DATABASE-ASSISTED spectrum sharing is a promising

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1 1 Optimal Pricing and Admission Control for Heterogeneous Secondary Users Cangkun Jiang, Student Member, IEEE, Lingjie Duan, Member, IEEE, and Jianwei Huang, Fellow, IEEE Abstract Tis paper studies ow to maximize a spectrum database operator s expected revenue in saring spectrum to secondary users, troug joint pricing and admission control of spectrum resources. A unique feature of our model is te consideration of te stocastic and eterogeneous nature of secondary users demands. We formulate te problem as a stocastic dynamic programming problem, and present te optimal solutions under bot static and dynamic pricing scemes. In te case of static pricing, te prices do not cange wit time, altoug te admission control policy can still be time-dependent. In tis case, we sow tat a stationary (time-independent) admission policy is in fact optimal under a wide range of system parameters. In te case of dynamic pricing, we allow bot prices and admission control policies to be time-dependent. We sow tat te optimal dynamic pricing can improve te operator s revenue by more tan 30% over te optimal static pricing, wen secondary users demands for spectrum opportunities are igly elastic. Index Terms Spectrum Pricing, Heterogeneous Demands. I. INTRODUCTION DATABASE-ASSISTED spectrum saring is a promising approac to improve te utilization of limited spectrum resources [2], [3]. In suc an approac, primary licensed users (PUs) report teir spectrum usage patterns to a spectrum database, wic uses te primary activity records to coordinate te opportunistic spectrum access of secondary unlicensed users (SUs). Several government regulators, suc as te FCC in te US and te Ofcom in te UK, strongly advocate suc an approac (e.g., for te saring of TV wite space) due to its ig reliability compared to sensing. Under suc an approac, te database can effectively coordinate SUs accesses, by mitigating tese SUs mutual usage conflicts and controlling te potential conflicts wit PUs. Toug researcers ave made significant researc progress in addressing various tecnical issues of spectrum database (e.g., database system management and spectrum allocation [4] [6]), very few studies looked Manuscript received at xxx, Revised at xxx, Accepted at xxx, Date of publication xxx, 2016; date of current version xxx, Tis work was supported by te General Researc Funds (Project Number CUHK and CUHK ) establised under te University Grant Committee of te Hong Kong Special Administrative Region, Cina. Te associate editor coordinating te review of tis paper and approving it for publication was xxx. Part of te results appeared in IEEE WiOpt, May 2014 [1]. Cangkun Jiang and Jianwei Huang (corresponding autor) are wit te Network Communications and Economics Lab, Department of Information Engineering, Te Cinese University of Hong Kong ( {jc012, jwuang}@ie.cuk.edu.k); Lingjie Duan is wit te Engineering Systems and Design Pillar, Singapore University of Tecnology and Design, 8 Somapa Road, Singapore ( lingjie duan@sutd.edu.sg). Color versions of one or more of te figures in tis paper are available online at ttp://ieeexplore.ieee.org. Digital Object Identifier xxx at te economic issue of spectrum database (e.g., [4] [6]). Witout a proper economic mecanism, te database operator may not ave enoug incentives to coordinate te spectrum saring process. Tis motivates us to explore te revenue maximization problem for a spectrum database, in particular, te admission of SUs and pricing of idle spectrum resources. Tere are two key callenges wen considering suc a revenue maximization problem for te database operator. First, SUs demands can be eterogeneous in terms of spectrum occupancy. For example, a large file (e.g., video) downloading takes minutes or even ours to finis (ence we call eavytraffic), wile sending a sort text message or accessing location-based services can be completed in seconds (ence we call ligt-traffic). Second, SUs demands are often randomly generated over time. Te eterogeneity and randomness make it difficult for te operator to accurately predict future demands and make proper allocation decisions. Yet te two issues ave not been fully considered in te previous studies (e.g., [4] [6]). To address te above two callenges, we propose a joint spectrum pricing and admission control sceme for te database operator to maximize its expected revenue. Te optimization is over te time period during wic te spectrum cannel is available for SUs to opportunistically access due to te lack of PU activities. Te period is divided into several time slots, and te database operator needs to determine te optimal prices for different types of SUs (e.g., in eavyand ligt-traffic types) in eac time slot. Tese prices can be fixed (static pricing) or vary over time (dynamic pricing), and will affect ow SUs request to access te spectrum. However, pricing alone may not be enoug to mitigate te conflicts between multiple SUs wo want to access to te limited spectrum at te same time. Te operator also needs to determine te optimal admission control policy to control te total demand. Te pricing and admission decisions need to be jointly optimized to acieve te maximum performance. Tere are several recent results focusing on te spectrum pricing issues of a spectrum database (e.g., [4] [6]). Tese studies focused primarily on te static pricing wit complete information in te spectrum database system, witout considering te eterogeneous and stocastic SU demands as in our work. Under static pricing, te pricing decisions do not cange over time (e.g., [4] [11]). In particular, Duan et al. in [9] considered te static pricing of spectrum under supply uncertainty, and extended to te competitive duopoly pricing in [10]. Te literature on general dynamic pricing focuses on dynamic pricing decisions of selling a given stock of items by a deadline (e.g., [12], [13]), and in particular, pricing decisions of airline seats and otel rooms booking (e.g., [14], [15]).

