Offering Supplementary Wireless Technologies: Adoption Behavior and Offloading Benefits

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by  Carlee Joe-Wong

Complementary Wireless Network Technologies: Adoption
Behavior and Offloading Benefits
Carlee Joe-Wong Soumya Sen Sangtae Ha
[email protected] [email protected] [email protected]
PrincetonUniversity,Princeton,NJ08544
2
ABSTRACT hasalreadybeguntoofferfemtocellsinordertosupplement
1
its4Gnetworkcapacity[3],whileAT&ThasdeployedWiFi
0 To alleviate the congestion caused by rapid growth in de-
hotspots in New York to manage persistent 3G congestion
2 mand for mobile data, ISPs have begun encouraging users
to offload some of their traffic onto a supplementary, bet- [4]. Though AT&T’s WiFi is currently free, as mobile de-
p mand keeps growing, ISPs may soon begin to charge con-
ter quality network technology, e.g., offloading from 3G or
e sumersforaccesstothesesupplementalnetworks[5]. Many
S 4GtoWiFiandfemtocells. Withthegrowingpopularityof
ISPs, in fact, have already begun to do so: for instance, T-
suchofferings,adeeperunderstandingoftheunderlyingeco-
2 nomicprinciplesandtheirimpactontechnologyadoptionis Mobile charges their subscribers an extra $20 a month for
2 WiFi access, while Virgin Mobile charges an extra $15 per
necessary. To this end, we develop a model for user adop-
month [6, 7]. Given these developments, ISPs will soon re-
tion of a base wireless technology and a bundle of the base
] quire economic models that help them understand how to
I plus a supplementary technology. In our model, individual
N users make their adoption decisions based on several fac- price access to such supplementary technologies and what
implications theirpricing decisions may have.
. tors,includingthetechnologies’intrinsicqualities,through-
s put degradation due to congestion externalities from other We develop an analytical framework to understand user
c adoptiondecisionsbetweenabasetechnologyandabundled
subscribers, and the flat access rates that an ISP charges.
[
offering of a base plus supplemental technology; users may
We study the adoption dynamics and show that they con-
adopt the base technology, no technology, or the bundle of
1 vergetoauniqueequilibriumforagivenset ofexogenously
v determinedsystemparameters. Inparticular,wecharacter- bothtechnologies. Weassumethattheusershaveheteroge-
4 izetheoccurrenceofinterestingadoptionbehaviors,includ- neousvaluationsofeachtechnology’squalityandaccountfor
0 ingapossibledecreaseintheadoptionofthesupplementary the negative externalities of congestion effects as a technol-
0 technologyasitscoverageincreases. Similarbehaviorsoccur ogy’sadoptionincreases. Weusethisframework toidentify
5 the impact of various factors, such as, coverage and pricing
at anISP’sprofit-maximizingpricesandtheoptimalcover-
. on theequilibrium adoption levels and an ISP’sprofit.
9 age area for the supplementary technology. To account for
Our work is inspired by two research areas: the study
0 thepotentialbenefits from offloading inpractice, wecollect
of user technology adoption and that of network offloading.
2 3G and WiFi usage and location data from twenty mobile
Thoughbothareashaveseparatelyreceivedconsiderableat-
1 users. We then use this data to numerically investigate the
tentionfromeconomicsandnetworkingresearchers,ourcon-
: profit-maximizingadoptionlevelswhenanISPaccountsfor
v tributionliesinincorporatinguseradoptionmodelstostudy
its cost of deploying the supplemental technology and sav-
i tradeoffs between deployment costs and offloading benefits
X ings from offloading traffic onto thistechnology.
forasupplementarytechnology. Wegiveanalyticconditions
r underwhichnon-intuitiveadoptionbehaviorsoccurandcol-
a 1. INTRODUCTION
lect usage data to substantiate our study of an ISP’s profit
Overthepastfewyears,Internetserviceproviders(ISPs) andsavingsfromoffloading. Inparticular,ourmodelshows
havebeguntoexperiencetheeffectsofaprojected 78% an- interestingoutcomesinthefollowingscenarios(Section4.1):
nual growth rate in the demand for mobile data over the • An ISP wishes to expand its femtocell network to of-
next five years [1]. Yet existing mobile data networks are floadmore4Gdatatraffic,butcannotchangeitspric-
increasingly unable to accommodate this growth, leading ing structure due to exogenous factors, e.g., the pres-
ISPs to search for ways to curb network congestion with- ence of a major competitor. We show that while in-
out hurting their profit margins [2]. For instance, Verizon creasingfemtocellcoveragecanincreasethevolumeof
offloaded traffic, it may also decrease femtocell adop-
tion: increased congestion induces some users to drop
thebundledfemtocellserviceandonlysubscribeto4G.
We then specify conditions under which this decrease
Permission tomake digital orhardcopies ofall orpartofthis workfor
occurs even when the ISP offers revenue-maximizing
personalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesare
prices.
notmadeordistributedforprofitorcommercialadvantageandthatcopies
bearthisnoticeandthefullcitationonthefirstpage.Tocopyotherwise,to
• An ISP tries to induce heavy users to leave its 3G
republish,topostonserversortoredistributetolists,requirespriorspecific
network by increasing the access price. However, by
permissionand/orafee.
Copyright20XXACMX-XXXXX-XX-X/XX/XX...$10.00. doing so, an ISP may actually increase user adoption
of 3G: increasing the price of 3G and the 3G + WiFi sharing decisions of femtocell providers. In contrast, we do
bundle by the same amount can lead some users to notconsideranentrant-incumbentcompetitionscenario;we
drop their WiFi subscriptionsand adopt only 3G. model the adoption of a supplementary technology offered
byamonopolistproviderandtheresultingtradeoffsbetween
Given these adoption behaviors, we then consider an ISP’s
the deployment costs and the savings from offloading. Our
optimaloperatingpoint–at what accesspricesandcoverage
work is thusfocused on traffic offloading, unlike[13].
area does an ISP maximize its profit, and what adoption
levelsdothesecorrespondto? Indeterminingtheprofit,we
2.2 Traffic Offloading
consider the ISP revenue, the cost of deploying the supple-
mental technology, and the ISP’s savings from offloading. Shetty,ParekhandWalrandanalyzeuseradoptionofsplit-
Usingempirical datatoestimate theoffloading benefits,we and common-spectrum 4G and femtocell networks and use
consider thefollowing scenarios (Section 5.2): their results to study ISPrevenue maximization [14]. They
considertheutilityofheterogeneoususersunderbothspec-
• Suppose that an ISP seeks to optimize its profit with
trum sharing schemes and account for congestion effects
respect to femtocell coverage and access prices for its
with detailed throughput models. However, [14] does not
bundled 4G and femtocell offerings. We show that
consider ISP costs or savings from offloading and relies on
adoption of both 4G and the 4G + femtocell bundle
numericalstudiesduetothecomplexityoftheirthroughput
mayincreasewhenthemarginalsavingsintheamount
models.
of traffic offloaded increases.
