Investment and Pricing with Spectrum Uncertainty: A Cognitive Operators Perspective

1 In v estment and Pr icing with Spectr um Uncer tainty: A Cognitiv e Oper ator’ s P erspectiv e Lingjie Duan, Student Member , IEEE, Jianwei Huang, Member , IEEE, and Biying Shou Abstract —This paper studies the optimal inv estment and pr icing decisions of a cognitive mobile vir tual network operator (C-MVNO) under spectrum supply uncer tainty . Compared with a tr aditional MVNO who oft en leases spectrum via long-term cont racts, a C-MV N O can acquire spect r um dynamically in shor t-term by both sensing the empt y “spectrum holes” of licensed bands and dynamically leasing from the spectrum owner . As a result, a C-MVNO can make fle xible inv estment a nd pricing decisions to match the current demands of the secondary unlicensed users. Compared to dynamic spectrum leasing, spectrum sensing is typically cheaper , but the obtained useful spectrum amount is r and om due to primar y licensed users’ stochastic traffic. The C-MVNO needs to determine the optimal amounts of spectru m sensing and leasing by ev aluating the trade off be tween cost a nd uncer tainty . The C-MVNO also needs to determine the optimal pri ce to sell the spectrum to the secondar y unli censed users, taking into account wireless heterogeneity of users such as different maximum transmission power le vels and channel gains. We model and anal yze the interactions between the C-MVNO and secondar y unl icensed users as a Stac kelberg game. We show se veral interesting proper ties of the network equilibr ium, including thres ho ld st r uctures of the optimal in vestment and pricing decisions, the independence of the optimal price on users’ wireless characteristics , and guaranteed fair and predictable QoS among users. We pro ve that these proper ties hold for general SNR regime and general continuous distributions of sensing uncer tainty . W e show that spectrum sensing can significantly i mprov e the C-MVNO’ s ex p ected profit and users’ pay offs. Index T erms —Cognitive radio , spectrum trading, spectrum sensing, dynamic spectrum leasing , spectrum pricing, Stac kelberg game , Subgame P erfect equilibrium. ✦ 1 I N T R O D U C T I O N W I R E L E S S spectrum is typically considered as a scarce resource, and is tr aditionally allocated through static licensing. Field mea surements show that, however , most spectrum bands a re often under-utilized even in densely popula te d ur ban areas ( [2]). T o achieve more efficient spectrum utilization, people ha v e pro- posed various dynamic spectrum access approaches in- cluding hierarchical-access and dyn a mic exclusive use ( [3]–[7]). Hierarchical-access allows a secondary (unli- censed) network opera tor or users to opportunistically access the spectrum without affecting the normal oper- ation of the spec trum owner who serves the primary (licensed) users. Dynamic exclusive use a llows a spe c - trum owner to d ynamica lly transfer and trade the usage right of its licensed s pe ctrum to a th ird party (e.g., a secondary network operator or a secondary end-user) in the spectrum market. This pa p er considers a secondary operator who obtains spectrum resource via both spec- trum sensin g as in the hierarchical-access approach a nd dynamic spectrum leasing a s in the dynamic exclusive use approach. • Lingjie Duan and Jianwei Huang are with the Department of Information Engineering, The Chinese University of Hong Kong, Hong Kong. E-mail: { dlj008, jwhuang } @ie.cuhk.edu.hk. • Biying Shou i s with the Department of Management Sciences, City University of Hong Kong, Hong Kong. E-mail: biying.shou@cityu.edu.hk. Part of the results has appeared in IEE E INFOCOM, San Diego , USA, March 2010 [1]. Spectrum sensing obtains a wareness of the spectrum usage and existence of primar y users, by using geoloca- tion and datab a se, beacons, or cognitive rad ios (e.g., [8]– [11]). The pr imary users are oblivious to the presence of secondary cognitive network operators or users. The sec- ondary network operator or users can sense and utilize the unused “spectrum holes” in t he licensed spectrum without violating the usage rights of the primary users (e.g., [4], [7]) . Since the seconda ry operator or users does not know the pr imary users’ activities bef ore sensin g, the amount of useful spectrum obtained through sensing is uncertain (e.g. [12]–[15]). W ith dynamic spectrum leasing, a spectrum owner allows secondary users to operate in their temporarily unused part of spectrum in exchange of economic return (e.g., [5], [7], [16]). The dynamic spectrum leasing can be short-term or eve n real-time (e . g., [17]–[19]), and ca n be at a similar time scale of the spectrum sensing opera tion. In this paper , we study the operation of a cognitive radio network that consists a cognitive mobile vi r tual network oper ator ( C - MVNO) and a group of secondar y unlicensed users. The word “ virtual” refers to the fac t that the operator does not own th e wireless spectrum bands or even the physical network infrastructure. The C-MVNO serves a s the interfa ce between the spectrum owner a nd the sec ondary end-users. The word “cog- nitive” refers to the fact that the opera tor can obtain spectrum resource through both spectrum sensing using the cognitive radio technology and dynamic spectrum leasing from the spectrum ow ner . The operator then 2 resells the obtained spectrum (bandwidth) to secondary users to max imize its profit. The proposed model is a hybrid of the hierarchical-access and dynamic exclusive use models. It is applicable in var ious network sce- narios, such as achieving efficient utilization of the TV spectrum in IEEE 802.2 2 standard [20]. This standa rd suggests that the secondary system should operate on a point-to-multipoint ba sis, i.e. , the communications will happen between secondar y base stations and secondary customer-premis e s equipment. The base stations can be operated by one or severa l C-MVNOs introduced in this paper . Compared with a traditional M VNO who only lea ses spectrum through long-term contracts, a C-M VNO ca n dynamically adjust its sensing and leasing decisions to match the changes of user s’ demand at a short time scale. Moreover , sensing often offers a cheaper way to obtain spectrum compared with leasing. The cost of sensing mainly includes the sensing time and energy , and does not include explicit cost paid to the spectrum owner . W ith a mature spectrum sensing technology , sensing cost should be reasonable low (otherwise there is no point of using cogn itive radio). Spe ctrum leasing, ho weve r , involves direct negotiation with the spectrum owner . When the spectrum owner determines the cost of leasing, it needs to calculate its opportunity cost, i.e., how much revenue the spectrum can provide if the spec trum owner provides services directly over it. It is reasonable to believe that the le a sing cost is more expensive than the sensing cost in most cases 1 . Although sensing is cheap er , the amount of spectrum o bta ined th rough sensing is often uncertain d ue to the stochastic nature of primary users’ traffic. It is thus critical f or a C-M VNO to find the right balance between cost and uncertainty . Our key results and contributions are summarized as follows. For simplicity , we refer to the C-MVNO as “operator ”, secondary users a s “user s” , a nd “dynamic leasing” as “lea sing”. • A Stackelbe rg game model : W e model and analyze the interactions betwee n the opera tor and the users in the spectrum market as a Stackelberg ga me. As the leade r , the operator makes the sensing, leasing, and pricing decisions sequentially . As the followers, users then purchase bandwidth from the opera tor to maximize their pa yoffs. By using bac kward in- duction, we prove the existence a nd uniqueness of the equilibrium, and show how various system parameters (i.e ., sensing and