2 2 Te literature on dynamic pricing of wireless resources only emerged recently (e.g., [16] [18]). Song et al. in [16] studied te network revenue maximization problem by using dynamic pricing for multiple flows in a wireless multi-op network. Ha et al. in [17] proposed time-dependent pricing to decrease customers congestion cost. Ma et al. in [18] proposed time and location based pricing for mobile data traffic. However, in our work spectrum as its unique features to be priced and used. Unlike a traditional product, te unused spectrum resource cannot be stored and is immediately wasted. Moreover, SUs demands are often eterogeneous and random over time. Te caracterizations of bot te time-sensitivity of spectrum and te SUs demand eterogeneity and uncertainty make our model and analysis different from prior studies. We tackle tese issues by jointly considering te admission control of SUs and te dynamic pricing of spectrum in a dynamic setting over time, and caracterize te conditions under wic te often complicated optimal pricing and admission decisions degenerate to te stationary pricing and admission scemes. To our best knowledge, tis is te first work tat jointly prices and allocates te spectrum resource in a dynamic setting to serve eterogeneous and stocastic SU demands. We formulate te operator s revenue maximization problem as a stocastic dynamic programming problem, wic is in general callenging to solve. Our main results and key contributions are summarized as follows. Optimal static pricing and dynamic admission policies. We first constrain ourselves to te simple and widely used approac of static pricing, meanwile allowing dynamic time-dependent admissions. We sow tat te complex optimal dynamic admission policy often degenerates to a tresold-based stationary (time-independent) policy under a wide range of system parameters. Optimal dynamic pricing and dynamic admission policies. We furter allow te prices to dynamically cange over time based on different SUs stocastic demands. Altoug te optimal prices and admission decisions are coupled, we are able to compute te optimal policy troug a proper price-and-admission decomposition in eac time slot. Similarly, we sow tat te optimal admission policy often degenerates to a stationary admission policy under a wide range of system parameters. By comparing te optimal pricing and admission policies under bot static and dynamic pricing scemes, we sow tat te dynamic pricing sceme can significantly improve te database operator s revenue (by more tan 30%) wen SU s demands are igly elastic. Te rest of te paper is organized as follows. We introduce te model and problem in Section II. In Section III, we formulate and solve te optimal static pricing and dynamic admission control problem. In Section IV, we furter consider te joint optimization of dynamic pricing and dynamic admission control. In Section V, we extend our model and results to te more general cases of SU types. We sow te simulation results in Section VI and conclude te paper in Section VII. II. SYSTEM MODEL AND PROBLEM FORMULATION We consider a database operator wo records PUs activities and knows a cannel tat will not used by PUs during a set N = {1,,N} of consecutive time slots, similar as in [2], [3]. Te database operator wants to maximize its revenue troug selling te temporary spectrum opportunities to te SUs. Te duration of tis wole time period depends on te type of PU traffic, and is known in advance as te PUs need to register all traffic wit te database (e.g., [2], [3]). SUs randomly arrive and request cannel access at te beginning of eac time slot. To gain clear insigts into te admission policies of SUs, we first assume tat tere are two types of SUs depending on te lengt of te cannel access time. A ligt-traffic SU only needs to use te cannel for one time slot, and a eavy-traffic SU needs to occupy two consecutive time slots. In Section V, we will extend our analysis to te case were a eavy-traffic SU occupies more tan two consecutive time slots, and we will sow tat our main results do not cange. Furtermore, we will furter consider te general case were (i) tere are an arbitrary number of SU types, and (ii) eac type of SU may access a cannel for an arbitrary number of time slots. If an SU is admitted in n N, te database operator will carge te SU eiterr l (n) orr (n), depending on weter it is a ligt- or eavy-traffic SU. SUs are price-sensitive, and teir demand probabilities of requesting te spectrum after arriving are non-increasing in te prices. 1 Since we consider a single cannel case, te database operator can admit at most one SU in any time slot. Once an SU s service request is rejected by te database, it will leave te system witout waiting. Tis corresponds to te case were SUs ave delay-intolerant applications suc as VoIP and video conferencing. Fig. 1 summarizes te database s operations in our model. At te beginning of eac time slot n N, te database operator first announces prices r l (n) and r (n) for te ligtand eavy-traffic SU types, respectively. Ten SUs observe te price update and randomly arrive wit te probabilities affected by te prices. 2 Finally, te database operator admits at most one SU to te cannel (if te cannel is available) and rejects te oter SUs (if any). After te tree pases, te admitted SU will transmit data over te cannel during te rest of te slot. To maximize te expected revenue, te database operator wants to jointly optimize spectrum prices and admissions over all N time slots. In tis optimization problem, te database operator s decision of admitting a eavy-traffic SU will prevent admitting a ligt/eavy-traffic SU (if available) in te next time slot, ence te operation decisions over time are coupled. We will model te problem as a stocastic dynamic programming problem, and propose te optimal admission policies under 1 We in fact consider two different arrival processes. Te first process describes ow te SUs arrive at te system, wic can be any process suc tat tere is at least one arrival at te beginning of eac time slot. Te second process caracterizes ow te arrived SUs request spectrum access from te database. Suc a process depends on te prices set by te operator, as a iger price will reduce te demand from SUs. 2 In addition to prices, te wireless cannel condition will also influence eac SU s request for te spectrum. Due to page limit, we put te detailed cannel modeling, analysis, and simulation results into Appendix A of [23].

