Economic Analysis of ework Upgrade Lingjie Duan, Jianwei Huang, and Jean Walrand Absrac As he successor o he sandard, provides much higher daa raes o address cellular users ever-increasing demands for high-speed mulimedia communicaions. This paper analyzes he cellular operaors iming of nework upgrades and models ha users can swich operaors and services. Being he firs o upgrade o service, an operaor increases his marke share bu akes more risk or upgrade cos because echnology maures over ime. This paper firs sudies a monopoly marke wih one dominan operaor and some small operaors, where he monopolis decides his upgrade ime by rading off increased marke share and upgrade cos. The paper also considers a compeiion marke and develops a game heoreic model for sudying operaors ineracions. The analysis shows ha operaors selec differen upgrade imes o avoid severe compeiion. One operaor akes he lead o upgrade, using he benefi of a larger marke share o compensae for he larger cos of an early upgrade. This resul maches well wih many indusry observaions of asymmeric upgrades. The paper furher shows ha he availabiliy of upgrade may decrease boh operaors profis due o increased compeiion. Perhaps surprisingly, he profis can increase wih he upgrade cos. I. ITRODUCTIO The hird generaion of cellular wireless neworks was launched during he las decade. I has provided users wih high-qualiy voice channels and moderae daa raes up o Mbps. However, service canno seamlessly inegrae he exising wireless echnologies e.g., GSM, wireless LA, and Blueooh [], and canno saisfy users fas growing needs for high daa raes. Thus, mos major cellular operaors worldwide plan o deploy he fourh-generaion neworks o provide much higher daa raes up o hundreds of megabis per second and inegrae heerogeneous wireless echnologies. The echnology is expeced o suppor new services such as high-qualiy video cha and video conferencing [3]. One may expec compeiive operaors in he same cellular marke o upgrade o a service a abou he same ime. However, many indusry examples show ha symmeric upgrades do no happen in pracice, even when muliple op- L. Duan is wih he Engineering Sysems and Design Pillar, Singapore Universiy of Technology and Design email: swing.dlj@gmail.com. J. Huang is wih he ework Communicaions and Economics Lab, Deparmen of Informaion Engineering, The Chinese Universiy of Hong Kong email: jwhuang@ie.cuhk.edu.hk. J. Walrand is wih he Deparmen of Elecrical Engineering and Compuer Sciences, Universiy of California a Berkeley, California- 947 email: wlr@eecs.berkeley.edu. This work is suppored by SUTD-MIT Inernaional Design Cener Gran Projec no.: IDSF6OH and SUTD Sar-up Research Gran Projec no.: SRG ESD 4. This work is also suppored by he General Research Funds Projec umber CUHK 47 and CUHK 45 esablished under he Universiy Gran Commiee of he Hong Kong Special Adminisraive Region, China, and he SF-eSE grans 438 and 97. eraors have obained he necessary specrum and echnology paens for upgrade [3], [4]. In Souh Korea, for example, Korean Telecom ook he lead o deploy he world s firs nework using WiMAX echnology in 6, whereas SK Telecom sared o upgrade using more maure LTE echnology in. In US, Sprin deployed he firs WiMAX nework in lae 8, Verizon waied unil he end of o deploy his LTE nework, and AT&T planed o deploy his LTE nework a he end of [3]. In China, China Mobile and China Unicom are he wo dominan cellular operaors, and China Mobile has decided o firs deploy LTE nework during -3 [4]. Thus, he key quesion we wan o answer in his paper is he following: How do he cellular operaors decide he iming o upgrade o neworks? In his paper, we analyze he iming of operaors upgrades in differen models, including boh a monopoly marke and a compeiion marke. Operaors need o pay he cos of upgrade, which decreases over ime as echnology maures. There are wo key facors ha affec he operaors upgrade decisions: namely, upgrade cos and user swiching cos. An exising user can swich o he service of he same operaor or of a differen operaor, depending on how large he swiching cos is. In a monopoly marke where only a dominan operaor can choose o upgrade o, his operaor can use he service o capure a larger marke share from small operaors. In a compeiion marke where muliple operaors can choose o upgrade, we analyze he operaors ineracions as a non-cooperaive game. We sudy how he users iner-nework swiching cos affecs he operaors upgrade decisions, and our findings are consisen wih he asymmeric upgrades observed in he indusry. Our key resuls and conribuions are as follows. A revenue-sharing model beween operaors: Mos exising works only sudy a single nework s revenue by exploring he nework effec e.g., [7], and he resuls may no apply in a compeiive marke. In Secion III, we sudy wo inerconneced neworks, where heir operaors share he revenue of he iner-nework raffic. Monopolis s opimal iming of upgrade: By upgrading early, he monopolis in Secion IV obains a large marke share and a large revenue because of s Qualiy of Service QoS improvemen, bu i canno benefi from he cos depreciaion over ime. When he upgrade cos is relaively low, he upgrades a he earlies available ime; oherwise he pospones his upgrade. Compeiive operaors asymmeric upgrades: In Secions V and VI, users can swich operaors. By upgrading early, an operaor capures a large marke share and