leasing costs , users’ transmission power and channel conditions) affect the equilibrium behavior . • Threshold s tructures of the optimal investment and pricing d ecisions : At the e quilibrium, the operator will sense the spectrum onl y if the sensing cost is cheaper than a thr eshold. Furthermore, it will lease some spectrum only if the resource obtained 1. Th e analysis of this paper also covers the case where sen sing is more expensive than leasing, which is a trivial case to study . through sensing is below a threshold. Finally , the operator will cha rge a constant price to the users if the total ba ndwidth obtained through sensing and leasing does not exceed a threshold. The thresholds are easy to compute and the corresponding deci- sions rules a re easy to implement in pr actice. • Fair and predictable QoS : The operator ’s optimal pr ic- ing d ecision is independent of the users’ wireless characteristics. Each user receives a pa yoff tha t is proportional to its c ha nnel ga in and transmission power , which leads to the same si gnal- to-noise (SNR) for all users. • Impact of spect rum sensing : W e show that the avail- ability of sensing always increases the opera tor ’s profit in the exp ected sense. The a ctual realization of the profit at a particula r time hea vily depends on the spectrum sensing results. Users always get better payoffs when the opera tor performs spectrum sensing. Section 2 introduces the network mo d e l a nd pr ob- lem formulation. In Section 3, we analyze the game model through backward induction. W e discuss various insights obtained from the equilibrium analysis and present some numerical results in Section 4. In Section 5, we show the impact of spectrum sensing on both the operator and users. W e conclude in Section 6 and outline some f uture research directions. 1.1 Related W ork There is a growing interest in studying the investment and pricing decisions of cognitive network o p e rators recently . Several auction mechanisms have been pro- posed to s tud y the investment problems of cogn itive network operators (e. g., [21], [22]). Other recent results studied the pricing decisions of the cognitive network operators who interact with a group of secondary users (e.g., [23]–[30]). [2 1] considered users’ queueing dela ys and obtained most results through simulations. [23] pre- sented a recent survey on the spectrum sharing games of network o p e rators and cogn itive radio networks. [24] studied the competition among multiple service providers without modeling users’ wireless details. [2 5] considered a pr icing competition game of two operators and adopted a simplified wireless model for the users. [26] derived users’ demand f unctions based on the a c- ceptance probability model for the users. [27] ex plored demand functions based on both quality-sensitive and price-sensitive buyer population models. [ 28] formulated the interaction between one primary user ( monopolis t) and multiple secon d ary users a s a Sta ckelberg game. The primary user uses some secondary users as re- lays and leases its bandwidth to those relays to collect revenue. [29] studied a multiple-level spe c trum market among primary , secondary , and tertiary service s where global information is not ava ilable. [3 0] considered the short-term spectrum trading be twee n multiple primary users and multiple secondary users. The spectrum buy- 3  Spectrum  Owner’s  Service  Band Spectrum  Owner’s  T ransf erence  Band Channels  PUs’  Activity  Band  O perator’s  Sensed  B and Operat or ’s  Leased  Band 1 t ( ) f HZ 2 t 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2425 26 27 28 293031 32 33 34 Fig. 1. Operator’ s In vestment in Spectrum Sensing and Leasing ing behaviors of s e condary users are modeled as an evolutionary game, while selling behaviors of primary users a re modeled as a noncooperative game. [26]–[30] obtained most interesting results through simulations. There a re only few papers (e .g., [19], [29], [ 31]) that jointly considered the spectrum investment and service pricing p roblem as this pa p e r . None of the above work considered the impact of supply uncertainty due to spectrum sensing. Our model of spectrum uncerta inty is related to the random-yield model in supply chain management (e.g., [32]–[34]). The unique wireless aspects of the system model lead to new solutions a nd insights in our problem. Our pa per represents a first a ttempt of understanding how spectrum uncertainty impacts the ec onomic deci- sions of an cognitive radio operator . T o obtain sharp insights, we f ocus on a stylized model where a monopo- list operator fa ces a group of secondar y users. There are many more interesting research issues in this area. Some are fur ther discussed in Section 6. 2 N E T W O R K M O D E L 2.1 Bac kground on Spectrum Sensing and Leasing T o illustrate the opportunity and trade-off of spectrum sensing and lea sing, we consider a spectrum owner who divides its licensed spectrum into two types: • Service Band : This band is reserved for serving the spectrum owner ’s primary users (PU s). Since the PUs’ traffic is stochastic, there will be some unused spectrum which changes d ynamically . The opera tor can sense and utilize the unused portions. There a re no explicit communications between the spec trum owner and the operator . • T ransferen c e Ba nd : The spectrum owner temporarily does not use this band. The operator can lease the bandwidth through explicit communications w ith the spec trum owner . No sensing is allowed in this band. Due to the short-term property of both sensing and leasing, the operator needs to make both the sensing and leasing decisions in ea ch time slot. The exa mple in Fig. 1 demonstrates the dynamic opportunities for spectrum sensing, the uncertainty of sensing outcome, and the impac t of sensing or leasing T ABLE 1 K ey Notation s Symbol Physical M eaning B s Sensing b an dwidth B l Leasing band width C s Unit sensing cost C l Unit leasing cost α ∈ [0 , 1] Sensing realization factor I = { 1 , · · · , I } Set of secondary users π Unit price w i User i ’s bandwidth allocatio n r i User i ’s d ata rate P max i User i ’s maximum transmission power h i User i ’s channel gain n 0 Noise power density g i = P max i h i /n 0 User i ’s wireless characteristic SNR i = g i /w i User i ’s SNR G = P i ∈I g i Users’ aggregate wireless characteristics R Operator ’s profit decisions. The spectrum owner ’s entire band is divided into small 34 channels 2 . • T ime slot 1: PUs use channels 1 − 4 and 11 − 15 . The operator is unaware of thi s and senses channels 3 − 8 . As a result, it obtains 4 unused cha nnels ( 5 − 8 ). It lease s add itional 9 channels ( 2 0 − 2 8 ) from the transference band. • T ime slot 2 : PUs change their behavior and use channels 1 − 6 . T he operator senses channels 5 − 14 and obta ins 8 unused channels ( 7 − 14 ). It leases additional 5 channels ( 23 − 27 ) from the transference band. In this paper , we will only study the opera tor ’s deci- sions within a single time slot. W e choose the time slot length such that primary users’ activities remain roughly unchanged within a single time slot. This mea ns that it is enough for the operator to sense at the beginning of each time slot. For tra ffic types such a s TV programs, data transfer , and even V oIP voice sessions, the length of the time slot can be reasonable long. For read ers who are interested in the optimization of the time slot length to balance sensing and data tra nsmission , see [1 4]. 