4
Region A Region C.II (Bo+Bo)N λ1/α−1
0.3 T = 1 2 f S , (14)
N
m
0.2
and we have the following conclusions:
0.1
1.IfB >T,thenSWNE ≤SW∗ <SW∗ .Thefirstinequality
w w wo
0 isbinding,i.e,SWNE =SW∗ <SW∗ ,ifandonlyifequation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 w w wo
B10,S (10) holds.
2. If B ≤ (B1o+B2o)Nfλ1S/α−1, then SWNE ≤ SW∗ = SW∗ .
Fig. 3. Nash equilibrium regions for 2 SPs as the bandwidth regulations Nm w w wo
vary. The first inequality is binding, i.e., SWNE =SW∗ =SW∗ , if
w w wo
and only if the following condition is met:
theoremstatesthatthelossinsocialwelfareislowerbounded, Bn ∈(cid:104)B− B2oNfλ1S/α−1,B1oNfλ1S/α−1(cid:105),Bn =B−Bn.
andtheworstpointoccursatthescenariowheretheregulatory 1 N N 2 1
m m
(15)
constraints require both SPs to allocate all bandwidth only to
small-cells. Theorem4statesthatifthetotalamountofnewlyavailable
bandwidth is too large, no matter if the two competing SPs
Theorem 3 (Social Welfare): Compared to the case with-
maximize revenue or social welfare, we always have some
out the regulatory constraints, social welfare loss is incurred
social welfare loss compared to the case without regulatory
when the following inequality is true:
constraints. This can be explained as follows. Using the set
Nfλ1S/α−1 (cid:88)B < (cid:88)B0 . (12) of α-fair utility functions, without regulatory constraints the
N λ1/α−1+N i i,S socially optimal bandwidth allocation strategy is to allocate
f S m i∈N i∈N bandwidth to macro- and small-cells based on a fixed pro-
We have: portion. If the total amount of newly available bandwidth is
SWNE (cid:16) N λ1/α−1 (cid:17)α not large, simply following the original allocation satisfies
w ≥ f S , (13)
SW∗wo Nfλ1S/α−1+Nm tHhoewreevgeurl,atwiohnenretqhueiaremmoeunnttaonfdneiswthbaenredfworiedthsobceiaclolymeosptliamrgael.,
where the bound is tight exactly when Bi0,S =Bi,∀i∈N. since the new bandwidth is required to be allocated to small-
Inpractice,aspectrumregulator,suchastheFCC,mayseek cells only, the original optimal proportion would no longer be
tofindanoptimalwaytoallocatenewlyavailablespectrumso valid given the small-cell bandwidth constraints. As a result
that the market equilibrium yields the largest social welfare. of this, social welfare loss relative to the original allocation
Wenextuseourresultstoanalyzethecasewherethespectrum scheme becomes inevitable. Further, note that the bandwidth
regulator needs to allocate a total available new bandwidth B threshold at which this loss begins occurring is proportional
to two competitive SPs. SP 1 and 2 each have initial licensed to Nf , so that when there are more fixed users willing to use
bandwidth Bo and Bo, and get a proportion of the new smaNlml-cells,thethresholdincreases.Itisalsoincreasinginλ ,
1 2 S
bandwidth, denoted as Bn and Bn. The initial bandwidth is the gain in spectral efficiency of small-cells and in the initial
1 2
freetouseforeithermacro-cellsorsmall-cells.Incontrast,the allotment of licensed bandwidth.
newbandwidthcanonlybeusedforsmall-cells.Asmentioned Theorem 4 also indicates that when the amount of newly
before this is motivated by the 3.5GHz band, where FCC available bandwidth is below the threshold, there exists a
regulates the power constraint to be very small, and therefore bandwidth split that achieves the optimal benchmark social
it can only be used for small-cell deployment [1]. welfare. This result suggests that if a spectrum controller is
Thespectrumregulatorneedstodeterminetheoptimalsplit planning to enforce bandwidth regulations on newly released
ofthenewbandwidthsuchthatthesocialwelfareundermarket bands, it should consider the possible impacts on the market
equilibrium is maximized. We consider the following three equilibrium without regulations carefully. In particular, if the
scenarios for any possible bandwidth partition (Bn,Bn): amount of newly available bandwidth is too large, imposing
1 2
1)Theoptimalsocialwelfarewithoutregulatoryconstraints, suchregulationsmightleadtosocialwelfarelosscomparedto
SW∗ . Note, from [13], this is the same as the equilibrium the scenario where the regulations were not imposed. On the
wo
socialwelfarewithoutregulatoryconstraints.Thiswillbeused other hand, if the amount of new spectrum is small compared
as a benchmark. to the existing bands already licensed to SPs in the market,
the influence on the market equilibrium from the introduction forinnovation,spectrumcaps,andthemanagementofharmful
of bandwidth regulations on the new bands is minor and interference.