he s QoS improvemen which can compensae for he large upgrade cos. The oher operaor, however, pospones his upgrade o avoid severe compeiion and benefis from cos reducion. The availabiliy of upgrade may decrease boh operaors profis because of he increased compeiion, and paradoxically, heir profis may increase wih he upgrade cos. II. RELATED WORK A. ework Effec and ework Value In elecommunicaions, he nework effec is he added value ha a user derives from he presence of oher users [3]. In a nework wih users, each user perceives a value ha increases wih. A reasonable model was suggesed by Briscoe e al. [7], where each user perceives a value of order log. In ha model, a user ranks he oher users in decreasing order of imporance and assigns a value /k o he k-h user in ha order, for a oal value +/+ +/ log. The resuling oal nework value is log, which is appropriae for cellular neworks shown by quaniaive sudies [7]. B. ework Upgrade Recenly, here has been a growing ineres in sudying he economics of nework upgrades [8] []. Musacchio e al. [8] sudied he upgrade iming game beween wo inerconneced Inerne Service Providers ISPs, where one ISP s archiecure upgrade also benefis he oher because of he nework effec. This free-riding effec may make he second operaor pospone his upgrade or even never decide o upgrade. Jiang e al. [9] sudied a nework securiy game, where one user s invesmen upgrade can reduce he propagaion of compuer viruses o all users. In our problem, however, one operaor may benefi from he oher s upgrade only when he also upgrades, leing his users communicae wih exising users in oher neworks. Moreover, our model characerizes he dynamics of users swiching beween operaors and/or services. These dynamics imply ha an operaor can obain a larger marke share by upgrading earlier, and his weakens he free-riding effec. Sen e al. [] sudied he users adopion and diffusion of a new nework echnology in he presence of an incumben echnology. Our work is differen from ha sudy in ha we are no focusing on echnology compeiion o arac users, bu on he operaors compeiion in upgrade iming o obain greaer profis. Moreover, he swiching cos is no considered in [], whereas i is an imporan parameer of our model. A. Value of Cellular eworks III. SYSTEM MODEL In his paper, we adop he log Law, where he nework value wih users is proporional o log. The operaor of a cellular nework prefers a large nework value; his is because he revenue he obains by charging users can be proporional o he nework value. oice ha he value of a nework is larger han a nework even when wo neworks have he same number of users. This is because he communicaion beween wo users is more efficien and more frequen han beween wo users. Because he average daa rae in he service is 5- imes faser han he boh downlink and uplink, a nework can suppor many new applicaions. We denoe he efficiency raio beween and services as γ,. Tha is, by serving all his users via QoS-guaraneed raher han services, an operaor obains a larger normalized revenue log insead of γ log. oe ha his resul holds for a single operaor s nework ha is no conneced o oher neworks. ex we discuss he revenues of muliple operaors whose neworks e.g., wo neworks are inerconneced. For he purpose of illusraion, we consider wo neworks ha conain and users, respecively. The whole marke covers = + users. We assume ha wo operaors and laer services are equally good o users, and he efficiency raio γ is he same for boh operaors. The raffic beween wo users can be inra-nework when boh users belong o he same operaor or iner-nework when wo users belong o differen operaors, and he revenue calculaions in he wo cases are differen. We assume ha he user who originaes he communicaion session irrespecive of wheher he same nework or o he oher nework pays for he communicaion. This is moivaed by he indusry observaions in EU and many Asian counries. Before analyzing each operaor s revenue, we firs inroduce wo pracical conceps in cellular marke: erminaion rae and user ignorance. When wo users of he same operaor communicae wih each oher, he calling user only pays operaor. Bu when an operaor s user calls an operaor s user, operaor charges a erminaion rae for he incoming call []. 3 We denoe he wo operaors revenue-sharing porion per iner-nework call as η, where he value of η, depends on he agreemen beween he wo operaors or on governmens regulaion on erminaion rae. User ignorance is a unique problem in he wireless cellular nework, where users are ofen no able o idenify which specific nework hey are calling. Mobile number porabiliy furher exacerbaes his problem [4]. Thus a ypical user s evaluaion of wo inerconneced neworks does no depend on which nework he belongs o, and equals γ log where = +. We assume a call from any user erminaes a a user in nework i {, } wih a probabiliy of i / as in [4]. The operaors revenues when hey are boh providing services are given in Lemma. Lemma : When operaors and provide services, heir revenues are γ log and γ log, respecively. The proof of Lemma is given in our online echnical repor We assume ha an operaor s operaional cos proporional o nework value has been deduced already, and hus he revenue in his paper represens a normalized one. Our model can also be exended o he case where boh involved users in a communicaion session pay for heir communicaion. This is wha happening in US cellular marke. 3 In he US, erminaion rae follows Bill and Keep and is low. Then operaor can keep mos of he calling user s paymen. In EU, however, erminaion rae follows Calling Pary Pays and is much higher. Then mos of he calling user s paymen o operaor is used o compensae for he erminaion rae charged by operaor [].