Otherworkshavealsostudiedtrafficoffloading,butwith-
• As the ISP maximizes its profit in densely populated out developing an analytical model of user adoption deci-
areas, femtocell adoption may increase with the cost sions. Forinstance, [15] considers theproblem of offloading
of deployment. 3G traffic to WiFi networks, focusing on the implications
for ISPrevenue. Useradoption is heremodeled usinggiven
We give an overview of related work on user adoption
demand functions, which depend on the prices of 3G and
and offloading models in Section 2. In Section 3, we intro-
WiFi. Offloadingonto femtocell networks isstudied in [16],
duce our model and characterize the equilibrium adoption
which considers ISP revenue and social welfare under flat
levels. Section4usestheseequilibriatoderiveanalyticcon-
and usage-based pricing of both open and closed femtocell
ditions under which the equilibrium adoption can behave
networks. Ourworkcontributestotheseeffortsbyproviding
non-intuitively. We then investigate the equilibrium adop-
a fairly generic analytical framework, complemented with
tion levels when theISP maximizes its profit for a range of
datacollectedfromrealusers,tostudytheroleofeconomic
different cost structures (Section 5), and conclude the pa-
and technological decisions on the possible outcomes of the
per in Section 6. All the proofs, omitted here due to space
adoption process.
constraints, can befound in AppendixA.
2. RELATED WORK 3. TECHNOLOGY ADOPTIONMODEL
Our work relates to two topics in the literature: tech- In this section, we introduce an analytic framework to
nology adoption dynamics and the economics of offloading model the dynamics of user adoption based on the user’s
traffic onto a supplemental network (e.g., 3G/4G traffic to utility of subscribing to the base and supplemental tech-
WiFi/femtocell networks). nologies, denoted as Technologies 1 and 2, respectively. We
consider a monopolist ISP,i.e., one ISPthat does not com-
2.1 Technology Adoption peteforuserswithotherserviceproviders. Usersmaychoose
Many works in economics have studied technology adop- to adopt only the base technology (Technology 1), adopt a
tion in variouscontexts[8]. KatzandShapiro,forinstance, bundleofthebaseandsupplementaltechnologies(Technolo-
study the adoption dynamics of competing network tech- gies (1+2)), or to not adopt either technology. This choice
nologies with positive externalities in a homogeneous user isgovernedbythevaluethateachoftheaboveoptionspro-
population [9], while Cabral [10] presents a diffusion model videstotheuser,asdescribedinSection3.1. Users’choices
forasingletechnologyadoptionbyuserswithheterogeneous evolveover time in response to changes in thetechnologies’
network valuations. Economides and Viard [11] provide a adoption and congestion levels; we analytically formulate
static analysisfortheadoptionoftwocomplementary tech- these dynamics and characterize the steady-state equilib-
nologieswithpositiveexternalitiesandheterogeneityinuser rium adoption levels. In Section 3.2, we show that exactly
evaluations. Sen,et. al. [12]studythedynamicsofcompeti- one asymptotically stable equilibrium exists for any given
tionbetweentwogenericnetworktechnologieswithpositive set of exogenous system parameters.
network externalities, focusing on the role of converters in
3.1 UtilityFunctions
affecting theequilibrium outcomes.
While our work follows these in accounting for user het- A user’s value or utility from subscribing to a particular
erogeneity,itdiffersinthat(a)theexternalityinourmodel wireless technology depends on several factors, such as the
is thedominant negative externality of network congestion, intrinsic quality of the technology (e.g. the user’s mone-
and (b) we consider a non-competitive scenario in which a tary valuation of the maximum throughput), the negative
supplementarytechnologyisofferedbyanISPasabundled externalityofcongestion(i.e.,reducedthroughput),andthe
servicetorelievethebasetechnology’scongestion. Weinves- access price charged by the service provider. Following [12]
tigatetheimpact ofprofit-maximizingpricingandcoverage and [14], we account for thesefactors in definingtheutility
decisions on equilibrium adoption outcomes. functions associated with each technology adoption option.