3 3 1 n Pricing SUs Demand Admission SU Access (S 1,X 1,Y 1) Pricing SUs Demand Admission SU Access (S n,x n,y n) N Pricing SUs Demand Time Horizon Admission SU Access (S N,X N,Y N) Fig. 1. Database operation in N time slots. At te beginning of eac time slot, te database operator announces prices for incoming SUs. After observing te realized demands, te database operator ten makes te admission decision and inform te selected SU to access. Te notation (S n,x n,y n) denotes te resultant cannel occupancy and two SU types demand realizations (will be explained in Subsection III-A). static pricing in Section III and under dynamic pricing in Section IV, respectively. In bot sections, we allow dynamic admission decisions over time. Notice tat static pricing is a special case of dynamic pricing, and is widely used in industry due to its simplicity and low complexity. Hence, we are interested in exploring te benefits tat te flexible dynamic pricing may bring beyond te simplified static pricing. Ten we can provide insigts into wic pricing and admission sceme te operator sould coose and under wat conditions. III. OPTIMAL STATIC PRICING AND DYNAMIC ADMISSION We first consider te case of static pricing, were prices do not cange over time. It will serve as a bencmark and elp us quantify te performance gain by using dynamic pricing in Section IV. Wit static pricing, te database only needs to optimize and announce prices once at te beginning of time slot 1, and keeps te prices fixed for te rest N 1 time slots, i.e., r l (n) = r l and r (n) = r for eac time slot n N. We will formulate te revenue maximization problem wit static pricing and dynamic admission as a stocastic dynamic programming problem. In Subsections III-A to III-C, we will formulate and solve te optimal admission control problem troug backward induction, given any fixed prices. In Subsection III-D, we will optimize te static prices, considering te admission policies developed in Subsections III-A to III-C. A. Admission Control Formulation under Fixed Prices Given fixed prices r l and r, we now optimize te cannel admission decision in eac time slot. Suc optimization not only considers te cannel availability and SU demands in te current time slot, but also considers SU demands in future time slots. We will formulate it as a stocastic dynamic programming problem. We first define te system state as follows. Definition 1 (System State): Te system state in time slot n is (S n,x n,y n ). Here, S n denotes te number of remaining occupied time slots at te beginning of time slot n. Since S n {0,1}, S n also indicates te binary cannel state, were S n = 0 denotes tat te cannel is available for admission in time slot n, and S n = 1 oterwise. Te parameter X n = 1 denotes tat at least one ligt-traffic SU arrives at te beginning of te time slot (and is willing to pay for price r l ), and X n = 0 oterwise. Te parametery n is defined similarly as X n but for te eavy-traffic SUs. We define te SU demand probabilities in time slot n as p l = Pr{X n = 1} and p = Pr{Y n = 1}, respectively. As prices are uncanged over time, p l and p are te same for all time slots. Te system state canges over time, depending on te cannel admission decisions and SUs demand realizations over time. Te feasible set of admission actions in eac time slot depends on te current system state. Formally, we define te state-dependent feasible admission action set as follows. Definition 2 (Admission Action Set): Te set of feasible admission actions in time slot n is a state-dependent set A n (S n,x n,y n ). Wen S n = 1, i.e., te current time slot is not available for new admission as we are still serving te eavy-traffic SU from te last time slot, we ave A n (1,X n,y n ) = {0} for all possible (X n,y n ). Wen S n = 0, te admission action set depends on wic type of SUs demands in te current time slot. If no SUs request in time slot n (i.e., (X n,y n ) = (0,0)), te set of actions is still A n = {0}. If bot ligt- and eavy-traffic SUs demand, i.e., (X n,y n ) = (1,1), ten we can eiter serve no SU, a ligttraffic SU, or a eavy-traffic SU, and tus te set of actions is A n = {0,1,2}. To summarize, A n (0,X n,y n ) = {0}, if (X n,y n ) = (0,0), {0,1}, if (X n,y n ) = (1,0), {0,2}, if (X n,y n ) = (0,1), {0,1,2}, if (X n,y n ) = (1,1). We furter define te specific admission decision in time slot n as a n A n (S n,x n,y n ). Now we are ready to introduce te state dynamics. Wen S n = 1, we will not admit any SU, ence in te next time slot S n+1 = S n 1 = 0, as te remaining occupied time slots decreases by one. Wen S n = 0, te cannel availability of te next time slot only depends on te action a n. If we admit te ligt-traffic SU wit a n = 1, ten te cannel is available in te next time slot (as te remaining occupied time slot is 0), i.e., S n+1 = a n 1 = 0. If we admit te eavy-traffic SU wit a n = 2, it will occupy two time slots (time slots n and n+1). Tis means tat at te beginning of time slot n+1, we will ave te number of remaining occupied time slot to be 1, i.e., S n+1 = a n 1 = 1. At te beginning of time slot n+2, tere is no SU occupying te cannel, ence S n+2 = S n+1 1 = 0 and time slot n+2 is available for admission. To summarize, we derive te following state dynamics. Lemma 1 (State Dynamics): Te dynamics of te system state component S n for eac time slot n N satisfies te following equation: (1) S n+1 = (S n +a n (1 S n ) 1) +, (2) were (x) + := max(0,x), and S n {0,1} for eac n N. Lemma 1 captures te cange of remaining occupied time slots. Te system state components (X n,y n ) are te realizations of SU demands in te current time slot, and do not depend on te action a n in previous time slots. Te key notations we introduced so far are listed in Table I.