[]. Boh operaors revenues are linear in heir numbers of users or marke share, and are independen of he sharing porion η of he iner-nework revenue. Inuiively, he inernework raffic beween wo neworks is bidirecional: when a user originaes a call from nework o anoher user in nework, his iner-nework raffic generaes a fracion η of corresponding revenue o operaor ; when he oher user calls back from nework o nework, he generaes a fracion η of he same amoun of revenue o operaor. Thus an operaor s oal revenue is independen of η. Laer, in Secion IV, we show ha such independence on η also applies when he wo operaors boh provide services or provide mixed and services. B. User Churn during Upgrade from o Services When service becomes available in he marke offered by one or boh neworks, he exising users have an incenive o swich o he new service o experience a beer QoS. Such user churn does no happen simulaneously for all users, his is because differen users have differen sensiiviies o qualiy improvemens and swiching coss []. We use wo parameers λ and α o model he user churn wihin and beween operaors: Inra-nework user churn: If an operaor provides in addiion o his exising service, his users need o buy new mobile phones o use he service. The users also spend ime o learn how o use he service on heir new phones. We use λ o denoe he users swiching rae o he service wihin he same nework. Iner-nework user churn: If a user wans o swich o anoher nework s service, he eiher wais ill his curren conrac expires, or pays for he penaly of immediae conrac erminaion. This means ha inernework user churn incurs an addiional cos on op of he mobile device updae, and hus he swiching rae will be smaller han he inra-nework user churn. We use αλ o denoe he users iner-nework swiching rae o service, where α, reflecs he ransacion cos of swiching operaors. We illusrae he process of user churn hrough a coninuous ime model. The saring ime = denoes he ime when he specrum resource and he echnology are available for a leas one operaor see Secion IV for monopoly marke and Secion VI for compeiion marke. We also assume ha he porion of users swiching o he service follows he exponenial disribuion a rae λ for inra-nework churn and αλ for iner-nework churn. As an example, assume ha operaor inroduces a service a ime = T while operaor decides no o upgrade. The numbers of operaor s users and users a any ime are and, respecively. The number of operaor s users a ime is. As ime increases from T, users in boh neworks sar o churn o service, and = e λ max T,, = e αλ max T,, Operaor s users: Operaor s users: T Operaor s users: Fig.. The numbers of users in he operaors differen services as funcions of ime. Here, operaor upgrades a T and operaor does no upgrade. and operaor s service gains an increasing marke share, = e λ max T, e αλ max T,. We illusrae and in Fig.. We can see ha operaor s early upgrade aracs users from his compeior and increases his marke share. oice ha increases wih α, hus operaor capures a large marke share when α is large i.e., he swiching cos is low. C. Operaors Revenues and Upgrade Coss Because of he ime discoun, an operaor values he curren revenue more han he same amoun of revenue in he fuure. We denoe he discoun rae over ime as S, and he discoun facor is hus e S a ime according o [6]. We approximae one operaor s upgrade cos as a oneime invesmen. This is a pracical approximaion, as an operaor s iniial invesmen of wireless specrum and infrasrucure can be much higher han he mainenance coss in he fuure. For example, specrum is a very scarce resource ha is allocaed aucioned infrequenly by governmen agencies. Thus an operaor canno obain addiional specrum frequenly afer his upgrade. To ensure a good iniial coverage, an operaor also needs o updae many base saions o cover a leas a whole ciy all a once. Oherwise, users would be unhappy wih he service, and his would damage he operaor s repuaion. Tha is why Sprin and Verizon covered many markes in heir iniial launch of heir services [8]. More specifically, we denoe he upgrade cos a = as K, which discouns over ime a a rae U. Thus if an operaor upgrades a ime, he needs o pay an upgrade cos Ke U according o [6]. We should poin ou ha he upgrade cos decreases faser han he normal discoun rae i.e., U > S. This happens because he upgrade cos decreases because of boh echnology improvemen and ime discoun. Very ofen he advance of echnology is he dominan facor in deermining U, and his is discussed furher in Secion VI. Based on hese discussions on revenue and upgrade cos, we define an operaor s profi as he difference beween his revenue in he long run and he one-ime upgrade cos. Wihou loss of generaliy, we will normalize an operaor s revenue rae a any ime, oal revenue, and upgrade cos by log, where is he oal number of users in he marke. 4 4 Our model and analyical resuls laer can be exended o he case where increases over ime. As new users prefer service o service, and can easily swich a rae λ, operaors will have more incenives o upgrade earlier.