AnotherworkrelatedtooursisbyRen,Park,andvander For the two options, base and the base plus supplementary
Schaar [13], who consider the market entry and spectrum plans, the respective utility functions are given by (1) and
(2); theutility of non-adoptionis assumed to bezero. when U =0, i.e.,
1
p −T (x +(1−η)x )
U1(t)=θq1+T1(x1+(1−η)x1+2)−p1 (1) θ = 1 1 1 1+2 . (3)
(1,0) q
U (t)=(1−η) θq +T (x +(1−η)x ) 1
1+2 1 1 1 1+2
Thethresholdθ foradoptingTechnology2inaddition
+η θq2+(cid:0)T2(ηx1+2) −p2−p1. (cid:1) (2) to Technology 1(1o+c2c,u1)rs when U =U ≥0, i.e.,
1+2 1
Theaboveutilit(cid:0)yfunctionshaveth(cid:1)reeseparatevaluecom- T (ηx )−T (x +(1−η)x )− p2
ponents, as we discuss here. The intrinsic qualities (e.g., θ(1+2,1) ≥ 2 1+2 1 q1−q 1+2 η . (4)
monetary values of the maximum achievable throughput in 1 2
the absence of congestion) of Technologies 1 and 2 are de- Finally,wesolveforthethresholdθ(1+2,0) abovewhichusers
noted by q , i = 1,2, and we assume that the supplemen- will prefer to adopt both technologies, rather than have no
i
tal technology has a higher intrinsic quality than the base connectivity. This occurs when U1+2 =0, or
(q > q ). For example, femtocells and WiFi typically de-
2 1 −(1−η)T (x +(1−η)x )−ηT (ηx )
liver much higher maximum throughput than the base 4G θ = 1 1 1+2 2 1+2
(1+2,0) (1−η)q +ηq
or 3G networks. The valuation of this intrinsic quality is 1 2
p +p
weighted byarandom variableθ∈[0,1] toaccount for het- + 2 1 . (5)
erogeneity in users’ valuation of each technology’s quality. (1−η)q1+ηq2
Users who stream a lot of video, for instance, might have a In the remainder of this paper, we take the throughput
highθ value,whilethosewhomainlysurftheweb willhave degradation functions T and T to be linear, i.e., T (x) =
a low θ value.1 −γ x,i=1,2,whereγ 1andγ a2re(positive)approximi ation
i 1 2
Weassumethatthesupplementaltechnology,Technology constants.3 With thelinear T and T , (3-5) become
1 2
2, has a limited coverage area (e.g., users are not always
within range of a hotspot or a femtocell) that determines θ =p1+γ1(x1+(1−η)x1+2). (6)
the “coverage factor” η, such that a fraction η of traffic (1,0) q1
from adopters of the technology bundle travels over Tech- −ηγ x +γ (x +(1−η)x )− p2
nology2’snetwork. Weassumethatusersarehomogeneous θ(1+2,1) = 2 1+2 1 q1−q 1+2 η . (7)
in their usage volumes and that they are distributed uni- 1 2
formly throughout the coverage area; then the amount of θ =(1−η)γ1(x1+(1−η)x1+2)+η2γ2x1+2
trafficoffloaded toTechnology2isproportional tothefrac- (1+2,0) (1−η)q1+ηq2
tion of users adoptingthe technology bundle,multiplied by + p2+p1 . (8)
η. We let x1(t) denote the fraction of users adopting only (1−η)q1+ηq2
Technology 1 at time t and x (t) the fraction of users
1+2 For given adoption levels x and x , the ordering of
adopting both technologies, and note that x (t), x (t), 1 1+2
1 1+2 these threshold values (6-8) determines whether a user of
and x (t)+x (t)∈[0,1]. Thus, the amount of traffic on
1 1+2 type θ is willing to adopt a particular technology. Thus,
Technology 1 is x (t)+(1−η)x (t), while the amount
1 1+2 we can determine the fraction of users H x (t),x (t)
offloaded to Technology 2is ηx (t). 1 1 1+2
1+2
andH x (t),x (t) willingtoadoptTechnology1and
WeusedecreasingfunctionsT1 x1(t)+(1−η)x1+2(t) and 1+2 1 1+2 (cid:0) (cid:1)
Technologies (1 + 2) respectively. In doing so, we recall
T2 ηx1+2(t) to represent the th(cid:0)roughput degradation(cid:1)as a that θ ∈ (cid:0)[0,1]; for insta(cid:1)nce, if θ < 0, all users receive
function of the traffic volume for Technologies 1 and 2 re- (1,0)
(cid:0) (cid:1) positive utility from adopting Technology 1. We thus let
spectively, normalized to monetary units(e.g., thedecrease
in the technologies’ monetary value).2 The wireless service [·][0,1] denotetheprojection onto the[0,1] interval.4
Wefirstconsiderthecaseθ <θ ,i.e.,thethresh-
provider prices the access for the two options at p for the (1,0) (1+2,0)
1 old for preferring the base technology to no adoption is
base technology and p +p for the base plus supplemen-
1 2 smallerthanthatofpreferringbothtechnologiestonoadop-
tal technology bundle(i.e., p is theextra price that a user
2 tion. We show that θ > θ ; thus, if a user re-
pays for the bundled option). For notational convenience, (1+2,1) (1+2,0)
ceivespositiveutilityfrom Technology1andincreasesitby
thetimeargumentofx (t)andx (t)willbeassumedfrom
1 1+2 adopting Technology 2 as well (θ < θ < θ), she
here on to beimplicit in theutility functions (1) and (2). (1,0) (1+2,1)
cannotreceivenegativeutilityfromadoptingbothtechnolo-
Given these functions, we can find the threshold value of
gies (θ <θ <θ<θ ):
θ, θ , for which users will prefer to adopt Technology 1 (1,0) (1+2,1) (1+2,0)
(1,0)
(i.e.,U >0). Similarly,wecanalsofindthevalueofθ
1 (1+2,1) Proposition 1. Ifθ <θ ,thenθ <θ .
for which Technology 1 users will prefer the bundleof both (1,0) (1+2,0) (1+2,0) (1+2,1)
If θ <θ , then θ <θ .
Technologies (1+ 2) (i.e., U > U > 0). We note that (1+2,0) (1,0) (1+2,1) (1+2,0)
1+2 1
each θ threshold is a (time-dependent) function of x and
1
x1+2. Thus, if θ(1,0) < θ(1+2,0), the fraction of users H1 will-
The threshold θ for preferring Technology 1 occurs ing to adopt Technology 1 equals the fraction for whom
(1,0)
θ > θ > θ , and the fraction of users H will-
(1+2,1) (1,0) 1+2
1The exact values of the qi parameters depend on the par- ing to adopt Technologies (1 + 2) equals the fraction for
ticular technology being considered, while the distribution which θ > θ . For simplicity, we assume that users’
(1+2,1)
of θ values can be estimated from established techniquesin
marketing research, e.g., conjoint analysis [17]. 3This assumption is often used in theliterature on network
2WenotethatwelimitourmodeltoscenariosinwhichTech- technologyadoption [13]. InAppendixB,wederiveanalyt-
nology 2’s throughput is unaffected by the users on Tech- icalboundsontheapproximationerrorfortypicalthrough-
nology 1, e.g., non-interfering wireless technologies, as in a put functions.
split-spectrum 4G and femtocell deployment. 4That is, [y] =y if y∈[0,1], 0 if y<0, and 1 if y>1.
[0,1]
x
Table 1: Expressions for H and H in different 1 + 2
regions of (x1,x1+2). 1 1+2 θ(1, 0) = 0 θ(1, 0) = 1
Conditions on θ H H
1 1+2
θ <θ <0
a (1+2,0) (1+2,1) 0 1
θ <θ <0
(1+2,1) (1+2,0)
b θ(1,0) <0<θ(1+2,1) <1 θ(1+2,1) 1−θ(1+2,1) θ(1 + 2, 1) = 1 f d θ(1 + 2, 0) = θ(1, 0)

c 0<θ <θ <1 θ −θ 1−θ
(1,0) (1+2,1) (1+2,1) (1,0) (1+2,1)
d 0<θ <1<θ 1−θ 0
0<θ(1,0) <1<(1θ+2,1) (1,0) θ(1 + 2, 1) = 0 b c g
e (1+2,0) (1,0) 0 1−θ
0<θ <θ <1 (1+2,0)
(1+2,0) (1,0)
f θ <0<1<θ 1 0
g (11,0<) θ(1,0) <θ(1+(12+,02),1) 0 0 a e θ(1 + 2, 0) = 1
1<θ <θ
(1+2,0) (1,0)
Region c !"#&#’#$#!"# θ(1 + 2, 0) = 0 x1
!"#&#’#$#%# Figure 2: Visualizationof Table1’s regions in terms
!"#$#%# of the adoption levels.