4 4 Symbols TABLE I KEY NOTATIONS Pysical Meaning N = {1,,N} Set of time slots (S n,x n,y n) System state in time slot n a n(s n,x n,y n) and a n Admission action in time slot n A n(s n,x n,y n) Set of feasible admission actions in time slot n r(a n) Immediate revenue by te admission action a n R n(s n,x n,y n,a n) Total revenue from time slot n to N E[R n (Sn,Xn,Yn)] and R n (Sn) Optimal expected future revenue from n to N π (S n,x n,y n) Optimal admission strategy in time slot n π = {π (S n,x n,y n),n N} Optimal admission policy for all time slots R n (r l,r ) Total revenue from n to N as a function of prices r l (n),r (n) Price for ligt/eavy-traffic SUs in time slot n R n(r l (n),r (n)) Expected future revenue from time slot n to N p l (r l (n)),p (r (n)) Probabilities of aving at least one ligt- and eavy-traffic SU requesting spectrum in n k l,k Demand elasticity of ligt/eavy-traffic SUs I = {1,,I} Set of SUs types in te multiple types case X (i) n, i I Demand of type-i SUs in time slot n We are now ready to introduce te revenue maximization problem. We define a policy π = {a n (S n,x n,y n ), n N} as te set of decision rules for all possible states and time slots, and we let Π = {A n (S n,x n,y n ), n N} be te feasible set of π. Given all possible system state vectors S = {S n, n N}, X = {X n, n N}, and Y = {Y n, n N}, te database operator aims to find an optimal policy π (from te set of all admission policies Π) tat maximizes te expected total revenue from time slot 1 to N. Formally, we define Problem P1 as follows. P1: Revenue Maximization by Dynamic Admission maximize E π X,Y [R(S,X,Y,π)] (3) subject to a n (S n,x n,y n ) A n (S n,x n,y n ), n N, (4) S n+1 = (S n +a n (1 S n ) 1) +, n N \{N}, (5) variables π = {a n (S n,x n,y n ), n N}, (6) were te expectation in te objective function is taken over SUs random requests (X,Y ). We proceed to analyze Problem P1 by using backward induction [19]. After SUs demands X n and Y n are realized in time slot n, te operator makes te admission action a n to maximize te total revenue by considering future SU demands. We define te total revenue from time slot n to N as R n (S n,x n,y n,a n ). Te total revenue computed in time slot n as two parts: i) te immediate revenuer(a n ) for te current admission action a n, were r(a n ) = 0, r l, or r if a n = 0, 1, or 2, respectively; and ii) te expected future revenue from time slot n+1 to N, i.e., E[R n+1 (S n+1,x n+1,y n+1 )], were te expectation is taken over te SUs possible demands in te next time slot n+1, i.e., (X n+1,y n+1 ). Ten te optimization problem of time slot n in te backward induction process is R n(s n,x n,y n ) = max an A n R n (S n,x n,y n,a n ), (7) were te revenue s dynamic recursion is R n (S n,x n,y n,a n ) = r(a n )+E[R n+1 (S n+1,x n+1,y n+1 )]. (8) As a boundary condition in te last time slot N, we ave R N (S N,X N,Y N,a N ) = r(a N ), as tere is no future spectrum opportunity and revenue collection after time slot N. Te maximum expected revenue from time slot n to N is denoted by E Xn,Y n [R n (S n,x n,y n )], wic is a part of te revenue and will be utilized for admission decision-making in previous time slots. Since te expectation E Xn,Y n [R n (S n,x n,y n )] is taken over all possible SU demand combinations (X n,y n ), we rewrite it as R n(s n ), n N for simplicity. We derive te expected total revenue R n (S n,x n,y n,a n ) by adding te immediate revenue as a result of action a n and te corresponding expected future revenue R n+1 (S n+1) (if a n = 0 or 1, i.e., no admission or admitting a ligt-traffic SU) or R n+2 (S n+2) (if a n = 2, i.e., admitting a eavy-traffic