IV. MOOPOLY MARKET We firs look a he case where only operaor can choose o upgrade from o, while he oher operaors one or more always offer he service because of he lack of financial resources or he necessary echnology. This can be a reasonable model, for example, for counries such as Mexico and some Lain American ones, where America Movil is he dominan cellular operaor in he marke. As he world s fourh-larges cellular nework operaor, America Movil has he advanage over oher small local operaors in winning addiional specrum via aucions and obaining LTE paens, and he is expeced o be he monopolis in ha area [5]. The key quesion in his secion is how operaor should choose his upgrade ime T from he service o he service. T = means ha operaor upgrades a he earlies ime ha he specrum and echnology are available, and T > means ha operaor chooses o upgrade laer o ake advanage of he reducion in he upgrade cos. Because of user churn from he o he service, he operaors marke shares and revenue raes change afer ime T. For ha reason, we analyze periods T and > T separaely. Before upgrade T : Operaor s and oher operaors marke shares do no change over ime. Operaor s revenue rae a ime is π = γ, which is independen of ime. His revenue during his ime period is π, T = T π e S d = γ S e ST. 3 Afer upgrade > T : Operaor s marke share increases over ime, and he oher operaors oal marke share denoed by / decreases over ime. We denoe operaor s numbers of users and users as and, respecively, and we have + + =. This implies ha = e αλ T, = e λ T, and = e λ T e αλ T. oe ha a user s communicaion wih a or a user is sill based on he sandard, and only he communicaion beween wo users can achieve a high sandard QoS. Operaor s revenue rae is γ π = + + γ γ, + which is independen of he revenue sharing raio η beween he calling pary and receiving pary. Operaor s revenue during his ime period is hen π,>t = T π e S d, 4 where is an approximaion of he long-erm service provision e.g., one decade before he emergence of he nex generaion sandard. This approximaion is reasonable since he revenue in he disan fuure becomes less imporan because of discoun. Figure illusraes how he numbers of users of operaors differen services change over ime. Before operaor s upgrade e.g., T in Fig., he number of oal users in each nework does no change; 5 afer operaor s upgrade, operaor s and he oher operaors users swich o he new service a raes λ and αλ, respecively. By considering 3, 4, and he decreasing cos Ke UT, operaor s long-erm profi when choosing an upgrade ime T is π T = π, T + π,>t Ke UT = e ST S + γ + γ λ + S αλ + S e ST γ λ + S + γ Ke UT αλ + S K mono h = γ + e ST γ + αλ + S + γ S. 5 e ST We can show ha π T in 5 is sricly concave in T, hus we can compue he opimal upgrade ime T by solving he firs-order condiion. The opimal upgrade ime depends on he following upgrade cos hreshold in he monopoly marke, λ+s + αλ+s λ+s + +αλ+s U/S S αλ+s + γ γ. 6 U Theorem : Operaor s opimal upgrade ime in a monopoly marke is: Low cos regime upgrade cos K Kh mono : operaor upgrades a T =. High cos regime K > Kh mono : operaor upgrades a T = K U S log Kh mono >. 7 Inuiively, an early upgrade gives operaor a larger marke share and enables him o ge a higher revenue via he more efficien service. Such advanage is especially obvious in he low cos regime where he upgrade cos K is small. ex we focus on he high cos regime, and explore how he nework parameers affec operaor s upgrade ime. Observaion : Operaor s opimal upgrade ime T increases wih he upgrade cos K, and decreases wih α i.e., increases wih he users iner-nework swiching cos. The proofs of Observaion and he following observaions are given in our online echnical repor []. Observaion Figure : When h < K < U S Kmono h, T firs increases and hen decreases in U. When K U S Kmono h, T monoonically decreases in U. Figure shows T as a funcion of U and K. When K is large or U is large, operaor wans o pospone his upgrade unil he upgrade cos decreases significanly. A larger U hus K mono 5 I is possible for users o swich beween neworks in he same service class, bu such minor swiching does no change he average marke shares.