0 θ(1, 0) θ(1+2, 0) θ(1+2, 1) 1 3.2 Convergence andStability
Technology 1 Adopted 1 + 2 Adopted Wenowexaminethestabilityoftheequilibriumpointsin
Tables 2 and 3:
!#$#%#
Region e "
!"#&#’#$#%# Proposition 2. Assuming that an equilibrium point ex-
! #$#!#
"#&#’ " ists, it is asymptotically stable.
0 θ(1, 1+2) θ(1+2, 0) θ(1, 0) 1
Technologies 1 + 2 Adopted
While we assume that throughput degradation (T and
1
Figure 1: Visualizationof θ andH valuesfor regions T ) is linear in the previous section, our stability analysis
2
c and e in Table 1. dependsonlyontheJacobian ofthedynamics(11)atgiven
adoptionlevels(x ,x ). Sinceouronlyassumptiononthe
1 1+2
valuations θ are uniformly distributed in the interval [0,1]; slopesγ andγ ofthethroughputdegradationT andT is
1 2 1 2
then positivity,theJacobianexpressionsarenotaffectedbynon-
linear forms of the T . Thus, if T and T are continuously
H x ,x = θ − θ , i 1 2
1 1 1+2 (1+2,1) [0,1] (1,0) [0,1] differentiable and strictly decreasing, Prop. 2’s conclusion
H1+2(cid:0)x1,x1+2(cid:1)=1(cid:2)− θ(1+(cid:3)2,1) [0,1(cid:2)]. (cid:3) (9) still holds.
In other words, for any set of exogenous parameters val-
If thethresh(cid:0)oldsare r(cid:1)eversed,(cid:2)i.e., θ (cid:3) <θ , thenwe
(1+2,0) (1,0) uesandinitialadoptionlevels,theadoptiondynamicsmust
may useProp. 1 toderive
converge tosome stable equilibrium:
H x ,x =0, H x ,x =1− θ .
1 1 1+2 1+2 1 1+2 (1+2,0) [0,1] Proposition 3. With the adoption dynamics (11), no
(10)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:2) (cid:3) periodic orbit can exist: for any initial values x (0) and
Followingstandardeconomicmodels,weassumeanegligible 1
x (0), x (t)andx (t)converge toanequilibriumpoint.
cost of switching between theadoption choices [12, 13]. 1+2 1 1+2
We can explicitly write out (9 - 10) by dividing the dy-
namical space into 7 different regions, as shown in Table 1. Moreover, only onesuch equilibrium point exists:
Figure1visuallyrepresentstheadoptionexpressionsintwo
regions, and Fig. 2 uses the equations for the θ threshold Theorem 1. For given values of the system parameters
values (6 - 8) to map them to the adoption levels x1 and q1, q2, η, γ1, γ2, p1, and p2, the adoption levels x1(t) and
x1+2.5 x1+2(t) converge to a unique, asymptotically stable equilib-
The userdynamics can then bewritten as rium that does not depend on the initial values x1(0) and
x (0).
1+2
x˙ (t)=ρ H x (t),x (t) −x (t)
1 1 1 1+2 1
x˙2(t)=ρ(cid:2)H1+(cid:0)2 x1(t),x1+2((cid:1)t) −x1+(cid:3)2(t) , (11) In the remainder of the paper, we use x1 and x1+2 to
denotetheuniqueequilibrium adoption levels.
where ρ ∈ (0,1] d(cid:2)enotes(cid:0)the rate of ad(cid:1)option. At(cid:3)any time
t, the fraction of users adopting each technology equals the
fraction willing to adopt, less those who have already done 4. ADOPTIONBEHAVIORS
so. Given these dynamics, we now derivethe possible equi-
Inthissection,weinvestigatethedependenceoftheequi-
libriumpointsineachregion,i.e.,thevaluesofx andx
1 1+2 librium adoption on both prices and the coverage factor.
forwhichH x ,x =x andH x ,x =x for
1 1 1+2 1 1+2 1 1+2 1+2 Section4.1highlightsnon-intuitiveadoptionbehaviors,such
theH expressionsinTable1. Tables2and3summarizethe
(cid:0) (cid:1) (cid:0) (cid:1) as the possibility of a decrease in the adoption of both the
expressions for possible equilibria in each region.
bundled technologies and the total adoption level when the
5A qualitatively similar figure with the same adjacent re- coverage factor of the supplementary technology increases.
gions will beobtained even for nonlinear T and T . In Section 4.2, we consider the ISP’s revenuemaximization
1 2
Table 2: Equilibrium points (x x ) of the different regions in Table 1.
1 1+2
(x ,x ) Region Constraints
1 1+2
p +p <−(1−η)2γ −η2γ
1 2 1 2
a (0,1)
p <η((1−η)γ −ηγ )
2 1 2
(η(γ +γ )−q +q )p +γ p <−ηγ γ +(1−η)γ (q −q )
b η((1−η)γ1−ηγ2)−p2 , p2−ηγ1+η(q1−q2) 1 2 1 2 1 1 2 1 2 1 1 2
η(q1−q2)−η2(γ1+γ2) η(q1−q2)−η2(γ1+γ2) η((1−η)γ −ηγ )<p <η(γ +q −q )
1 2 2 1 2 1
(cid:16) (cid:17)
c SeeTable 3. SeeTable 3.
d q1−p1,0 −γ <p <q , p + ηγ1p1 >η q −q + γ1q1
q1+γ1 1 1 1 2 q1+γ1 2 1 q1+γ1
(cid:16) (cid:17) −(1−η)2γ −η2γ <p +p <(cid:16) (1−η)q +ηq(cid:17)
e 0, (1−η)q1+ηq2−p1−p2 1 2 1 2 1 2
(1−η)q1+ηq2+(1−η)2γ1+η2γ2 η(q −q −(1−η)γ +ηγ )p −(q +(1−η)γ )p >η2q γ −η(1−η)γ q
2 1 1 2 1 1 1 2 1 2 1 2
(cid:16) (cid:17)
f (1,0) p <−γ , p >η(q +γ −q )
1 1 2 2 1 1
p >q
1 1
g (0,0)
p +p >(1−η)q +ηq
2 1 1 2
Table 3: Equilibrium points (x ,x ) of region c in Table 1.