2.2 Notations and Assumptions W e consider a cognitive network with one operator and a set I = { 1 , . . . , I } of users. The operator has the cognitive capability and can sense the unused spectrum. One way to realize this is to let the operator construct a sensor network that is dedicate d to sensing the ra dio environment in space and time [35]. The opera tor will collect the sensing information from the sensor network and pro vide it to the unlicensed us e rs, or pr oviding “sensing as service”. If the operator owns several base stations, then e a ch base station is responsible for collect- ing sensing information in a certain geographical area. As mentioned in [35], there has been significant current 2. Chann el 16 is the guard ban d between the service and transference bands. 4  Stage I: Operator determ i nes sensing amount   realize available bandwidth    Stage II: Operator determ in es leasing amount   Stage III: Operator announces pri ce  to market Stage IV : End-users determine the dem ands for bandwidth from the operator Fig. 2. A Stac kelberg Game research efforts in the context of an E uropean project SENDORA [3 6], which aims a t developing techniques based on sensor networks for supporting coexistence of licensed a nd unlicensed wireless users in a sa me area. The users are equipped with software defined radios and can tune to transmit in a wide range of frequencies as instructed by the operator , but do not necessar ily have the cognitive sensing capacity . Since the seconda r y user s do not worry about sensing, they can spend most of their time and energy on ac tua l data tra nsmissions. Such a network structure puts most of the implementation complexity a t the operator side and reduces the user equipment complexity , and thus might be easier to im- plement in practice than a “full” cognitive network. The key notations of this paper a re listed in T able 1 with some explanations as follows. • Investment decisions B s and B l : the operator ’s sensing and leasing bandwidths, respectively . • Sensing realization facto r α : when the operator senses a total bandwidth of B s , only a proportion of α ∈ [0 , 1] is unused and can be used by the operator . α is a random va riable and depends on the primary users’ a ctivities. W ith perfect sensing results , users can use bandwidth up to B s α without generating interferences to the primary users. • Cost parameters C s and C l : the opera tor ’s fixed sens- ing a nd lea sing costs per unit bandwidth, respec- tively . Sensing cost C s depends on the operator ’s sensing technologies. When the operator senses spectrum, it needs to spend time and energy on channel sampling and signal processin g ( [3 7]). Sens- ing over different channels often needs to be d one sequentially due to the potentially la rge number of channels open to opportunistic spectrum acce ss and the limited power/hardware ca pacity of cognitive radios ( [38]). The larger sensing bandwidth and the more channels, the longer time a nd higher energy it requires ( [39]). For simplicity , we assume that total sensing cost is linear in the sensing bandwidth B s . Leasing cost C l is d etermined through the negotia- tion between the operator and the spectrum owner and is assumed to be la rger than C s . • Pricing d ecision π : the operator ’s choice of price per unit bandwidth to the users. 2.3 A Stac kelberg Game W e consider a Stac kelberg Game between the opera tor and the users as shown in Fig. 2. The operator is the Stackelberg leader: it first decides the sensing a mount B s in S tage I, then de cides the leasing a mount B l in Stage II (based on the sensing result B s α ), and then announces the price π to the users in S tage III ( based on the total supply B s α + B l ). Finally , the users choose their bandwidth demands to maximize their individual payoffs in Stage IV . W e note that “sensing followed by leasing” is optimal for the operator to ma ximize pr ofit. Since sensing is cheaper than leasing, the opera tor shou ld lease only if sensing does not provide enough resour c e. If the operator determines sensing and lea sing simultaneously , then it is likely to “over-lease” (compared with “sensing followed by leasing”) to avoid having too little resource when α is small. “ Leasing before sensing” can not improve the operator ’s pr ofit either d ue to the same reason. 3 B A C K WA R D I N D U C T I O N O F T H E F O U R - S TAG E G A M E The Stac kelberg game falls into the class of dynamic game, and the common solution concept is the S ubgame Perfect Equilibrium (SPE, or simply as equilibrium in this paper). A general technique for determining the SPE is the ba c kwa rd induction ( [40]). W e will start with Stage IV and a nalyze the users’ behaviors given the operator ’s investment and pricing decisions. Then we will look a t Stage III and analyze how the opera tor makes the pricing decision given investm e nt decisions and the possible reactions of the users in Sta ge IV . Finally we proceed to derive the opera tor ’s optimal leasing decision in Stage II and then the optimal sensing decision in Sta ge I. The backward induction captures the sequential dependence of the decisions in four stages. 3.1 Spectrum Allocation in Stage IV In Stage IV , end-users determine their bandwidth de- mands given the unit price π announced by the operator in stage III. Each user can represent a tr ansmitter-receiver node pair in an a d hoc network, or a node that tra nsmits to the operator ’s base sta tion in an uplink scenario. W e assume that users ac c ess the spectrum provided by the opera- tor through FDM (Frequency-division multiplexing) or OFDM (Orthogonal frequency-division multiplexing) to avoid mutual interf erences. User i ’s achievable rate (in nats) is 3 : r i ( w i ) = w i ln  1 + P max i h i n 0 w i  , (1) 3. W e assume that the operator only provides bandwidth without restricting the application types. This assumption has be e n commonly used in d ynamic spectrum sharing literature, e.g., [16], [19], [24], [41]. 5 where w i is the a llocated bandwidth from the oper ator , P max i is user i ’s maximum transmission power , n 0 is the noise power per unit ba ndwidth, h i is user i ’s channel gain (between user i ’s own transmitter and receiver in an ad hoc network, or between user i ’s transmitter to the operator ’s ba se station in an uplink scenario). T o obtain rate in (1), user i spreads its maximum transmission power P max k across the entire allocated bandwidth w i . T o simplify the notation, we let g i = P max i h i /n 0 , thus g i /w i is the user i ’s signal-to-noise ratio ( S NR). Here we f ocus on best-effort users who are interested in maximizing their data rates. Each user only knows its local information (i.e. , P max i , h i , and n 0 ) and does not know anything about other users. From a user ’s point of view , it does not matter whether the ba ndwidth has been obtained by the opera- tor through spectrum sensing or dynamic leasing. Each unit of allocated bandwidth is perfectly reliable for the user . T o obtain c losed-form solutions, we first focus on the high SNR regime wh ere SNR ≫ 1 . This is motivated by the fact that users often have limited choices of modulation and c oding schemes, a nd thus may not be able to decode a transmission if the S NR is below a threshold. In the high SNR regime, the r ate in ( 1) can be approximated as r i ( w i ) = w i ln  g i w i  . (2) Although the analytical solutions in Section 3 are derived based on (2), we e mphasize that all the major engineering insights remain true in the general SNR regime. A formal proof is in Section 4. A user i ’s payoff is a function of the allocated band- width w i and the price π , u i ( π , w i ) = w i ln  g i w i  − πw i , (3) i.e., the difference between the data rate and the linear payment ( π w i ). Pa yoff u i ( π , w i ) is concave in w i , and the unique bandwidth demand that maximizes the payoff is w ∗ i ( π ) = ar g max w i ≥ 0 u i ( π , w i ) = g i e − (1+ π ) , (4) which is a lwa ys positive, linear in g i , and decreasing in price π . Since g i is linear in channel ga in h i and transmission power P max i , then a user with a better channel condition or a larger transmission power has a larger demand. Equation (4 ) shows that ea ch user i a chieves the same SNR: SNR i = g i w ∗ i ( π ) = e (1+ π ) . but a different payoff that is linear in g i , u i ( π , w ∗ i ( π )) = g i e − (1+ π ) . W e de note users’ aggregate wireless char a cteristics as G = P i ∈I g i . The users’ total demand is X i ∈I w ∗ i ( π ) = Ge − (1+ π ) . (5) 1 ( ) S S 2 ( ) S S 3 ( ) S S S 0 1 ( ) D S Fig. 3. Diff erent intersection cas es of D ( π ) and S ( π ) Next, we will consider how the operator makes the investment (sensing and leasing) and pricing d ecisions in Stages I-III ba se d on the total demand in eq. (5) 4 . In particular , we will show that the operator will a lways choose a price in Sta ge III such that the total dema nd (as a function of price) does not excee d the total supply . 