Opimal upgrade decision T.6.4..8.6.4. K=.4 K=.8 K=..5.5 3 3.5 4 Discouning rae of upgrade cos U Fig.. Operaor s opimal upgrade ime T as a funcion of cos K and he discoun rae U of cos. Oher parameers are =, = 5, S =, λ =, and α =.5. Opimal upgrade ime T.5.4.3...9.8 /=3% /=5% /=7%..4.6.8 Efficiency raio beween and services γ Fig. 3. Operaor s opimal upgrade ime T as a funcion of he efficiency raio beween and services γ and his original marke share. Oher parameers are =, K =, S =, λ =, and U =. a faser cos-decreasing rae srenghens his willingness o pospone. When K is small and U is small, he upgrade cos is small and does no decrease fas enough. In his case, operaor chooses o upgrade early if he revenue increase can compensae for he small upgrade cos. Observaion 3 Figure 3: When operaor s original marke share is large, T increases in he efficiency raio γ beween and. When is small, T decreases in γ. Figure 3 shows T as a funcion of γ and. When is large, operaor canno arac many users from oher operaors and has smaller incenives o upgrade. As γ increases and hus he QoS gap beween and shrinks, he is less ineresed in service and hus pospones his upgrade. When is small, operaor has limied marke share and can arac many more users by upgrading early. As γ increases, operaor obains a higher revenue beween his exising users and he increasing number of users. Operaor, however, does no benefi much from his, as he loses his marke share due o operaor s services. Thus T decreases in γ. V. COMPETITIO MARKET: DUOPOLY MODEL AD GAME FORMULATIO In his secion, we focus on he compeiion beween muliple operaors who can choose o upgrade o services. To make he analysis racable and o derive clear engineering insighs, we focus on he case of wo operaors duopoly in his paper. This analysis serves as he firs sep in undersanding he more general oligopoly case. This duopoly model is reasonable in a counry like China, where China Mobile and China Unicom are he wo dominan cellular operaors in he marke. A similar siuaion exiss in several oher Asian and European counries as well. The focus of his and he following secion is o undersand why in so many exising indusry examples e.g., [3] [5] operaors choose o upgrade o services a differen imes even hough hey have he resources o upgrade simulaneously. In paricular, we examine wheher such asymmeric upgrades emerge even when he wo operaors are similar e.g., having similar marke shares before upgrades e.g., Verizon has 6.3 million users and AT&T has 98.6 million users in he US. In our online echnical repor [], we also examine he case where neworks are heerogeneous in naure, and we show ha asymmeric operaors e.g., wih differen marke shares have more incenives o upgrade a differen imes. Thus in he following analysis we can consider wo operaors ha have he same marke shares before he upgrades =, he same upgrade cos K, and he same cos discoun rae U. We will firs derive he operaors profis under any upgrade decisions, and hen analyze he duopoly game where each operaor chooses he bes upgrade ime o maximize his profi. A. Operaors Long-erm Profis Le us denoe wo operaors upgrade imes as T and T, respecively. Because he wo operaors are symmeric, wihou loss of generaliy, we assume in he following example before Lemma ha operaor upgrades no laer han operaor i.e., T T. To calculae he operaors profis, we firs need o undersand how users churn from o services, and how his affecs he operaors revenue raes over ime. Figure 4 shows ha user churn is differen in hree phases, depending on how many operaors have upgraded. Phase I T : o operaor has upgraded and boh operaors marke shares do no change. Phase II T < T : Operaor has upgraded o service bu operaor has no. The users of wo operaors swich o operaor s service a differen raes. The numbers of users in he operaors differen services are = e λ T, = e αλ T, and = e λ T e αλ T. Phase III > T : Boh operaors have upgraded, and users only swich o he service of heir curren operaor. The numbers of users in operaors differen services are = e λ T, = e λ T e αλt T, = eαλt T λ T,