1 1+2
x −ηγ2q1+(1−η)γ1q2+p1(ηγ2−(1−η)γ1+q2−q1)+p2(−(1−η)γ1−q1)/η
1 −γ1q2−ηγ1γ2+q1((1−η)γ1−ηγ2+q1−q2)
x −γ1q2+q1(q1−q2)+p1γ1+p2(γ1+q1)/η
1+2 −γ1q2−ηγ1γ2+q1((1−η)γ1−ηγ2+q1−q2)
p (ηγ −(1−η)γ +q −q )+p (−(1−η)γ −q )/η<ηγ q −(1−η)γ q
1 2 1 2 1 2 1 1 2 1 1 2
Constraints p γ +p (γ +q )/η<γ q −q (q −q )
1 1 2 1 1 1 2 1 1 2
p (ηγ +ηγ +q −q )+p γ >−ηγ γ +(1−η)γ (q −q )
1 2 1 2 1 2 1 1 2 1 1 2
problem, and find that this behavior persists underthe op- gies (x ∈(0,1)). Then x decreases with η if
1+2 1+2
timal prices.6
(1−η)2γ q +(1−η2)γ q +η(η−2)γ q −η2γ q
1 1 1 2 2 1 2 2
4.1 Observations +(p1+p2)(q2−q1−2(1−η)γ1+2ηγ2)<0. (12)
We first consider the adoption behavior for a range of IfsomeusersadoptTechnology1,someadoptbothtechnolo-
coveragefactors(η),e.g.,Section1’sexampleofanISPthat gies, and some neither (x >0, x >0 and x +x <
1 1+2 1 1+2
increases its femtocell coverage to offload more traffic from 1), then total adoption x +x increases with η.
1 1+2
4G, but cannot changeits access prices duetothepresence
of a competitor. Figure 3a shows the equilibrium adoption
Wenotethatthemathematical condition(12) isdecreas-
levelsforasetofexogenoussystemparameters. Atlarge(>
ing in ηγ ; Technology 2’s throughput degradation coeffi-
0.7) values of η, adoption x of the bundled technologies 2
1+2
cient γ , multiplied by the coverage factor η, must be suf-
decreases with η, even though the coverage area increases. 2
ficiently high for x to decrease. On the other hand,
As η increases, a larger portion of traffic ηx is offloaded 1+2
1+2
the presence of the positive (p +p )(q −q ) term indi-
onto Technology 2, and theresulting increase in congestion 1 2 2 1
cates that if the marginal difference in the intrinsic quality
canloweradoptionofTechnologies(1+2). Thendepending
between the two technologies (q −q ) is large, users may
ontheadoptionx ofTechnology1,thetotaladoptionx + 2 1
1 1
adoptthebundledtechnologiesevenifTechnology2isvery
x may increase or decrease as x decreases. In Fig.
1+2 1+2
congested.
3a, x is positive and increasing, and the total adoption
1
AnotherinterestingfeatureofFig. 3bistheabruptswitch
also increases. Figure 3b shows an example in which x
1+2
from all users adopting the base technology to all users
decreases with η, but x =0. Thus, x +x =x and
1 1 1+2 1+2
adopting the bundled technologies when η < 0.1. In this
the total adoption may decrease as the coverage increaes. In
example, the access price p of Technology 2 is relatively
fact, x is crucial to thebehaviorof x +x : 2
1 1 1+2
low, as is its throughput degradation coefficient γ when
2
compared to the intrinsic quality q . Thus, as η increases
2
Proposition 4. Suppose that nousers adopt Technology slightly, the utility of adopting Technologies (1 + 2) in-
creases quickly: theuserneednotpaymuchmoreforTech-
1 (x =0), and that some, but not all, adopt both technolo-
1 nology 2, which provides higher quality service with rel-
atively little throughput degradation. Thus, many users
6In Appendix C, we show that similar behaviors occur adopt the supplemental technology in addition to the base
whentheuserheterogeneityvariableθ isnon-uniformlydis- one. Asηgrowsfurtherto0.08,theutilityofadoptingTech-
tributed. nologies (1 + 2) becomes larger than that of adopting only
80 70 70
Base
70 60 Base 60 BToutnadlle
%)60 Base %)50 BToutnadlle %)50
Adoption Levels (23450000 BToutnadlle Adoption Levels (234000 Adoption Levels (234000
10 10 10
00 0.1 0.2 0.3 0.4 0η.5 0.6 0.7 0.8 0.9 1 00 0.1 0.2 0.3 0.4 0η.5 0.6 0.7 0.8 0.9 1 00 20 40 60 p 80 100 120 140
1
(a) Adoption levels, x >0 for η>0.7.(b) Adoptionlevels, x =0 for η>0.1. (c) Adoptionas p increases.
1 1 1
Figure 3: As the supplementaltechnology’s coveragearea η increases, (a) x decreasesfor large η, while(b)
1+2
total adoptionmayalsodecrease. As(c) the basetechnology’saccessprice p increases,the basetechnology’s
1
adoption increases. Parameter values are (a) q =200, q =250, γ =50, γ =20, p =40, p =10; (b) q =100,
1 2 1 2 1 2 1
q =300, γ =50, γ =100, p =40, p =10; and (c) q =200, q =225, γ =150, γ =50, p =30, η=0.5.