3.2 Optimal Pricing Strategy in Stage III In S ta ge I II, the opera tor dete rmines the optimal pricing considering users’ total dema nd (5), given the band- width supply B s α + B l obtained in Stage II. The operator profit is R ( B s , α, B l , π ) = min π X i ∈I w ∗ i ( π ) , π ( B l + B s α ) ! − ( B s C s + B l C l ) , (6) which is the difference be twee n the revenue and total cost. The min oper ation denotes the fa ct that the operator can only satisfy the demand up to its available supply . The objective of Stage III is to find the optimal price π ∗ ( B s , α, B l ) that maximizes the profit, that is, R I I I ( B s , α, B l ) = max π ≥ 0 R ( B s , α, B l , π ) . (7) The subscript “III” denotes the best profit in Sta ge III. Since the bandwidths B s and B l are given in this stage, the total cost B s C s + B l C l is already fixed. The only optimization is to choose the optimal price π to maximize the revenue, i.e., max π ≥ 0 min π X i ∈I w ∗ i ( π ) , π ( B l + B s α ) ! . (8) The solution of problem (8) depends on the bandwidth investment in Stages I and II. Let us define D ( π ) = π P i ∈I w ∗ i ( π ) and S ( π ) = π ( B l + B s α ) . Figure 3 shows three possible r ela tionships betwee n these two ter ms, where S j ( π ) (for j = 1 , 2 , 3 ) represents each of the three possible c hoices of S ( π ) depending on the bandwidth B l + B s α : • S 1 ( π ) (exce ssive supply): No intersection with D ( π ) ; 4. W e assume that t h e operator know s the value of G through proper feedback me chanism from the users. 6 T ABLE 2 Optimal Pricing Decision and Profit in Stage III T otal Bandwidth Obtained in Stages I and II Optimal Price π ∗ ( B s , α, B l ) Optimal Profit R I I I ( B s , α, B l ) Excessive Supply Regime: B l + B s α ≥ Ge − 2 π E S = 1 R E S I I I ( B s , α, B l ) = Ge − 2 − B s C s − B l C l Conservative Supply Regime: B l + B s α < Ge − 2 π C S = ln  G B l + B s α  − 1 R C S I I I ( B s , α, B l ) = ( B l + B s α ) ln  G B l + B s α  − B s ( α + C s ) − B l (1 + C l ) • S 2 ( π ) (excessive supply): intersect once with D ( π ) where D ( π ) has a non-negative slope; • S 3 ( π ) (conservative supply ) : intersect once with D ( π ) where D ( π ) has a negative slope. In the excessive supply regime, max π ≥ 0 min ( S ( π ) , D ( π )) = max π ≥ 0 D ( π ) , i.e., the max-min solution occurs at the maximum value of D ( π ) with π ∗ = 1 . In this regime, the total supply is la rger than the total demand at the be st price choice. In the conservative supply regime, the max-min solution occurs at the unique intersection point of D ( π ) and S ( π ) . The above observations lead to the f ollowing result. Theorem 1: The optimal pricing decision and the corre- sponding optimal profit at Sta ge III can be characteriz e d by T a b le 2. The proof of Theorem 1 is given in Appendix A . Note that in the excessive supply regime, some ba ndwidth is left unsold (i.e., S ( π ∗ ) > D ( π ∗ ) ). This is because the acquired ba ndwidth is too la rge, and selling all the bandwidth will lead to a very low price that decreases the revenue (the product of price and sold bandwidth). The profit can be apparently improved if the oper a tor acquires less bandwidth in Stage s I and II. L ater analysis in S tages II and I will show that the equilibrium of the game must lie in the conservative supply regime if the sensing cost is non-negligible. 3.3 Optimal Leasing Strategy in Stage II In Stage II, the operator decides the optimal leasing amount B l given the sensing result B s α : R I I ( B s , α ) = max B l ≥ 0 R I I I ( B s , α, B l ) . (9) W e decompose problem (9) into two subproblems based on the two supply regimes in T a ble 2, 1) Choose B l to reach the excessive supply regime in Stage III: R E S I I ( B s , α ) = max B l ≥ max { Ge − 2 − B s α, 0 } R E S I I I ( B s , α, B l ) . (10) 2) Choose B l to reach the conservative supply regime in Stage III: R C S I I ( B s , α ) = max 0 ≤ B l ≤ Ge − 2 − B s α R C S I I I ( B s , α, B l ) , (11) T o solve subproblems (10) and (11), we need to con- sider the bandwidth obtained f rom sensing. • Excessive Supply ( B s α > Ge − 2 ): in this case, the feasible sets of both subproblems (10) a nd (11) are empty . In fact, the ba ndwidth supply is already in the excessive supply regime as d efined in T able II, and it is optimal not to lease in Stage II. • Conservative Supply ( B s α ≤ Ge − 2 ): first, we can show that the unique optimal solution of subproblem (10) is B ∗ l = Ge − 2 − B s α . This means that the optimal objective va lue of subproblem (10) is no larger than that of subproblem (11), a nd thus it is enough to consider subproblem (11) in the conservative supply regime only . Base on the above observations a nd some further analysis, we can show the following: Theorem 2: In S tage II, the optimal lea sing decision and the corresponding optimal profit are summarized in T able 3. The proof of T heorem 2 is given in Appendix B. T able 3 contains three cases ba sed on the v alue of B s α : ( CS1), (CS2), and ( ES3). T he first two cases in volve solving the subproblem (11) in the conservative supply regime, and the la st one corresponds to the excessive supply regime. Although the decisions in cases (CS2) and (E S 3) are the same (i.e., zero leasing amount), we still treat them separa tely since the profit expressions are different. It is clear that we have an optimal threshold leasing policy here: the operator wants to achieve a total band- width equal to Ge − (2+ C l ) whenever possible. When the bandwidth obtained through sensing is not enough, the operator will lease additional bandwidth to reach the threshold; otherwise the ope r ator will not lease. 3.4 Optimal Sensing Strategy in Stage I In S tage I, the operator will decide the optimal sensing amount to maximize its ex p ected profit by taking the un- certainty of the sensing realization fa ctor α into a ccount. The operator needs to solve the f ollowing problem R I = ma x B s ≥ 0 R I I ( B s ) , where R I I ( B s ) is obtained by ta king the expectation of α over the profit functions in Stage II (i.e., R C S 1 I I ( B s , α ) , R C S 2 I I ( B s , α ) , and R E S 3 I I ( B s , α ) in T able 3). T o obtain closed-form solutions, we assume that the sensing realization f actor α follow s a uniform distri- bution in [0 , 1] . In Sec tion 4 .1, we prove that the major engineering insights also hold under any general distribution. T o avoid the trivia l case where sensing is so cheap that it is optimal to sense a huge amount of ba ndwidth, 7 T ABLE 3 Optimal Leasing Decision and Profit in Stage II Given Sensing Result B s α A fter Stage I Optimal L easing Amount B ∗ l Optimal Profit R I I ( B s , α ) (CS1) B s α ≤ Ge − (2+ C l ) B C S 1 l = Ge − (2+ C l ) − B s α R C S 1 I I ( B s , α ) = Ge − (2+ C l ) + B s ( αC l − C s ) (CS2) B s α ∈  Ge − (2+ C l ) , Ge − 2  B C S 2 l = 0 R C S 2 I I ( B s , α ) = B s α ln  G B s α  − B s ( α + C s ) (ES3) B s α > Ge − 2 B E S 3 l = 0 R E S 3 I I ( B s , α ) = Ge − 2 − B s C s 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.005 0.01 0.015 0.02 0.025 B s R II (B s ) ( C l , C s )=(2,0.8) ( C l , C s )=(2,1.2) Fig. 4. Expected profit in Stage II under diff erent sensing and leasing costs we further assume that the sensing cost is non-negligible and is lower bounded by C s ≥ (1 − e − 2 C l ) / 4 . T o de r ive f unction R I I ( B s ) , we will consider the following three interva ls: 1) Case I: B s ∈ [0 , Ge − (2+ C l ) ] . In this case, we a lways have B s α ≤ Ge − (2+ C l ) for any value α ∈ [0 , 1] , which corresponds to case (CS1) in T a ble 3. The expected profit is R 1 I I ( B s ) = E α ∈ [0 , 1]  R C S 1 I I ( B s , α )  = Ge − (2+ C l ) + B s  C l 2 − C s  , which is a linear function of B s . If C s > C l / 2 , R 1 I I ( B s ) is linearly decreasing in B s ; if C s < C l / 2 , R 1 I I ( B s ) is linearly increasing in B s . 