- - - T T Phase I - - - T T Phase II - - - T T Phase III Fig. 4. Users swiches over he operaors services: Phase I wih services only, Phase II wih operaor s service, and Phase III wih boh operaors services. Operaor s users: Operaor s users: T Operaor s users: Operaor s users: T Fig. 5. The numbers of users in he operaors differen services as funcions of ime. Here, operaor chooses his upgrade ime a T and operaor chooses T wih T T. and = eαλt T e λ T. Figure 5 summarizes how users churn in he hree phases. Similar o Secion IV, we can derive operaors revenue raes based on users churn over ime. By inegraing each operaor s revenue rae over all hree phases, we obain ha operaor s long-erm revenue. Recall ha an operaor s profi is he difference beween his revenue and he one-ime upgrade cos. By furher considering he symmeric case of T T, we have he following resul. Lemma : Consider wo operaors i, j {, } wih i j upgrading a T i and T { j. Operaor i s long-erm profi is π ER T i, T j, if T i T j ; π i T i, T j = π LT 8 T j, T i, if T i T j, where π ER T i, T j and π LT T j, T i are given in 9 and, respecively. oe ha an operaor s profi π i T i, T j is coninuous in his upgrade ime T i. When operaor i s upgrade ime T i is less han T j, he increases his marke share a rae αλ during he ime period from T i o T j ; bu when T i > T j, operaor i loses his marke share a rae αλ during he period from T j o T i. This explains why we need wo differen funcions π ER T i, T j and π LT T j, T i o compleely characerize he long-erm profi for each operaor. B. Duopoly Upgrade Game ex we consider he non-cooperaive game heoreical ineracions beween wo operaors, where each of hem seeks o maximize his long-erm profi by choosing he bes upgrade ime. Upgrade Game: We model he compeiion beween wo operaors as follows: Players: Operaors and. Sraegy spaces: Operaor i {, } can choose upgrade ime T i from he feasible se T i = [, ]. 6 Payoff funcions: Operaor i {, } wans o maximize his profi π i T i, T j defined in 8. oice ha we consider a saic game here, where boh operaors decide when o upgrade a he beginning of ime. This is moivaed by he fac ha operaors usually make long-erm decisions in pracice raher han changing decisions frequenly, as many upgrade operaions e.g., financial budge and echnological rials need o be planned and prepared. As we consider ha each operaor has complee informaion abou his compeior s and users parameers, he will no deviae from his iniial decision as ime goes on. Also, i should be poined ou ha in many compeiion markes operaors can obain available resource for upgrade a a similar ime as we menioned a he beginning of his secion. Afer making decision a =, one operaor does no change his decision laer on. This is reasonable when operaors can predic he fuure marke adopion. 7 In Secion VI, we analyze he duopoly upgrade game under differen swiching coss i.e., he value of α,. ash equilibrium is a commonly used soluion concep for a saic game. A a ash equilibrium, no player can increase his payoff by deviaing unilaerally [9]. We are ineresed in characerizing he condiions under which an asymmeric upgrade equilibrium emerges beween symmeric operaors. VI. COMPETITIO MARKET: PRACTICAL ITER-ETWORK SWITCHIG RATE In his secion, we consider he case of α >, i.e., users may swich o he service of a differen operaor. The equilibrium analysis in his general case depends on he relaionship beween U upgrade cos discoun rae and S money depreciaion rae. We assume ha U is much larger han S, i.e., U > S + αλ. This represens he pracical case where he advance of echnology is he dominan facor in deermining U, and no many users choose o swich operaors when he service is jus deployed i.e., small α [7]. For example, Sprin deployed he firs nework in US by using WiMAX echnology in 8 when LTE echnology 6 oe ha T i = means ha operaor i never upgrades. 7 Operaors can predic he fuure marke adopion by exploring hisorical records of he marke and some rial of deploymen. In he fuure we will sudy he incomplee informaion case, where an operaor may learn more informaion as he ime goes and revise his upgrade decision if he has no upgraded ye.
π ER T i, T j = γ + γe ST i e αλt j T i ST j S + γ e ST i + e +αλt j T i ST j λ + S γ λ + S e αλt j T i ST j π LT T j, T i = e αλt i T j ST i + γe ST i e αλt j T i ST j αλ + S + e ST i e +αλt j T i ST j + αλ + S e λt j T i + e αλt j T i γ e ST i λ + S S γ λ + S γ e λt i T j + e αλt i T j λ + S λ + S + e ST i e αλt j T i ST j αλ + S Ke UT i. 9 + γ S e ST j γ αλ + S e ST j e αλt i T j ST i Ke UT i. was no maure ye. Only wo years laer, in, LTE could already offer a much lower cos per bi han WiMAX [3]. From, LTE is expeced o be he leading echnology choice for neworks. This example moivaes ha U is much larger han S, and we will sudy wheher he operaors symmeric upgrades happen in his scenario. Recall ha by upgrading a T and T, he operaors receive he profis given in 8. In game heory, one operaor s bes response funcion is his upgrade ime ha achieves he larges long-erm profi, as a funcion of a fixed upgrade ime of he oher operaor [9]. A fixed poin of he wo operaors bes response funcions is he ash equilibrium, and in general here can be more han one such fixed poin. We can show ha he operaors bes responses funcions T bes T and T bes T depend on he