2 1 2 1 2 1 2 1 2 2
the base technology,save for those users who adopt neither adoptionintheseregions. UsingTable4,weprovethatrev-
technology duetolow valuation levels θ. enueisgreatestunderpartialadoptionofbothtechnologies:
Finally,weconsideradoptionbehaviorsforfixedTechnol-
ogy 2 access price p and coverage factor η. For instance, Proposition 6. Ifηisfreetovary,itsrevenue-maximizing
2
as proposed in the introduction, the ISP may increase the value is η = 1, i.e., full coverage of the supplemental tech-
access price p of its base technology in an attempt to in- nology. For any fixed η, if
1
duce heavy users to leave their network. We find that in γ q
some cases, increasing p actually increases Technology 1’s η 2 ≥(1−η) 2, (14)
1 γ q
1 1
adoption x . Figure 3c shows an example; we note that
1
thoughx increases,thetotaladoptionx +x decreases. therevenue-maximizingequilibriumadoptionlevelslieinre-
1 1 1+2
Wecaninfactfullycharacterizetheconditionsunderwhich gion c of Table 1: some users adopt Technology 1, some
this behavioroccurs: adoptTechnologies(1+2),andsomeadoptneithertechnol-
ogy(x p∗,p∗ >0,x p∗,p∗ >0,x p∗,p∗ +x p∗,p∗ <
1 1 2 1+2 1 2 1 1 2 1+2 1 2
1). If (14) does not hold, then no users adopt Technology
Proposition 5. SupposethatsomeusersadoptTechnolo- 1, but (cid:0)some a(cid:1)dopt Techn(cid:0)ologie(cid:1)s (1 + 2(cid:0)) (x (cid:1)p∗,p∗ (cid:0)= 0, (cid:1)
gies 1 and (1 + 2), while some adopt neither technology 1 1 2
x p∗,p∗ >0).
(x1 >0, x1+2 >0, x1+x1+2 <1). Then the base technol- 1+2 1 2 (cid:0) (cid:1)
ogy’s adoption x1 increases with the access price p1 if (cid:0) (cid:1)
The condition in Prop. 6 can be interpreted as stating
q −q <(1−η)γ −ηγ . (13)
2 1 1 2 thatwhenthequalityq ofTechnology1issufficientlyhigh
1
andthemarginalthroughputdegradationγ sufficientlylow
1
Qualitatively,(13)indicatesthatTechnology1’sadoption relative to Technology 2, then for a large coverage factor η,
x increaseswithp ifthequalitydifferential(q −q )from some users will adopt Technology 1 at the optimal prices.
1 1 2 1
adoptionofTechnology2isoutweighedbythemarginalsav- However, under these conditions the adoption x1+2 p∗1,p∗2
ingsinthroughputdegradationfromadoptingonlyTechnol- of Technologies (1 + 2) will decrease, as shown in Fig. 4’s
ogy 1 ((1−η)γ −ηγ ). As p increases, users adopt Tech- example.7 As in Fig. 3a, in Fig. 4 the volume (cid:0)of traf(cid:1)-
1 2 1
nology 1, rather than thebundledTechnologies (1+ 2). ficηx1+2 p∗1,p∗2 offloadedontoTechnology2increaseswith
η, leading some users to adopt only Technology 1. How-
4.2 Revenue Maximization ever, sinc(cid:0)e p∗ =(cid:1) η(q −q )/2 increases with η (cf. region
2 2 1
We now examine the behavior of Technologies (1 + 2)’s c in Table 4), ISP revenue increases with η. Overall adop-
equilibrium adoption x1+2 as the coverage factor η varies tion x1 p∗1,p∗2 +x1+2 p∗1,p∗2 also increases; theincrease in
and the ISP chooses prices so as to maximize its revenue x1 p∗1,p(cid:0)∗2 offs(cid:1)ets the d(cid:0)ecreas(cid:1)e in x1+2 p∗1,p∗2 . Formally,
p (x +x )+p x . WefirstuseTables2and3’sexpres-
1 1 1+2 2 1+2 (cid:0) (cid:1) (cid:0) (cid:1)
sions for the equilibrium x1 and x1+2 to find the revenue- Proposition 7. If(14)holds,thenadoptionx1+2 p∗1,p∗2
amsaxshimowizninginpTriacbelsep4∗1.anTdope∗2mapthaesaiczhe pthoesisribdleepeeqnudileinbcreiumon, xofTecph∗n,oplo∗gieinsc(r1ea+se2s)wdietchreηa.sesandtotaladoptionx(cid:0)1 p∗1,p(cid:1)∗2 +
price, in the remainder of this section we use the notation 1+2 1 2 (cid:0) (cid:1)
x1 p∗1,p∗2 andx1+2 p∗1,p∗2 todenotetheequilibriumadop- (cid:0) (cid:1)
tion levels given theoptimal prices p∗ and p∗.
(cid:0) (cid:1) (cid:0) (cid:1) 1 2 5. OPERATINGCOSTS ANDPROFIT
Weseefrom Table 4that theISPearnsnon-positiverev-
enueif it maximizes its revenueat equilibria in regions a, f, In addition to considering adoption under revenue max-
and g. Intuitively, in these regions, all users adopt at least imization, as in Section 4.2, ISPs must take into account
onetechnologyattheequilibrium(x +x =1inTable2).
1 1+2 7We note that the overall adoption level in Fig. 4 is low
Yet if users’ technology valuations θq are sufficiently close
i whencomparedwiththoseofFig. 3. Withdifferentparam-
tozeroduetoasmallθvalue,theirutilityfunctions(1)and
eters(e.g. q andγ values),theoveralladoptionlevelsmay
i i
(2) will be negative unless the prices are negative. Thus, change; we use the ones here to reflect current smartphone
theISPmustoffernegative pricesinordertoguaranteefull penetration rates in theU.S.[18].
Table 4: Revenue-maximizingprices assuming equilibrium adoption levels in regions a-g (cf. Tables 1-3).
p∗ p∗ Revenue
1 2
a <−(1−η)γ −(1−η)2γ −η2γ −p −(1−η)2γ −η2γ
1 1 2 1 1 2
b∗ −ηγ1γ2+(1−η2)(q1−q2) η(q2−q1) η4(q1−q2)2+(1−η)γ1(q1−q2)−ηγ1γ2
η(γ1+γ2)−q1+q2 2 η(γ1+γ2)+q2−q1
c† q1 η(q2−q1) q12ηγ2+q22ηγ1+q12(q2−q1)+ηq1(q1−q2)2
2 2 4(γ1q2+ηγ1γ2+q1(q2−q1+ηγ2−(1−η)γ1))
2
d q1 ≥η q −q + γ1q1 q1
2 2 1 2(q1+γ1) 4(q1+γ1)
e (1−η)q1+ηq2 −p ≤p ηq2+η2(cid:16)γ2 −η − η2q1γ2−(cid:17)η(1−η)γ1q2 ((1−η)q1+ηq2)2
2 2 1 q1+(1−η)γ1 q1+(1−η)γ1 4((1−η)q1+ηq2+(1−η)2γ1+η2γ2)
(cid:16) (cid:17)
f −γ ≥η(q +γ −q ) −γ
1 2 1 1 1
g >q ≥η(q −q ) 0
1 2 1
∗If2(1−η)γ −2ηγ >q −q ,the revenue-maximizingprices for region b are instead: p∗ = (1−η)γ1(−γ1+q1−q2),
1 2 2 1 1 η(γ1+γ2)+q2−q1
p∗ =(1−η)γ −ηγ , revenue= −ηγ1γ2+((1−η)2γ1+η2γ2)(q1−q2)−η((1−η)γ1−ηγ2)2
2 1 2 η(γ1+γ2)+q2−q1
†If ηγ q ≤(1−η)γ q , then at theoptimal prices x =0 and theequilibrium lies in region e.