SU), considering all possible SU demands (X n,y n ) in time slot n: R n (S n,x n,y n,a n ) = (1 p l )(1 p )[0+ R n+1(s n+1 )] +p l (1 p )[r l + R n+1 (S n+1)] +(1 p l )p [(0+ R n+1(s n+1 )) 1 {an=0} +(r + R n+2 (S n+2)) 1 {an=2}] +p l p [(r l + R n+1(s n+1 )) 1 {an=1} +(r + R n+2 (S n+2)) 1 {an=2}], (9) wic can be computed according to (7) and (8) recursively and backwardly from time slot N ton. Later, we will calculate R n (S n,x n,y n,a n ) by setting te specific values of a n in te last two terms of (9) according to te different admission strategies for time slot n. Next, we will solve Problem P1 using (7)-(9). B. Optimal Dynamic Admission Control By using backward induction [19], we start wit te final time slot N and derive te optimal decisions slot by slot back. In time slot n, te admission decision is made by comparing te corresponding total revenues R n (S n,x n,y n,a n ) for different admission a n in time slot n. Based on te above discussions, we propose te optimal dynamic admission control policy in Algoritm 1. More specifically, tis control policy π is developed by solving Problem P1 using standard backward induction mentioned earlier. In te following Cases I-III, we formally compare te immediate revenue plus te expected future revenue to make admission decisions (i.e., r + R n+2(s n+2 ), r l + R n+1(s n+1 ), and 0+ R n+1 (S n+1)): In Case I (lines 5-6) of Algoritm 1, it is more beneficial for te operator to admit a eavy-traffic SU (if it exists) tan a ligt-traffic SU. In Case II (lines 7-8) of Algoritm 1, it is more beneficial for te operator to admit a ligt-traffic SU (if it exists) tan a eavy-traffic SU. In Case III (lines 9-10) of Algoritm 1, it is more beneficial for te operator to only admit a ligt-traffic SU (if it exists).

5 5 Algoritm 1: Optimal Admission Control Policy 1: Set n = N, R N+1 = 0 2: Te optimal admission for N is a N = XN and R N = p lr l 3: for n = N 1,,2,1 do 4: Calculate R n+1 (Sn+1) using (9). 5: if r + R n+2 (Sn+2) r l + R n+1 (Sn+1) ten 6: if Y n = 1, ten a n = 2; if Y n = 0,X n = 1, ten a n = 1; oterwise a n = 0. 7: else if R n+1 (Sn+1) < r + R n+2 (Sn+2) < r l + R n+1 (Sn+1) ten 8: if X n = 1, ten a n = 1; if X n = 0,Y n = 1, ten a n = 2; oterwise a n = 0. 9: else 10: if X n = 1, ten a n = 1; oterwise a n = 0. 11: end if 12: end for 13: return te optimal admission policy π By te principle of optimality [19], π = {π (S n,x n,y n ),n N} is optimal, as sown in te following proposition. Proposition 1: Algoritm 1 solves Problem P1 and computes te optimal admission policy π. Te proof of Proposition 1 is given in Appendix B of [23]. Note tat te optimal policy π is a contingency plan, wic contains te optimal admission policy in eac time slot n N for any system state. After deriving te optimal policy, we can implement te policy forwardly from time slots 1 to N, after observing SUs demand realizations. C. Stationary Admission Policies Te optimal admission control solution in Algoritm 1 does not ave a closed-form caracterization and te system still needs to ceck a uge-size table created from te algoritm after knowing te realizations of SU random demands. Tis motivates us to focus on a class of low complexity stationary admission policies, were te admission rules do not cange over time (wile te actual admission decisions migt cange over time). We will caracterize te conditions under wic tese stationary admission policies are optimal. Recall tat tere are tree possible admission strategies in eac time slot, depending on te values of r + R n+2 (S n+2), r l + R n+1(s n+1 ), and 0+ R n+1(s n+1 ). For a particular time slotn, for example, ifr + R n+2 (S n+2) > r l + R n+1 (S n+1) > 0+ R n+1(s n+1 ), we prefer to serve te eavy-traffic SU type rater tan te ligt-traffic one or not serving anyone (i.e., te admission priority follows Λ(2) > Λ(1) > Λ(0)). Here, we define te function Λ(a n ) to capture te priority order of te admission action a n {0,1,2}. Due to te fact r l + R n+1 (S n+1) > 0 + R n+1 (S n+1) and serving a ligt-traffic SU is better tan serving no one, tere are a total of tree reasonable admission priority orders, i.e., Λ(2) > Λ(1) > Λ(0), Λ(1) > Λ(2) > Λ(0), and Λ(1) > Λ(0) > Λ(2). We discuss tem one by one next. Table II sows te tree stationary policies tat we will discuss. Recall tat wens n = 1 (i.e., cannel is still occupied in te current time slot), we ave a n = 0 (not admitting any SU) for any values of X n and Y n. Table II only focuses on te case of S = 0. Te tree rows/sub-tables, namely, Tab.II : a n, Tab.II LP : a LP n, and Tab.II LD: a LD n, represent te Heavy-Priority (i.e., Λ(2) > Λ(1) > Λ(0)), Ligt-Priority Admission Policies TABLE II THREE STATIONARY ADMISSION POLICIES System states (S n,x n,y n) (0,0,0) (0,0,1) (0,1,0) (0,1,1) Tab.II : a n Tab.II LP : a LP n Tab.II LD: a LD n (i.e., Λ(1) > Λ(2) > Λ(0)), and Ligt-Dominant (i.e., Λ(1) > Λ(0) > Λ(2)) admission policies, respectively. For eac policy, we will derive te conditions of te static prices r l and r, under wic te policy acieves te optimality of Problem P1. We first analyze te Heavy-Priority admission policy (in Tab.II : a n, n N ). Under tis policy, we will serve a eavy-traffic SU (a n = 2) wenever possible (Y n = 1), and only serve a ligt-traffic SU (a n = 1) wen tere is only a ligt-traffic SU (X n = 1 and Y n = 0). 3 Suc a stationary policy is optimal if te following two conditions old for eac and every time slot n {1,,N 1}, r + R n+2(0) 0+ R n+1(0), (10) r + R n+2 (0) r l + R n+1 (0). (11) Inequality (10) sows tat serving a eavy-traffic SU wo occupies two consecutive time slots leads to a iger expected total revenue tan serving no SU in te current time slot. Inequality (11) sows tat serving a eavy-traffic SU leads to a iger expected total revenue tan serving a ligt-traffic SU in te current time slot. Since (11) ensures (10), we only need to consider (11). Similarly, we can derive te condition under wic te Ligt-Priority admission policy (in Tab.II LP : a LP n ) is optimal, i.e., 0 + R n+1 (0) < r + R n+2 (0) < r l + R n+1 (0) for all n {1,,N 1}. Under tis policy, we will admit a ligt-traffic SU wenever possible (X n = 1), and admit a eavy-traffic SU oterwise (X n = 0 and Y n = 1). Finally, we can derive te condition under wic te Ligt- ) is optimal, (0) for all n {1,,N 1}. Under tis policy, we will coose to admit a ligt-traffic SU (a n = 1) wenever possible (X n = 1), and will never admit any eavy-traffic SU, as it leaves no room to accept a ligttraffic SU in te next time slot. To summarize, we ave te following teorem. Recall tat r /r l denotes te ratio between te prices carged to te eavy-traffic and te ligt-traffic SUs, and p l and p are te demand probabilities defined in Subsection III-A. Teorem 1: A stationary admission policy becomes te optimal policy to solve Problem P1 if one of te following condition is true: Dominant admission policy (in Tab.II LD: a LD n i.e., r + R n+2 (0) 0+ R n+1 Te Heavy-Priority admission policya n in Tab.II for alln N is optimal ifr /r l 2p l +(1 p l )/(1 p ). Te Ligt-Priority admission policy a LP n in Tab.II LD for all n N is optimal if p l r /r l 1+p l. 3 Note tat te discussion is only meaningful for time slot 1 to N 1, as in te last time slot N we will admit a ligt-traffic SU wenever possible.