upgrade cos K, and in paricular, hey depend on wo cos hresholds K comp h < K comp h ha lead o hree cos regimes: low, medium, and high. When he upgrade cos K is less han he firs hreshold i.e., low cos regime, boh operaors will upgrade a = o maximize he revenue from he service. By solving π LT, T i Ti= =, i {, }, T i K comp h we have K comp h = αλ γ λ + S + αλ S γ λ + 3αλ + S λ + S λ + S U. When he upgrade cos K is larger han K comp h i.e., medium or high cos regimes, a leas one operaor pospones his upgrade unil he upgrade cos decreases sufficienly. In paricular, when K is larger han he second hreshold K comp h i.e., high cos regime, boh operaors pospone heir upgrades. When operaor i {, } upgrades a =, operaor j i pospones his upgrade o Tj bes, which is he unique soluion o π LT, T j T Tj=Tj bes =. j The hreshold K comp h can be obained by solving π ER T i, Tj bes Ti= =. 3 T i Operaor s upgrade ime T.5.4.3.. K=.6: E K=.58: E K=.58: E K=.6: E K=.58: T bes T K=.58: T bes T K=.6: T bes T K=.6: T bes T...3.4.5 Operaor s upgrade ime T Fig. 6. Two operaors bes upgrade responses o each oher according o differen cos values in he medium cos regime. Oher parameers are α =.5, =, U =, S =, λ = and γ =.5 such ha U > S + αλ. ex we illusrae numerically how he wo operaors bes response funcions T bes T and T bes T change wih he upgrade cos K. Figure 6 shows ha each operaor s bes response funcion is disconinuous in he medium cos regime, and he wo bes response funcions wih he same value of K inersec a wo poins: = T < T and symmerically = T < T. To illusrae his siuaion, consider operaor s bes response T bes T in he case K =.6. If operaor upgrades early such ha T is less han.5, operaor does no upgrade a he same ime o avoid a severe compeiion. If operaor upgrades laer such ha T is larger han.5, operaor chooses o upgrade earlier han operaor o increase his marke share. Thus T bes T is disconinuous a T =.5. Figure 7 shows ha each operaor s bes response funcion is disconinuous in he high cos regime, and he wo funcions wih he same value of K inersec a wo poins, equilibria < T < T and symmerically < T < T. Unlike Fig. 6, he high cos here prevens any operaor from choosing he upgrade ime =. Figure 8 summarizes how operaors upgrade equilibrium changes as cos K increases: saring wih Ti = Tj = in low cos regime, hen = Ti < Tj wih increasing T j in medium cos regime, and finally < Ti < Tj wih increasing T and T in high cos regime. In he following heorem, we prove ha he operaors do no choose symmeric upgrades as long as he cos is no low. Theorem : The wo operaors upgrade equilibria sa-
Operaor s upgrade ime T.7.6.5.4.3. K=.7: E. K=.7: E K=.87: E K=.87: E K=.7:T bes T K=.7:T bes T K=.87:T bes T K=.87:T bes T...3.4.5.6.7 Operaor s upgrade ime T Operaor s equilibrium profi π.56.54.5.5.48.46.44.4 E : T T E : T T Medium cos regime: π <π as K High cos regime: π <π as K Low cos regime: π =π as K.4.45.5.55 Operaor s equilibrium profi π Fig. 7. Two operaors bes upgrade responses o each oher according o differen cos values in he high cos regime. Oher parameers are α =.5, =, U =, S =, λ = and γ =.5 such ha U > S + αλ. Fig. 9. Two operaors equilibrium profis π,π change as cos K increases under large γ =.5. Oher parameers are α =.5, =, U =, S =, and λ = such ha U > S + αλ. Operaor s equilibrium ime T.3.5..5..5 High cos regime: <T <T : as K Medium cos regime: =T <T as K E : T T E : T T Low cos regime:.5..5..5.3 =T =T as K Operaor s equilibrium ime T Fig. 8. Two operaors equilibrium T,T changes as cos K increases. Oher parameers are α =.5, =, U =, S =, λ =, and γ =.5 such ha U > S + αλ. isfy he following properies: Low cos regime K K comp h : Boh operaors upgrade a T = T =. Medium cos regime K comp h < K K comp h : Operaors do no upgrade a he same ime, and only one operaor may upgrade a =. The possible equilibria can only be T < T and symmerically T < T. High cos regime K > K comp h : Operaors do no upgrade a he same ime, and none of hem upgrade a =. The possible equilibria can only be < T < T and symmerically < T < T. The proof of Theorem is given in our online repor []. To undersand he inuiion behind he asymmeric srucure, we summarize he advanages of earlier and laer upgrades wih α > as follows: Earlier upgrade gives an operaor he advanage o arac more users from he oher operaor, and enables he operaor o collec a higher revenue from he service. Laer upgrade allows an operaor o incur a reduced upgrade cos and o ake advanage of he nework effec in he marke wih more exising users when he upgrades. In order o fully enjoy he wo advanages of earlier or laer upgrades, operaors will avoid symmeric upgrade. If one operaor upgrades much earlier o capure a larger