controllable,andthereforewillnotincuranylossinthesocial
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In this paper we considered the impact of bandwidth 2016,SanFrancisco,May2016.
regulations on resource allocation in a HetNet. We showed
that by imposing a required minimum bandwidth allocation
to small-cells, the optimal bandwidth allocation strategies of
SPs can change dramatically. While this change is relatively
straightforward in the monopoly scenario, it turns out to be
much more complicated in the competitive scenario with two
SPs. Specifically, the existence and uniqueness of Nash equi-
libriaarestillpreservedafteraddingtheregulatoryconstraints.
However, the equilibria can exhibit very different structures
andcharacteristicsastheconstraintsvary.Wealsoshowedthat
the introduction of such regulations may shift the equilibrium
away from an efficient allocation, thus incurring some social
welfare loss. Our results suggest that adding such regulatory
constraints complicates the resource allocation schemes in
the HetNets. While these constraints may be introduced by
the spectrum regulator to address other concerns, they can
ultimatelyreducethesocialwelfareachieved.Forfuturedirec-
tions,weareplanningtostudyotherpolicyconsiderationsthat
wedidnottakeintoaccountinthispaper,liketheimplications
APPENDIXA Then at the Nash equilibrium without constraints, we have:
PROOFOFTHEOREM1
B R
D =λ u(cid:48)(R )+λ2 1,S 0u(cid:48)(cid:48)(R )
TheproofofTheorem1isstraightforwardandwecanapply 1 S S S N S
f
theresultsin[10]directly.Inparticular,iftheoriginaloptimal B R
−u(cid:48)(R )− 1,M 0u(cid:48)(cid:48)(R ) (18)
bandwidthallocationstillholdswiththeaddedregulationcon- M N M
m
straints, then we are done. Otherwise we have to increase the (cid:104) (cid:105) (cid:104) (cid:105)
=λ u(cid:48)(R )+R u(cid:48)(cid:48)(R ) − u(cid:48)(R )+R u(cid:48)(cid:48)(R )
small-cell bandwidth allocation. Both the revenue and social S S S S M M M
welfare are concave functions in the small-cell bandwidth −λ2B2,SR0u(cid:48)(cid:48)(R )+ B2,MR0u(cid:48)(cid:48)(R )=0. (19)
allocation, and at the original equilibrium point the marginal S N S N M
f m
revenue and social welfare increase with respect to per unit
At the equilibrium with constraints, similarly we have:
increase in bandwidth are equal for both macro- and small-
cells. Hence, when the small-cell bandwidth increases, we (cid:104) (cid:105) (cid:104) (cid:105)
D =λ u(cid:48)(R(cid:48) )+R(cid:48) u(cid:48)(cid:48)(R(cid:48) ) − u(cid:48)(R(cid:48) )+R(cid:48) u(cid:48)(cid:48)(R(cid:48) )
entertheregionwherethemarginalrevenueandsocialwelfare 1 S S S S M M M
increase with respect to per unit increase in bandwidth for B R
−λ2 2 0u(cid:48)(cid:48)(R(cid:48) )≤0. (20)
small-cells is smaller than that of macro-cells. As a result, the S N S
f
best option is to operate at the boundary point, i.e., allocating
exactly the required minimum amount of bandwidth to small- However, since u(cid:48)(r)+ru(cid:48)(cid:48)(r) decreases in r, u(cid:48)(cid:48)(r) < 0
cells. and increases in r, and the fact that RS(cid:48) < RS,RM(cid:48) > RM,
the inequality sign in (20) should be reversed. Therefore we
have a contradiction.