2 1 1 2 1
45
40
35
%)
s (30
el Base
Lev25 Bundle
on 20 Total
pti
do15
A
10
5
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
η
Figure 4: Adoption levels for η ∈[0,1] and revenue-
maximizing prices (q = 50, q = 100, γ = 50,
1 2 1
γ =100). As η increases, total adoption x p∗,p∗ +
2 1 1 2
x p∗,p∗ increases, driven by the increase in
1+2 1 2 (cid:0) (cid:1) (a)Hourly3GandWiFi. (b) Location-specific.
x p∗,p∗ .
1 1(cid:0) 2 (cid:1)
Figure 5: Screenshots of our usage monitoring app
the(cid:0)savin(cid:1)gsfromoffloadingtrafficandthecosttodeploythe
on the Android platform.
supplemental technology. In this section, we first use em-
piricalusagedatatoestimatetheamountoftrafficthatcan We implemented a simple data monitoring app and re-
beoffloaded ontothesupplementaltechnology’s network at leased it to users in the United States. Figure 5 shows app
times of peak usage on the base technology. We then use screenshots;ineachhour,werecordedthevolumeof3Gand
cost parameters appropriate for a WiFi deployment to in- WiFi usage and WiFi base station IDs. We find that the
vestigate user adoption underISPprofit maximization. probability of WiFi access in the hour of highest 3G usage
is82%oftheoverallprobabilityofWiFiaccess. Onaverage,
5.1 Trial Data
55% of 3G traffic occurs in these peak hours, corroborating
We gather 3G and WiFi usage data from 20 Android existing findings that 3G data usage exhibits severe peaks
smartphonesoversixdays. SinceISPcostisdrivenbypeak- duringtheday [19].
hourtraffic,wefocusonusagewhenthe3Gnetworkismost
heavily utilized [19]. Our goal is twofold: first, to estimate 5.2 Optimizing Profit
thefraction of 3Gtraffic thatoccurs at thispeak time;and
In addition to its revenue, ISP profit includes its savings
second, to estimate the probability of WiFi access at this
from offloading, less the cost of deploying a supplemental
time,giventheoverallWiFiaccessprobability. Wecanthen technology.8 Since these parameters depend on the market
estimatetheamountoftrafficthatwillbeoffloadedtoWiFi
at the peak time, given the WiFi adoption and coverage 8We assume that the deployment cost of the base technol-
factor. ogy is independent of the adoption levels, e.g., an already-
conditions,weconsiderthreescenarios: asmallcity;alarge, square miles, (a 130 meter radius), for the small city, 0.005
sparsely populated city (e.g., in California); and a large, square miles for the sparse city, and 0.002 square miles for
more densely populated city (e.g., New York or Philadel- thedensecity. Wethenfindc =6.2,4.9,and11.5forthe
AP
phia). We refer to the latter two cities as the“sparse”and small, sparse, and densecities respectively.
“dense”cities. In Figure 6, we show the adoption levels x∗ and x∗ at
1 1+2
We model the cost savings introduced by user offloading the ISP’s optimal operating point for a range of c and
WF
asalinearfunctionoftheamountoffloadedduringthepeak c values, obtained by varying the marginal savings from
AP
hour,i.e.,themarginalcostofpeaktraffic,multipliedbythe offloading and cost of one AP’s deployment. Though some
amount offloaded [19]. Wetakethismarginal cost tobe1.0 characteristicspersistforallscenarios–forinstance,adoption
¢/MBinthesmallcity,1.9¢/MBinthesparsecity,and2.9 x∗ of Technologies (1 + 2) increases with the marginal
1+2
¢/MB in the dense city; these values are based on AT&T’s offloadingsavingsc –therearesomenoticeabledifferences.
WF
and Verizon’s data plan overage charges in the U.S. From For the small and sparse cities, as c increases at small
WF
our trial data, we find that each user consumes on aver- values, x∗ does not noticeably increase, but adoption x∗
1+2 1
age 1200MB in each month,with 660MB occurring at peak of Technology 1 does. As c increases at larger values,
WF
hours of the day. As described in the previous section, the moreTechnology1usersalsoadoptTechnology2,i.e.,x∗
1+2
probability of peak-hour WiFi access is 82% of the overall increases and x∗ decreases for all three cities.
1
accessprobability;wetakethisoverallprobabilitytobethe Asthemarginalcostofdeploymentc increases,thecov-
AP
coverage factor η. Thus, each user offloads (0.82η)(660MB) erage factor η∗ decreases, as does the adoption x∗ of Tech-
1
= 541η MB at the peak hours over one month. Multiply- nology 1. For the small city, this decrease in η∗ induces
ing by the fraction of users adopting both technologies and behaviorsimilartothatofFig. 4: asη∗ decreases,adoption
the ¢/MB marginal savings from offloading, we write the x∗ of Technologies (1 + 2) first increases, thendecreases.