2) Case II: B s ∈  Ge − (2+ C l ) , Ge − 2  . Depending on the value of α , B s α ca n be in e ither case (CS1) or c a se (CS2) in T able 3 . The expected profit is R 2 I I ( B s ) = E α ∈ h 0 , Ge − (2+ C l ) B s i  R C S 1 I I ( B s , α )  + E α ∈ h Ge − (2+ C l ) B s , 1 i  R C S 2 I I ( B s , α )  = B s 2 ln  G B s  − B s 4 + B s 4  Ge − (2+ C l ) B s  2 − B s C s . R 2 I I ( B s ) is a strictly concave function of B s since its second-order deriva tive ∂ 2 R 2 I I ( B s ) ∂ B 2 s = 1 2 B s "  Ge − (2+ C l ) B s  2 − 1 # < 0 as B s > Ge − (2+ C l ) in this case. 3) Case III: B s ∈  Ge − 2 , ∞  . Depending on the value of α , B s α can be any of the three cases in T able 3. The expected profit is R 3 I I ( B s ) = E α ∈ h 0 , Ge − (2+ C l ) B s i  R C S 1 I I ( B s , α )  + E α ∈ h Ge − (2+ C l ) B s , Ge − 2 B s i  R C S 2 I I ( B s , α )  + E α ∈ h Ge − 2 B s , 1 i  R E S 3 I I ( B s , α )  =  G e 2  2 e − 2 C l − 1 4 B s − B s C s + G e 2 . Because its first-order de r ivative ∂ R 3 I I ( B s ) ∂ B s =  Ge − 2 B s  2 1 − e − 2 C l 4 − C s < 0 , as B s > Ge − 2 in this case, R 3 I I ( B s ) is decreasing in B s and achieves its maximum at B s = Ge − 2 . T o summarize, the operator needs to maximize R I I ( B s ) =      R 1 I I ( B s ) , if 0 ≤ B s ≤ Ge − (2+ C l ) ; R 2 I I ( B s ) , if Ge − (2+ C l ) < B s ≤ Ge − 2 ; R 3 I I ( B s ) , if B s > Ge − 2 . (12) W e can verify that Case II always achieves a higher optimal profit than Case III. This means that the optimal sensing will only lead to either case (CS1 ) or case (CS 2 ) in Stage II, which corresponds to the conservative supply regime in Stage III. This confirms our previous intuition that equilibrium is a lways in the conservative supply regime under a non-negligible sensing cost, since some resource is wa sted in the excessive supply regime (see discussions in Section 3.2). T able 4 shows that the sensing decision is made in the following two cost regimes: • High sensing cost regime ( C s > C l / 2 ): it is optimal not to sense. Intuitively , the coefficient 1 / 2 is due to the uniform distribution assumption of α , i.e. , on average obtaining one unit of available ba ndwidth through sensing costs 2 C s . • Low sensin g cost regime ( C s ∈ h 1 − e − 2 C l 4 , C l 2 i ): the optimal sensing a mount B L ∗ s is the unique solution to the following equation: ∂ R 2 I I ( B s ) ∂ B s = 1 2 ln  1 B s /G  − 3 4 − C s −  e − (2+ C l ) 2 B s /G  2 = 0 . (13) The uniqueness of the solution is due to the strict concavity of R 2 I I ( B s ) over B s . W e can further show that B L ∗ s lies in the interval of  Ge − (2+ C l ) , Ge − 2  8 T ABLE 4 Choice of Optimal Sensing Amount in Stage I Optimal S e nsing Decision B ∗ s Expected Profit R I High Sensing Cost Regime: C s ≥ C l / 2 B ∗ s = 0 R H I = Ge − (2+ C l ) Low Sensing Cost Regime: C s ∈  (1 − e − 2 C l ) / 4 , C l / 2  B ∗ s = B L ∗ s , solution to eq. (13) R L I in eq. (14) and is linear in G . Finally , the operator ’s optimal expected profit is R L I = B L ∗ s 2 ln  G B L ∗ s  − B L ∗ s 4 + 1 4 B L ∗ s  G e 2+ C l  2 − B L ∗ s C s . (14) Based on these observations, we c a n show the follow- ing: Theorem 3: In Stage I, the optimal sensing decision and the corresponding optimal profit are summarized in T able 4. The optimal sensing amount B ∗ l is linear in G . Figure 4 shows two possible cases for the function R I I ( B s ) . The vertical dashed line represents B s = e − (2+ C l ) . For illustration purpose, we assume G = 1 , C l = 2 , and C s = { 0 . 8 , 1 . 2 } . When the sensing cost is large (i.e., C s = 1 . 2 > C l / 2 ), R I I ( B s ) achieves its optimum at B s = 0 and thus it is optimal not to sense. When the sensing cost is small ( i.e., C s = 0 . 8 < C l / 2 ), R I I ( B s ) achieve s its optimum at B s > e − (2+ C l ) and it is optimal to sense a positive amount of spectrum. 4 E Q U I L I B R I U M S U M M A RY A N D N U M E R I C A L R E S U L T S Based on the discussions in Se ction 3, we summarize the operator ’s equilibrium sensing/leasing/pricing de- cisions and the equilibrium resource a llocations to the users in T able 5. Sev e ral interesting observations are as follows. Observation 1: Both the optimal sensing amount B ∗ s (either 0 or B L ∗ s ) and leasing amount B ∗ l are linear in the users’ aggr ega te wireless chara cteristics G = P i ∈I P max i h i /n 0 . The linearity enables us to normalize optimal sensing and leasing decisions by user s’ aggregate wireless c ha r- acteristics, and study the relationships between the nor- malized optimal decisions and other system para mete r s as in Figs. 5 and 6. Figure 5 shows how the normalized optimal sensing decision B ∗ s /G cha nges with the costs. For a given leas- ing cost C l , the optimal sensing d ecision B ∗ s decreases a s the sensing cost C s becomes more expensive, and drops to zero when C s ≥ C l / 2 . For a given sensing cost C s , the optimal sensing d ecision B ∗ s increases as the leasing cost C l becomes more expensive, in which case sensing becomes more a ttr a ctive. Figure 6 shows how the normalized optimal leasing decision B ∗ l /G depends on the costs C l and C s as well as the sensing realization fa ctor α in the low sensing cost regime (denoted by “ L ” ). In all cases, a higher value α means more bandwidth is obtained from sensing and there is a less need to lea se. Figure 6 confirms the threshold structure of the optimal lea sing decisions in Section 3.3, i.e., no lea sing is needed whenever the bandwidth obtained from sensing reaches a threshold. Comparing different curves, we can se e that the operator chooses to lease more a s leasing becomes cheaper or sensing becomes more expe nsive. For high sensing cost regime, the optimal lea sing amount only depends on C l and is inde p e ndent of C s and α , and thus is not shown here. Observation 2: The optimal pricing decision π ∗ in Stage III is independent of users’ aggregate wireless characteristics G . Observation 2 is closely related to Observation 1. S ince the total bandwidth is linear in G , the “a verage” reso urce allocation per user is “constant” at the equilibrium. This implies that the p r ice must be independent of the user population change, otherwise the resource allocation to each individua l user will change with the p r ice acc ord- ingly . Observation 3: The optimal pricing decision π ∗ in Stage III is non-increasing in α in the low sensing cost regime. First, in the low sensing cost regime where the sensing result is poor (i.e., α is small as the third column in T able 5), the operator will lea se additional resource such tha t the total bandwidth reaches the threshold Ge − (2+ C l ) . In this case, the price is a c onstant and is independent of the value of α . Second, when the sensing result is good (i.e., α is large as in the last column in T ab le 5), the total bandwidth is la rge enough. I n this case, as α increases, the amount of total bandwidth in creases, and the optimal price decreases to max imize the profit. Figure 7 shows how the optimal price changes with various costs a nd α in the low sensing cost regime. It is clea r that price is first a constant and then starts to decrease wh en α is larger than a threshold. The thresho ld decreases in the optimal sensing decision of B L ∗ s : a smaller sensing cost or a higher leasing cost will lead to a higher B L ∗ s and thus a smaller threshold. It is interesting to notice that the equilibrium price only changes in a time slot where the sensing realization factor α is large. This means that a lthough operator has the f reedom to change the price in every time slot, the actual variation of price is much less frequent. This leads to less overhead a nd makes it easier to implement in practice. Figure 8 illustrates this with different sensing costs and α realizations. The left two subfigures corre- spond to the realizations of α a nd the corresponding prices with C s = 0 . 