marke share ha can compensae for a large upgrade cos, he second operaor will no upgrade a he same ime o avoid severe compeiion in marke share; insead, he second operaor will wai unil his loss of users and revenue is compensaed by he reducion of upgrade cos wih U > S + αλ. ex we sudy how operaors equilibrium profis change wih cos K and he economic efficiency raio γ beween and services in he hree cos regimes. oe ha an operaor may or may no be able o charge a significanly higher price from a user hough does improve a lo over in QoS. Figures 9 and show operaors equilibrium profis under large and small γ values, respecively. We firs sudy he large γ scenario in Fig. 9 which means an operaor o charge a slighly higher price in service han. Wihou loss of generaliy, we focus on he case where operaor upgrades no laer han operaor i.e., T T. In he low cos regime, by upgrading a T = T =, he wo operaors profis are he same and decrease wih cos K. In he medium cos regime, Fig. 9 shows ha operaor receives a larger profi han operaor by upgrading a T =. Perhaps surprisingly, his profi increases wih K, whereas operaor s profi decreases wih K. Inuiively, he increase of K encourages operaor o furher pospone his upgrade and lose more users o operaor. The change of operaor s profi rades off he increases of his marke share and upgrade cos. As operaor s marke share increases, his growing users communicae more wih his users via he efficien service under large γ. Operaor s revenue increases because of a more efficien inra-nework raffic, which helps compensae for he upgrade cos. In he high cos regime, Fig. 9 shows ha boh operaors have o pospone heir upgrades and, surprisingly, boh operaors profis increase wih K. As K increases, operaor furher pospones his upgrade and operaor
Operaor s equilibrium profi π.3.5..5 E : T T E : T T High cos regime: π <π as K Low cos regime: π =π as K Medium cos regime: π <π as K...5..5.3 Operaor s equilibrium profi π selec asymmeric upgrade imes o avoid severe compeiion. Compared o duopoly, he operaor who is he las o upgrade loses more marke share o he ohers and bu enjoys he smalles upgrade cos and he larges nework effec. I is also ineresing o sudy operaors usage of plan announcemens before acual upgrades. One operaor who acually decides o upgrades laer han wha he announces can preven some of his users swiching o oher neworks. We can use a signaling game o sudy operaors announcemens. Moreover, we can sudy operaors mixed sraegies in upgrades, where one operaor chooses a probabiliy disribuion of upgrading a differen imes. Alhough heoreically ineresing, mixed sraegies are hard o implemen in pracice [9]. Fig.. Two operaors equilibrium profis π,π change as cos K increases under small γ =.. Oher parameers are α =.5, =, U =, S =, and λ = such ha U > S + αλ. s marke share decreases more slowly. Thus operaors compeiion in he marke share is posponed, and under large γ operaor can obain more revenue before operaor s upgrade. Operaor, on he oher hand, also benefis from his posponemen o decrease his upgrade cos. Since operaor also pospones his upgrade, operaor can sill capure a large marke share even hough he upgrades laer. As K, no operaor upgrades and operaors profis approach he symmeric profis. Under large γ, he service is no much beer han and he availabiliy of upgrade only inensifies operaors compeiion. Compared o radiional scenario, boh operaors profis decrease when he upgrade cos is high. In oher words, boh operaors will be beer off if echnology is no available in his case. Figure shows how operaors profis change wih K under small γ. The resuls are more inuiive; his is because he operaors profis decrease wih K in all hree cos regimes. Under small γ, he availabiliy of he upgrade significanly improves he revenue in each nework. A larger K reduces he benefi of upgrades. However, he operaors profis will no be smaller afer he upgrade under any value of K. VII. COCLUSIOS AD FUTURE WORK This paper presens he firs analyical sudy of operaors upgrade decisions. We firs analyze a monopoly marke, where he monopolis s opimal upgrade ime rades off an increased marke share and he decreasing upgrade cos. We hen develop a non-cooperaive game model o sudy he compeiion beween operaors. Our resuls show ha operaors selec differen upgrade imes o avoid severe compeiion in marke share. We furher show ha he availabiliy of upgrade may decrease boh operaors profis due o heir compeiion, and heir profis may increase wih he upgrade cos. There are some possible ways o exend he resuls in his paper. For example, we could consider an oligopoly marke wih more han wo compeiive operaors. 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