1+2
ISP’s monetary savings from offloading as c ηx , with A large η∗ implies that the traffic offloaded η∗x∗ is also
WF 1+2 1+2
c = 5.4, 10.6, or 15.8 for the small, sparse, and dense large, and the resulting congestion induces some users to
WF
cities respectively. adoptonlyTechnology1andleaveTechnology2. Thesame
We next consider the cost of deploying the supplemental effect is observed for the sparse and dense cities, without
technology. We assume that the ISP’s access point (AP) thefinaldecreasein x∗ forlarge c (small η∗). Thus,in
1+2 AP
deploymentineachtypeofcityissuchthatthethroughput citieswithdenserpopulations,adecreaseincoveragedueto
degradation is the same function of the fraction of users higher costs may in fact increase adoption of Technology 2.
on Technology 2’s network (i.e., equal γ values). In more Finally, we examine the adoption behavior at the profit-
2
densely populated cities, the ISP may utilize a denser AP maximizing pricesp∗ andp∗ forfixedcoveragefactor η and
1 2
deployment in order to accommodate the larger number of costparametersc andc . FromFig. 7,weseethatasη
AP WF
users in the sparse and dense cities. These additional APs increases,adoptionx p∗,p∗ ofTechnologies(1+2)first
1+2 1 2
do not increase the geographical coverage area, but rather increases and then decreases. For smaller values of η, the
(cid:0) (cid:1)
accommodate more users within thesame area. ISP induces users to adopt Technology 2 and offload traf-
Wemodelthecostofdeploymentasalinearfunctionofη: fic onto this network. As η grows, however, the ISP allows
c η. WethusmaximizetheISP’stotalprofitatequilibrium x p∗,p∗ todecrease. Sincealargecoveragefactorallows
AP 1+2 1 2
the ISP to charge a high access price p∗ for Technology 2,
p (x +x )+p x +c ηx −c η (15) (cid:0) (cid:1) 2
1 1 1+2 2 1+2 WF 1+2 AP ISPrevenueincreases,offsetingthedecreaseinsavingsfrom
offloading less traffic. We note, however, that this thresh-
with respect to the optimization variables η, p , and p .
1 2
In the remainder of the section, we use η∗, p∗, and p∗ to old η value is largest for the dense city at about 80%, and
1 2
smallest for the small city at about 64%. This observation
denotetheoptimal valuesof η,p ,and p ,respectively;the
1 2
corresponding adoption levels are denoted byx∗ and x∗ . is consistent with the dense city’s larger marginal savings
1 1+2
from offloading.
To find c , we use parameters appropriate for a WiFi
AP
deployment. We assume that each additional AP increases
6. CONCLUSION
thecoveragefactorηbyafixedamount∆ηandcoststheISP
afixedamountCAP permonth. From[20],weestimateCAP In this paper, we develop a model of user adoption for
asamonthlyoperationalcostof$20,pluscapitalinvestment base and supplemental wireless network technologies that
of $1200 spread over 12 months, so that CAP = $120. For accounts for both heterogeneity in users’ technology valu-
simplicity,weinterprettheWiFiaccessprobabilityη asthe ations, congestion effects, and pricing decisions. We show
physical area covered by APs, e.g., in the case of uniform that user adoption converges to a unique, stable equilib-
user mobility. The cost of covering an area η with access rium point, and derive analytical conditions under which
points is then CAP⌈η/∆η⌉ ≈ (CAP/∆η)η. Normalizing by non-intuitiveadoption behaviorsoccur. Wethen show that
theuser population, we findthat these may persist when ISPs maximize either their revenue
orprofit. Toderivearealisticprofitmodel,weuseempirical
C (Marketarea)
cAPη= (APcoverageAParea)(Marketpopulation)η usage data to characterize an ISP’s savings from offloading
traffic to the supplemental network. We find that the pop-
$120 ulation density of the ISP’s market can significantly affect
= η.
(APcoveragearea)(Populationdensity) theequilibrium adoption behavior.
Though we use empirical data to realistically study ISP
We use population densities of 2000, 5000, and 12000 peo-
savings from offloading traffic onto a supplementary net-
ple per square mile for the small, sparse, and dense cities
work, our parameters can only approximate true market
respectively. The AP coverage area is assumed to be 0.01
structures. Similarly, our user adoption model makes ap-
deployed 3G network. proximating assumptions, one of which is that users’ tech-
Small City Sparse City Dense City
60 60 60
%)50 %)50 %)50
els (40 els (40 els (40
ev ev ev Base
n L30 n L30 n L30 Bundle
o o o
pti20 pti20 pti20 Total
o o o
d d d
A10 A10 A10
0 0 0
0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60
c c c
WF WF WF
50 50 50
%)40 %)40 %)40
s ( s ( s (
vel30 vel30 vel30
e e e
L L L
on 20 on 20 on 20
pti pti pti
o o o
d10 d10 d10
A A A
0 0 0
0 5 10 15 20 0 2 4 6 8 2 4 6 8 10 12 14
c c c
AP AP AP
Figure 6: Adoption levels for the profit-maximizing prices and coverage factor (q = 50, q = 100, γ = 25,
1 2 1
γ =50) in different scenarios. Nominal cost parameters are (c ,c )=(5.4,6.2), (10.6,4.9) and (15.9,11.5) for
2 WF AP
the small, sparse, and dense cities respectively. Qualitatively, the dynamics as c and c vary are seen to
AP WF
depend on the user population density.
Small City Sparse City Dense City
50 50 50
%)40 %)40 %)40
s ( s ( s ( Base
el el el
ev30 ev30 ev30 Bundle
L L L
on 20 on 20 on 20 Total
pti pti pti
o o o
d10 d10 d10
A A A
0 0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
η η η
Figure 7: Adoption levelsfor the profit-maximizingprices and fixedcoverage factor (q =50, q =100, γ =25,
1 2 1
γ =50) in different scenarios. Nominal cost parameters are (c ,c )=(5.4,6.2), (10.6,4.9) and (15.9,11.5) for
2 WF AP
the small, sparse, and dense cities respectively. As η increases past a threshold value, adoption x p∗,p∗ of
1+2 1 2
Technologies (1 + 2) decreases despite the potential to offload more traffic as x p∗,p∗ increases.
1+2 1 2 (cid:0) (cid:1)
(cid:0) (cid:1)
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∂x 0 −1
New YorkUniversity,Tech. Rep., 2007. (cid:20) (cid:21)
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q1−q2 q1−q2
scpomecmtruunmicsahtiaorninsgmsachrkeemtse,”seIlEecEtEio/nAiCnMfemTrtaoncesl.lon " q1−−γ1q2 ηγ2−q1(−1−q2η)γ1 −1#
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Theeigenvaluesof thismatrix areλ=ηγ1+γ2 −1, λ=−1.
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Since we assume q < q , both eigenvalues are real and
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negativeandtheequilibriuminregionbisstable. Similarly,
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in region cthe Jacobian has theform
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theeconomics of GPRS networks with Wi-Fi
γ1 − γ1 −1 −ηγ2+(1−η)γ1 − (1−η)γ1
integration,”European Journal of Operational q1−q2 q1 q1−q2 q1 ,
Research, vol. 187, no. 3, pp.1459–1475, 2008. " q1−−γ1q2 ηγ2−q1(−1−q2η)γ1 −1 #