48 and C l = 1 . As the sensing 9 T ABLE 5 The Operator’ s and Users ’ Equil ibrium Behaviors Sensing Cost Regimes High Sensing Cost: C s ≥ C l 2 Low Sensing Cost: 1 − e − 2 C l 4 ≤ C s ≤ C l 2 Optimal S e nsing Amount B ∗ s 0 B L ∗ s ∈  Ge − (2+ C l ) , Ge − 2  , solution to eq. (13 ) Sensing Realization Factor α 0 ≤ α ≤ 1 0 ≤ α ≤ Ge − (2+ C l ) /B L ∗ s α > Ge − (2+ C l ) /B L ∗ s Optimal L easing Amount B ∗ l Ge − (2+ C l ) Ge − (2+ C l ) − B L ∗ s α 0 Optimal Pricing π ∗ 1 + C l 1 + C l ln  G B L ∗ s α  − 1 Expected Profit R I R H I = Ge − (2+ C l ) R L I in eq. (14) R L I in eq. (14) User i ’s SNR e (2+ C l ) e (2+ C l ) G B L ∗ s α User i ’s Payoff g i e − (2+ C l ) g i e − (2+ C l ) g i ( B L ∗ s α/G ) 0.2 0.3 0.4 0.5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Sensing Cost C s B s * /G C l =0.5 C l =0.7 C l =1 Fig. 5. Optimal sensing amount B ∗ s as a function of C s and C l . 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 Sensing Reliazation Factor α B l * /G L :( C l , C s )=(0.5, 0.2) L :( C l , C s )=(0.5, 0.1) L :( C l , C s )=(1, 0.1) Fig. 6. Optimal leasing amount B ∗ l as a function of C s , C l , and α . 0 0.2 0.4 0.6 0.8 1 0.8 1 1.2 1.4 1.6 1.8 2 Sensing Reliazation Factor α Optimal Price π * L:( C l , C s )=(0.5, 0.2) L:( C l , C s )=(0.5, 0.1) L:( C l , C s )=(1, 0.1) Fig. 7. Optima l pr ice π ∗ as a function of C s , C l , and α . 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 Time Slots t 1 with C s =0.48 and C l =1 α 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 Time Slots t 2 with C s =0.35 and C l =1 α 0 10 20 30 40 50 1 1.2 1.4 1.6 1.8 2 Time Slots t 1 with C s =0.48 and C l =1 Optimal Price π * ( α ( t 1 )) 0 10 20 30 40 50 1 1.2 1.4 1.6 1.8 2 Time Slots t 2 with C s =0.35 and C l =1 Optimal Price π * ( α ( t 2 )) Fig. 8. Optimal pr ice π ∗ ov er time with different sensing costs and α r ealizations cost C s is quite high in this case, the operator does not rely heavily on sensing. As a result, the variability of α (in the upper subfigure) has ver y s ma ll impa c t on the equilibrium price (in the lower subfigure). In fa ct, the pr ice only changes in 11 out 50 time slots, and the maximum amplitude variation is around 10% . The right two figures correspond to the case where C s = 0 . 35 and C l = 1 . As sensing cost is cheaper in this ca se, the operator senses more and the impa ct of α on price is larger . The price changes in 30 out of 5 0 time slots, a nd the variation in amplitude can be as large as 3 0% . Observation 4: The operator will sense the spectrum only if the sensing cost is lower tha n a threshold. Fur- thermore, it will lea se a dditional spectrum only if the spectrum obtained through sensing is below a threshold. Observation 5: Ea ch user i obtains the sa me S NR inde- pendent of g i and a payoff linea r in g i . Observation 5 shows that users obtains fair and pre- dictable resource a llocation at the equilibrium. In fact, a user does not need to know anything about the total number and payoffs of other users in the system. It ca n simply predict i ts QoS if it knows the cost structure of the network ( C s and C l ) 5 . Such property is highly desirable in practice. Finally , users achieve the same high S NR a t the equi- librium. The SNR va lue is either e (2+ C l ) or G/ ( B L ∗ s α ) , both of which are larger than e 2 . This mea ns that the approximation ratio ln( SNR i ) / ln(1+ SN R i ) > ln( e 2 ) / ln(1+ e 2 ) ≈ 94% . The ratio can even be close to one if the price π is high. In Sections 3.1 and 3 .4, we made the high SNR regime approximation and the uniform distribution assumption of α to obtain closed-form expressions. Next we show that relaxing both assumptions will not change any of the major insights. 5. Th e analysis of the gam e, howeve r , does n ot require the users to know C s or C l . 10 4.1 Robustness of the Observ atio ns Theorem 4: Observations 1-5 still hold under the gen- eral SNR regime ( a s in (1)) a nd any general distribution of α . Proof: W e represent a user i ’s pa yoff f unction in the general SNR regime, u i ( π , w i ) = w i ln  1 + g i w i  − πw i . (15) The optimal d emand w ∗ i ( π ) that maximizes (15) is w ∗ i ( π ) = g i /Q ( π ) , where Q ( π ) is the unique posi tive solution to F ( π , Q ) := ln(1 + Q ) − Q 1+ Q − π = 0 . W e find the inverse function of Q ( π ) to b e π ( Q ) = ln (1 + Q ) − Q 1+ Q . By applying the implicit f unction theorem, we can obtain the first-order der ivative of function Q ( π ) over π as Q ′ ( π ) = − ∂ F ( π , Q ) /∂ π ∂ F ( π , Q ) /∂ Q = (1 + Q ( π )) 2 Q ( π ) , (16) which is always positive. Hence, Q ( π ) is increasing in π . User i ’s optimal payoff is u i ( π , w ∗ i ( π )) = g i Q ( π ) [ln(1 + Q ( π )) − π ] . (17) As a result, a user ’s optimal SNR equals g i /w ∗ i ( π ) = Q ( π ) and is user-independent . The total dema nd from all users equals G/Q ( π ) , and the operator ’s investment and pric- ing problem is R ∗ = max B s ≥ 0 E α ∈ [0 , 1] [ma x B l ≥ 0 max π ≥ 0 (min  π G Q ( π ) , π ( B l + B s α )  − B s C s − B l C l )] . (18) Define f R ∗ = R ∗ G , f B l = B l G , and f B s = B s G . Then solving (18) is equivalent to solving f R ∗ = max f B s ≥ 0 E α ∈ [0 , 1] [ma x f B l ≥ 0 max π ≥ 0 (min  π Q ( π ) , π ( f B l + f B s α )  − f B s C s − f B l C l )] . (19) In Problem (19), it is clear that the operator ’s optimal decisions on leasing, sensing and pricing do not depe nd on users’ aggregate wireless char acteristics. This is true for a ny continuous distribution o f α . And a user ’s optimal payoff in eq. (17) is linear in g i since Q ( π ) is independent of users’ wir eless character istics. This shows that Observations 1, 2, and 5 hold for the general SNR regime a nd a ny genera l distribution of α . W e can also show that Observations 3 and 4 hold in the general case, with a de ta iled proof in Appendix C. 5 T H E I M P A C T O F S P E C T RU M S EN S I N G U N - C E RTA I N T Y The key difference between our model and most existing literature (e.g., [19], [21], [22], [24], [26], [ 27]) is the possibility of obtaining resource through the cheaper but uncertain a pproach of spectrum sensing. Here we will elaborate the impact of sensing on the performances of operator and users by comparing with the baseline case where sensing is not possible. Note that in the high sensing cost regime it is optimal not to sense, as a result, the performance of the operator and users will be the same as the ba seline case. Hence we will focus on the low sensing cost regime in T able 5. Observation 6: The opera tor ’s optimal expect ed profit always be nefits f rom the ava ilability of spectrum se nsing in the low sensing cost regime. Figure 9 illustrates the normalized optimal expected profit as a f unction of the sensing cost. W e assume leasing cost C l = 2 , a nd thus the low sensing cost regime corresponds to the case where C s ∈ [0 . 2 , 1] in the figure. It is clear that sensing ac hieve s a better optimal expected profit in this regime. In fact, sensing leads to 250% increase in profit when C s = 0 . 