APPENDIXB
PROOFOFTHEOREM2
APPENDIXC
As this is a concave game, to prove the existence and PROOFOFTHEOREM3
uniquenessoftheNashequilibrium,wecanusetheuniqueness
Applying the same arguments we used in proving Theorem
theorem (Theorem 6) in Rosen’s paper [16], which gives
1, we know that since increasing the small-cell bandwidth
sufficientconditionintermsofacertainmatrixbeingnegative
allocationbeyondtheoriginalequilibriumpointonlydecreases
definite. In our previous work [13] it was proved that the
the social welfare, then the worst case occurs at the point that
requiredmatrixisnegativedefiniteforthecorrespondinggame
all bandwidth is required to be allocated to small-cells.
without bandwidth restrictions. Here the only difference is
that we have additional linear constraints on the bandwidth
APPENDIXD
allocations, which does not have any effect on this result.
PROOFOFTHEOREM4
Therefore, the same arguments also apply here.
As for the second part of the theorem, denote RS and RS(cid:48) For scenario 2) and 3), as long as the sum of the small-
as the average service rate in small-cells with and without cell bandwidth allocations of the two SPs at the equilibrium
the regulatory constraints, respectively. Suppose at the Nash withouttheconstraintsislargerthanthesumoftheregulation
equilibrium with constraints, the sum bandwidth allocation to constraints, then they are the same. This requires:
small-cells is less than that without the regulatory constraints,
then we have: N λ1/α−1(Bo+Bn+Bo+Bn)
f S 1 1 2 2 ≥Bn+Bn, (21)
R(cid:48) <R . (16) N λ1/α−1+N 1 2
S S f S m
Denote Di = ∂∂BSi,iS, it follows that: which yields the following condition:
D +D =λ (cid:104)2u(cid:48)(R )+R u(cid:48)(cid:48)(R )(cid:105)− B ≤ (B1o+B2o)Nfλ1S/α−1. (22)
1 2 S S S S N
m
(cid:104) (cid:105)
2u(cid:48)(RM)+RMu(cid:48)(cid:48)(RM) . (17) Otherwise,iftheprecedingconditionisnotsatisfied,thesocial
welfare corresponding to the second scenario is also less than
Since 2u(cid:48)(r)+ru(cid:48)(cid:48)(r) decreases in r, and we know that that corresponding to the first scenario, i.e, SW∗ <SW∗ .
w wo
at the Nash equilibrium without constraints, D1 = D2 = 0, On the other hand, the only possible way for scenario 3) to
we can conclude that D1 +D2 > 0 at the equilibrium with achieve the optimal social welfare corresponding to scenario
constraints. As a result, at least one of D1 or D2 must 1)istoensuretheNashequilibriumisexactlythesameasthe
be greater than 0 at equilibrium. Without loss of generality, one without the regulation constraints. This requires:
suppose D >0 at the equilibrium with constraints.
2
Given D2 > 0 , it must be that B2(cid:48),S = B2, and D1 ≤ 0. Nfλ1S/α−1(B1o+B1n) ≥Bn,Nfλ1S/α−1(B2o+B2n) ≥Bn,
This is because if D1 > 0 also holds, B1(cid:48),S = B1 and it Nfλ1S/α−1+Nm 1 Nfλ1S/α−1+Nm 2
contradicts with the fact that R(cid:48) <R . (23)
S S
which can be simplified to:
(Bo+Bo)N λ1/α−1
B ≤ 1 2 f S , (24a)
N
m
(cid:104) BoN λ1/α−1 BoN λ1/α−1(cid:105)
Bn ∈ B− 2 f S , 1 f S . (24b)
1 N N
m m
When SW∗ <SW∗ , it means the required minimum sum
w wo
bandwidth allocation to small-cells is larger than the sum
bandwidthinsmall-cellsattheequilibriumwithoutconstraints.
Since we know that at the original equilibrium the social
welfare is maximized and the social welfare is a concave
function with respect to the sum bandwidth in small-cells,
in this case the social welfare maximizing point with the
constraints is therefore exactly the required minimum small-
cell bandwidth point, i.e, when B +B = B0 +B0 .
1,S 2,S 1,S 2,S
Theonlypossibilityforscenario3)toachievethisistoensure
B = B0 ,B = B0 at the Nash equilibrium with
1,S 1,S 2,S 2,S
constraints. As a result, equation (10) becomes exactly the
condition for SWNE =SW∗.
w w