6 6 0 II: Ligt-Priority Admission Policy p l I: Ligt-Dominant Admission Policy 1 + p l 2p l + 1 p l 1 p III: Algoritm 1 IV: Heavy-Priority Admission Policy Fig. 2. Optimal stationary admission policies for all price ratio r /r l values (regimes I, II, and IV). Te Ligt-Dominant admission policy a LD n Tab.II LP for all n N is optimal if r /r l < p l. Te proof of Teorem 1 is given in Appendix C of [23]. Te teorem sows tat eac of te tree stationary policies is optimal witin a particular range of te price ratio r /r l. Fig. 2 illustrates te results of Teorem 1 grapically. In tis figure, we divide te feasible range of te price ratio r /r l into four regimes, among wic in tree regimes (I, II, and IV) te stationary policies are optimal. We are able to furter caracterize te closed-form optimal total revenues for tese tree regimes, and te details can be found in Appendix C [23]. It is clear tat a larger value of r /r l gives a iger preference to te admission of a eavy-traffic SU. In regime III, we ave to use Algoritm 1 to compute te optimal admission policy. After analyzing te optimal admission control decisions from time slot N to 1 in backward induction, we now optimize te initial pricing decision at te beginning of time slot 1. D. Optimal Static Pricing Under static pricing, te database operator optimizes and announces te prices r and r l in time slot 1, and do not cange tese prices for te remaining N 1 time slots. As explained in Section II, we consider te general case were prices will affect SU demands during eac time slot. As a concrete example, we consider te widely used linear demand function in economics [20], were te probability of an SU of type i {l,} requesting te spectrum resource in a time slot is p i (r i ) = 1 k i r i, were 0 r i ri max = 1/k i. 4 Te parameters k l and k caracterize te demand elasticity of te ligt-traffic and te eavy-traffic SUs, respectively, and larger values of k l and k reflect iger price sensitivities. 5 By using te tree stationary admission policies in Teorem 1, we are able to derive tree closed-form objectivern,n N as a function of prices r l and r. Next we optimize te prices tat maximize te total revenue R1 in Problem P1. Proposition 2: Consider te case r /r l 2p l +(1 p l )/(1 p ), in wic te eavy-priority admission policy is optimal as sown in Teorem 1. Te optimal static pricing (rl,r ) is 4 Canging to common nonlinear functions are unlikely to cange te key results. Tis is because te optimal static pricing can be solved in Proposition 2, even for nonlinear functions, we can still searc te optimal static pricing. 5 In practice, te price elasticity parameters can be estimated according to te market survey or istorical data about demand responses (e.g., [21]). By doing independent repeated trials, te operator can estimate te parameters. r rl in te optimal solution to te following problem maximize R 1 (r l,r ), (12) subject to r /r l 2p l +(1 p l )/(1 p ), (13) 0 r l rl max,0 r r max, (14) variables r l,r, (15) were ( ) R1(r l,r ) = N pl r l +(1 p l )p r + 1 (p l p p ) ( (pl p p )(r p l r l ) 1 (p l p p ) ) (pl p p )(1 (p l p p ) N ) 1 (p l p p ). (16) Te proof of Proposition 2 is given in Appendix D of [23]. Te same conclusion olds for te oter two cases sown in Teorem 1, and te details are provided in [23]. Te function R 1 (r l,r ) turns out to be non-convex in general, and te optimal prices cannot be solved in closed form. However, notice tat te key benefit of static pricing is tat it does not need to be recomputed and updated frequently over time, tus we can compute te optimal static prices offline once and te ig computational complexity is not a major practical issue. IV. OPTIMAL DYNAMIC PRICING AND DYNAMIC ADMISSION In Section III, we ave considered te static pricing and dynamic admission control problem. Now we consider te case of dynamic pricing, were te prices vary over time. In te following, we will formulate te dynamic pricing and dynamic admission control problem, aiming at deriving te optimal dynamic pricing and admission policies. A. Dynamic Pricing-and-Admission Problem Formulation Now we furter study te general case of dynamic pricing, were te database operator as te flexibility of canging prices over time. Te database operator s goal is to compute te optimal prices r l = {rl (n),n N} and r = {r (n),n N}, and te optimal admission policy π = {a n(s n,x n,y n ),n N} for all time slots and system states to maximize its expected revenue, i.e., P2: Joint Dynamic Pricing and Dynamic Admission maximize E π X,Y [R(S,X,Y,π,r l,r )] (17) subject to a n (S n,x n,y n ) A n (S n,x n,y n ), n N, (18) S n+1 = (S n +a n (1 S n ) 1) +, n N \{N}, (19) 0 r l (n) rl max, n N, (20) 0 r (n) r max, n N, (21) variables {π,r l,r }. (22) We can again use backward induction to solve Problem P2 in eac time slot. Different from Section III, we need to jointly