2 . The benefit decreases as the sensing cost becomes higher . When sensing becomes too expensive, the operator will choose not to sense and thus a chieve the same profit as in the baseline ca se . Theorem 5: The operator ’s realized profit (i.e., the profit for a given α ) is a strictly increasing function in α in the low sensing cost regime. Furthermore, there exists a threshold α th ∈ (0 , 1) such that the operator ’s realized profit is larger than the baseline approach if α > α th . Proof: A s in T able 5, we have two cases in the low sensing cost regime: • If α ≤ Ge − (2+ C l ) /B L ∗ s , then substituting B L ∗ s into R C S 1 I I ( B s , α ) in T able 3 lead s to the realized profit R C S 1 I I ( α ) = Ge − (2+ C l ) − B L ∗ s C s + B L ∗ s αC l , which is strictly and linearly increasing in α . • If α ≥ Ge − (2+ C l ) /B L ∗ s , then substituting B L ∗ s into R C S 2 I I ( B s , α ) in T able 3 lead s to the realized profit R C S 2 I I ( α ) = B L ∗ s α  ln  G B L ∗ s α  − 1  − B L ∗ s C s . Because the first-order d erivative ∂ R C S 2 I I ( α ) ∂ α = B L ∗ s  ln  G B L ∗ s α  − 2  > 0 , as B L ∗ s ≤ Ge − 2 , R C S 2 I I ( α ) is strictly increasing in α . W e can also verify that R C S 1 I I ( α ) = R C S 2 I I ( α ) when α = Ge − (2+ C l ) /B L ∗ s . Therefore, the realized profit is a continuous and strictly increasing function of α . Next we prove the existence of threshold α th . First consider the extreme case α = 0 . Since the operator ob- tains no bandwidth through sensing but still incurs some cost, the profit in this case is lower than the baseline case. Furthermore, we ca n ve r ify that R C S 2 I I (1) > R H I in T able 5, thus the realized profit at α = 1 is a lways larger than the baseline case. T ogether with the continuity a nd strictly increasing nature of the realize d profit f unction, we have proven the e xistence of threshold of α th . Figure 10 shows the realiz ed profit as a function of α for different costs . The realized profi t is increasing in α in both c ases. The “crossing” feature of the two increasing curves is because the optimal sensing B ∗ s is larger under a chea per sensing c ost ( C s = 0 . 5 ), which leads to larger 11 0.2 0.4 0.6 0.8 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Sensing Cost C s Normalized Exp. Profit R I /G C l =2, with sensing C l =2, w./o. sensing Fig. 9. O perator’ s normalized op timal e x pected profit as a function of C s and C l . 0 0.2 0.4 0.6 0.8 1 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 Sensing Realization Factor α Normalized Realized Profit R II CS1/2 ( α )/G C l =2,C s =0.8 C l =2,w./o. sensing C l =2, C s =0.5 Fig. 10. Operator’ s normalized opti- mal realized profit as a function of α . 0 0.2 0.4 0.6 0.8 1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Sensing Realization Factor α Normalized Realized Payoff of User u i * ( α ) /g i C l =2, C s =0.8 C l =2, C s =0.5 C l =2, w./o. sensing Fig. 11. User i ’ s nor malized optimal realized pa yoff as a fun c tion of α . realized pro fit loss (gain, r e spectively) when α → 0 ( α → 1 , respectively). This shows the tradeoff be twee n improvement of exp e cted profit and the large variability of the realized profit. Theorem 6: Users always benefit from the availa b ility of spectrum sensing in the low sensing cost regime. Proof: In the baseline approach without se nsing, the operator always charges the price 1 + C l . As shown in T able 5, the equilibrium price π ∗ with sensing is always no larger than 1 + C l for any value of α . Since a user ’s payoff is strictly d e creasing in pr ice, the users always benefit from sensing. Figure 11 shows how a user i ’s normalized realized payoff u ∗ i /g i changes with α . The payoff linearly in- creases in α when α becomes larger than a threshold, in which case the equilibrium price becomes lower than 1 + C l . A smaller sensing cost C s leads to more aggressive sensing and thus more benefits to the users. 6 C O N C L U S I O N S A N D F U T U R E W O R K This paper represents som e initial results towards under- standing the new business models, opportunities, and challenges of the emerging cognitive virtual mobile net- work operators (C-M VNOs) under supply uncertainty . Here we focus on studying the trad e-off between the cost and uncertainty of spectrum investment thr ough sensing and lea sing. W e model the interactions between the operator and the users by a S tackelberg game, which captures the wireless heterogeneity of users in terms of maximum tra nsmission power levels and channel gains. W e have discovered several interesting fea tures of the game equilibrium. W e sho w that the operator ’s opti- mal sensing, leasing, and pricing d ecisions follow nice threshold structures. The availability of sensing a lways increases the oper a tor ’s expected profit, d e spite that the realized profit in each time slot will have some va r iations depending on the sensing result. Moreover , users always benefit in terms of pa yoffs when sensing is performed by the operator . T o keep the problem tra c ta ble, we have made several assumptions throughout this p a per . Some assumptions can be (easily) gener alized without affecting the main insights. • Imperfect sensing : we c an incorporate imperfe ct spec- trum sensing ( i.e., miss-detec tion and fa lse-positive) into the model, which will change the uncerta inty of the spectrum sensing. Given that our results work for any distribution of the sensing realiza tion α , it is likely that such generalization does not change the major insights. • Learning : we can also consider the interactions of multiple time slots, where the sensing realizations of previous time slots ca n be used to update the distributions of α in future time slots. Aga in, the per slot decision model introduced in this pa per is still a pplicable with a time-dependent α d istribution input. Generalizations of some other assumptions, however , lead to more challenging new problems. • Incomplete information : when the opera tor does not know the information of the users, the system needs to be modeled as a dynamic game with incomplete information. M ore elaborate economic models such as screening and signaling [ 42] be c ome releva nt. • T ime scale separat io n : it is possible that dynamic leasing is performed at a different ( much larger) time sca le compared with spectrum sensing. In that case, the operator has to make the leasing decision first, and then make several sequential sensing deci- sions. This leads to a dynamic decision model with more stages a nd tight couplings a cross sequential decisions. • Operator comp etition : There may be multiple C- MVNOs providing services in the same geographic area. In that case, the operators need to attract the users through price competition. Also, if they sense and lease from the same spectrum owner , the operators may have overlapping or conflicting resource requests. Although we ha ve obtained some preliminary results along this line in [ 43], mor e studies are de finitely de sira ble. Through the analytical and simulation study of an 12 idealized model in this pape r , we have obta ined var i- ous interesting engineering and e c onomical insights into the operations of C-MVNOs. W e hope that this paper can contribute to the further understanding of proper network architecture decisions and business models of future cognitive ra dio systems. A P P E N D I X A P R O O F O F T H E O R E M 1 Given the total bandwidth B l + B s α , the objective of Stage III is to solve the optimiza tion problem (8), i.e., max π ≥ 0 min( D ( π ) , S ( π )) . First, by examining the deriva- tive of D ( π ) , i.e., ∂ D ( π ) /∂ π = (1 − π ) Ge − (1+ π ) , we ca n see that the continuous function D ( π ) is increasing in π ∈ [0 , 1] and decreasing in π ∈ [1 , + ∞ ] , and D ( π ) is maximized when π = 1 . Since S ( π ) always increases in π and D ( π ) is concave over π ∈ [0 , 1] , S ( π ) intersects with D ( π ) if and only if ∂ D ( π ) ∂ π > ∂ S ( π ) ∂ π at π = 0 , i.e. , B l + B s α < Ge − 1 . Next we divide our discussion into the intersection case and the non-intersection ca se: 1) Given B l + B s α ≤ Ge − 1 , S ( π ) intersects with D ( π ) . By solving equation S ( π ) = D ( π ) the intersection point is π = ln  G B l + B s α  − 1 . There are two sub- cases: • when B l + B s α ≤ Ge − 2 , S ( π ) intersects with D ( π ) , and min( D ( π ) , S ( π )) is maximized at the intersection point, i.e., π ∗ = ln  G B l + B s α  − 1 . (See S 3 ( π ) in Fig. 3.) • when B l + B s α ≥ Ge − 2 , S ( π ) intersects with D ( π ) , and min( D ( π ) , S ( π )) is maximized at the maximum value of D ( π ) , i.e., π ∗ = 1 . (See S 2 ( π ) in Fig. 3.) 2) Given B l + B s α ≥ Ge − 1 , S ( π ) doesn’t intersect with D ( π ) . Then min( D ( π ) , S ( π )) is maximized at the maximum value of D ( π ) , i.e., π ∗ = 1 . ( See S 1 ( π ) in Fig. 3.) A P P E N D I X B P R O O F O F T H E O R E M 2 Given the sensing result B s α , the objective of Stage II is to solve the decomposed two subproblems (10) and ( 11), a nd select the best one with better optimal performance. S ince R E S I I I ( B s , α, B l ) in subproblem (10) is linear ly d e creasing in B l , its optimal solution always lies at the lower boundary of the feasible set (i.e., B ∗ l = max { Ge − 2 − B s α, 0 } ). W e compare the optimal profits of two subproblems (i.e ., R E S I I ( B s , α ) and R C S I I ( B s , α ) ) for different sensing results: 1) Given B s α > Ge − 2 , the obtained bandwidth after Stage I is already in e xcessive supply regime. T hus it is optimal not to lea se for subproblem (10) (i.e., B E S 3 l = 0 of case (ES3 ) in T able 3). 2) Given 0 ≤ B s α ≤ Ge − 2 , the optimal leasing deci- sion for subproblem (1 1) is B ∗ l = Ge − 2 − B s α and we h a ve R E S I I I ( B s , α, B l ) = R C S I I I ( B s , α, B l ) when B l = Ge − 2 − B s α , thus the optimal objective value of (1 0) is always no larger than that of (1 1) and it is enough to consider the conservative supply regime only . Since ∂ 2 R C S I I I ( B s , α, B l ) ∂ B 2 l = − 1 B l + B s α < 0 , R C S I I I ( B s , α, B l ) is concave in 0 ≤ B l ≤ Ge − 2 − B s α . Thus it is enough to e x amine the first-order condi- tion ∂ R C S I I I ( B s , α, B l ) ∂ B l = ln  G B l + B s α  − 2 − C l = 0 , and the boundary condition 0 ≤ B l ≤ Ge − 2 − B s α . This results i n optimal l e asing d ecision B ∗ l = max( Ge − (2+ C l ) − B s α, 0) and leads to B C S 1 l = Ge − (2+ C l ) − B s α and B C S 2 l = 0 of ca ses (CS1) and (CS2) in T able 3 . By substituting B C S 1 l and B C S 2 l into R C S I I I ( B s , α, B l ) in T able 2, we derive the corresponding optimal profits R C S 1 I I ( B s , α ) a nd R C S 2 I I ( B s , α ) in T a ble 3. R E S 3 I I ( B s , α ) can also be obtained by s ubstituting B E S 3 l into R E S I I I ( B s , α, B l ) . A P P E N D IX C S U P P L E M E N TA RY P R O O F O F T H E O R E M 4 In this section, we prove that Observations 3 and 4 hold for the genera ca se (i.e., the general S NR regime and a general distributions o f α ). W e first sh ow that Observation 4 holds for the general case. C.1 Threshold structure of sensing It is not difficult to show that if the sens ing cost is much larger than the leasing cost, the operator has no incentive to sense but will d irectly lease. Thus the threshold structure on the sensing decision in Stage I still holds for the general case. W e ignore the details d ue to space limitations. C.2 Threshold structure of leasing Next we show the threshold structure on lea sing in Stage II a lso holds. Similar as in the proof of Theorem 1, we define D ( π ) = π G Q ( π ) and S ( π ) = π ( B s α + B l ) . • W e first show that D ( π ) is increasing when π ∈ [0 , 0 . 468] and decreasing when π ∈ [0 . 46 8 , + ∞ ) . T o see this, we take the first-order derivative of D ( π ) over π , D ′ ( π ) = 2 Q ( π ) 2 + Q ( π ) − (1 + Q ( π )) 2 ln(1 + Q ( π )) Q ( π ) 3 , which is p ositive when Q ( π ) ∈ [0 , 2 . 163) and nega- tive when Q ( π ) ∈ [2 . 163 , + ∞ ) . Since eq. (16) shows that Q ( π ) is increasing in π and π ( Q ) | Q =2 . 163 = 0 . 468 , a s a result D ( π ) is increasing in π ∈ [0 , 0 . 468] 13  1 ( ) S S 0 0.468 S 2 ( ) S S 1 th B S Fig. 12. Different intersection cases of S ( π ) and D ( π ) in the general SNR regime .  0.462 2 ( ) th l B C G B G l C G ' ( / ) D B G 0 Fig. 13. The relation between the nor malize d total band- width B / G and the derivativ e of the rev enue D ′ ( B /G ) . and decreasing in π ∈ [0 . 46 8 , + ∞ ) . In other words, D ( π ) is maximized at π = 0 . 468 . • Next we derive the operator ’s optimal pricing d e - cision in Stage III. Fi gure 12 shows two pos sible intersection cases of S ( π ) and D ( π ) . B th 1 is defined as the total bandwidth obtained in Stages I and II (i.e., B s α + B l ) such that S ( π ) intersects with D ( π ) at π = 0 . 4 68 . Here is how the optimal pricing is determined: – If B s α + B l ≥ B th 1 (e.g., S 1 ( π ) in Fig. 12), the op- timal price is π ∗ = 0 . 468 . The total supply is no smaller (and often ex c eeds) the total demand. – If B s α + B l < B th 1 (e.g., S 2 ( π ) in Fig. 12), the optimal price occurs a t the unique intersection point of S ( π ) and D ( π ) (where D ( π ) has a negative first-order deriva tive). The total supply equals total demand. • Now we a re ready to show the threshold structur e of the leasing decision. – If the sensing result from Sta ge I satisfies B s α ≥ B th 1 , then the operator w ill not lease. Thi s is because lea sing will only increase the total cost without increasing the revenue, since the optimal price is fixed at π ∗ = 0 . 468 and thus revenue is also fixed at D ( π ∗ ) . – Let us f ocus on the case where the sensing result from Sta ge I satisfies B s α < B th 1 . L et us de fine B = B s α + B l , then we have B = G/Q ( π ) and π = ln(1 + G/B ) − G/ ( G + B ) . This enables us to rewrite D ( π ) a s a function of total resource B only , D ( B ) = B  ln  1 + G B  − G G + B  . The first-order de rivative of D ( B ) is D ′ ( B ) = ln  1 + 1 B / G  − 1 1 + B / G − 1 (1 + B / G ) 2 , (20) which de notes the increase of revenue D ( B ) due to unit increase in bandwidth B . Since obtaining each unit bandwidth has a cost of C l in S tage II, the operator will only lease positive amount of bandwidth if and only if D ′ ( B s α ) > C l . T o facil- itate the discussion s, we will plot the f unction of D ′ ( B /G ) in Fig. 13, with the under standing that D ′ ( B /G ) = D ′ ( B ) G . The intersection point of B /G = 0 . 462 in Fig. 13 c orresponds to the point of π = 0 . 468 in Fig. 12. The positive pa rt of D ′ ( B ) on the left side of B /G = 0 . 46 2 in Fig. 1 3 corresponds to the part of D ( π ) with a negative first-order deriva tive in Fig. 12. For any value C l , Fig. 13 shows that there exists a unique threshold B th 2 ( C l ) such that D ′ ( B th 2 ( C l ) /G ) = C l G , i.e., D ′ ( B th 2 ( C l )) = C l . Then the optimal leasing amount will be B th 2 ( C l ) − B s α if the bandwidth obtained from sensing B s α is less than B th 2 ( C l ) , otherwise it will be ze ro. C.3 Threshold structure of pricing and Observation 3 Based on the proofs a bove, we show that Observation 3 also holds for the genera l case as follows. Let us denote the optimal sensing decision a s B ∗ s , and consider two sensing realizations α 1 and α 2 in time slots 1 a nd 2, respectively . W ithout loss of generality , we a ssume that α 1 < α 2 . • If B ∗ s α 2 ≥ B th 1 , then the optimal price in time slot 2 is π ∗ = 0 . 468 (see Fig. 1 2). The optimal price in time slot 1 is always no smaller tha n 0 . 468 . • If B ∗ s α 1 < B ∗ s α 2 < B th 1 , then we need to consider three subcases: – If B ∗ s α 1 < B ∗ s α 2 ≤ B th 2 ( C l ) , then the operator will lease up to the threshold in b oth time slots, i.e., B ∗ l = B th 2 ( C l ) − B ∗ s α 1 in time slot 1 and B ∗ l = B th 2 ( C l ) − B ∗ s α 2 in time slot 2. 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