7 7 TABLE III THREE ADMISSION STRATEGIES IN TIME SLOT n Admission Strategies in time slot n Conditions Heavy-Priority Strategy (): a n = (2 X n)y n +X n r (n)+ R n+2 r l(n)+ R n+1 Ligt-Priority Strategy (LP): a n = r (n)+ R n+2 > 0+ R n+1 X n 1 {Yn=0} +(2 X n) 1 {Yn=1} r (n)+ R n+2 < r l(n)+ R n+1 Ligt-Dominant Strategy (LD): a n = X n r (n)+ R n+2 0+ R n+1 determine te prices and te admission decisions from time slot N to 1. Te subproblem in eac time slot n N is P3: Pricing-and-Admission Subproblem in time slot n maximize R n ( rl (n),r (n),a n (S n,x n,y n ) ) (23) subject to a n (S n,x n,y n ) A n (S n,x n,y n ), (24) 0 r l (n) rl max, (25) 0 r (n) r max, (26) variables {a n (S n,x n,y n ),r l (n),r (n)}. (27) Te expression of R n can be similarly derived by using te derivation procedure of (9), except tat te demand probabilities p l and p are functions te prices r l (n) and r (n), i.e., p l (r l (n)) and p (r (n)), respectively. Te key callenge of solving Problem P3 is te coupling between te pricing and admission decisions in eac time slot. Next we will propose a decomposition approac for te two decisions tat elps us solve Problem P3 in eac time slot n. B. Decomposition of Pricing and Admission in Eac Time Slot First we want to clarify te difference between an admission strategy and an admission policy. An admission strategy specifies te admission actions for a particular time slot n, wile an admission policy applies to all time slots in N (e.g., tose in Table II). Here we will focus on te admission strategy, as we only study Problem P3 for a particular time slot n. Next we consider all possible admission control strategies for a time slot n, as sown in Table III. In tis table, stands for eavy-priority strategy, LP stands for ligt-priority strategy, and LD stands for ligt-dominant strategy. Eac strategy is accompanied by a condition of te total revenue from time slot n to N. Te strategy is optimal for time slot n if te corresponding condition olds. Let us take te eavy-priority strategy () as an example to explain our decomposition approac. In tis strategy, we will serve a eavy-traffic SU (a n = 2) wenever possible (Y n = 1), and only serve a ligt-traffic SU (a n = 1) if tere is no eavytraffic SU (X n = 1 and Y n = 0). Summarizing tese cases togeter, te decision under te eavy-priority strategy can be written as a n = (2 X n )Y n + X n. Te corresponding condition for te eavy-priority strategy in Table III sows tat te total revenue of admitting a eavy-traffic SU is no less tan tat of admitting a ligt-traffic SU. Te conditions for te oter two admission strategies (LP and LD) can be derived similarly. Using te result in Table III, we can solve Problem P3 in te following two steps: Price optimization under a cosen admission strategy: Assume tat one of te tree admission strategies in Table III will be used in time slot n, we optimize prices r l (n) and r (n) to maximize te expected total revenue. Admission strategy optimization: Compare te maximized expected total revenues (from slot n to N) under te tree admission strategies wit te optimized prices, and pick te best admission strategy and pricing combination tat leads to te largest revenue. Notice tat te above decomposition metod is for eac time slot n N. Te above decomposition procedure guarantees tat we obtain te optimal solution of te joint problem P3 for te following reason. First, te tree possible admission strategies in eac time slot are exaustive and mutually exclusive, in te sense tat te optimal pricing decision in time slot n guarantees tat tere is only one strategy tat is optimal to adopt in tis time slot, depending on te conditions in Table III. Second, if one-out-of-te-tree admission strategies is optimal to adopt in time slot n, tere must exist an associated optimal pricing accordingly tat maximizes te total revenue. We tus conclude tat te two-step decomposition procedure is guaranteed to solve Problem P3 optimally. Next, we will derive te closed-form optimal pricing under eac of te tree admission strategies, respectively. We will conduct admission strategy optimization in Subsection IV-C. 1) Optimal Pricing under Heavy-Priority Strategy: Given strategy cosen in time slot n, we derive te expected ( r total revenue Rn l (n),r (n) ) by setting a n = 2 and a n = 2 in te last two terms of (9), respectively, were te (n) and p (r (n)) can also be modeled probabilitiesp l (rl as te linear function in Subsection III-D. Notice tat te database operator may not always use in future time slots. In order to optimize te prices, te database operator needs to solve te following problem. P4: Optimal Pricing for time slot n under maximize R n ( r l (n),r (n) ) (28) l (n)+ R n+1, (29) 0 rl (n) rl max, (30) 0 r (n) r max, (31) subject to r (n)+ R n+2 r variables r l (n),r (n). (32) Constraint (29) guarantees tat te eavy-priority admission strategy is optimal in time slot n, were R n+2 and R n+1 are determined by te optimal solutions to Problem P3 in time slots n + 2 and n + 1. Since te optimization problem P4 is a continuous function over a compact feasible set, te maximum is guaranteed to be attainable. It is easy to sow tat Problem P4 is not a convex optimization problem due to te tree-order polynomial objective function. Tus, a solution satisfying KKT conditions may be eiter a local optimum or a global optimum. Hence, we need to find all solutions satisfying KKT conditions, and ten compare tese solutions to find te global optimum. We will first examine te feasible region of Problem P4 based on any possible prices rl (n) and r (n). It turns out tat te feasible region is a polyedron in a two-dimensional

8 8 large r max small r max r (n) 0 F r max l r F (n) = r l (n) + R n+1 R n+2 rl (n) k kl < k TABLE IV OPTIMAL PRICING UNDER HEAVY-PRIORITY STRATEGY 4k l 3k 4k k l R n+1 R n+2 4k l 3k, 2 4k k l 1+ k kl k 2 1+ k kl I0 E1 E2 1 E2 kl < 3 N/A E k kl 3 N/A N/A E2 k Fig. 3. Te feasible region F of Problem P4. Tere are two possible sapes according to te values of r max. Te interior points, te red dots, and te line segments between tem are all possible solutions. plane. Fig. 3 sows te feasible region. According to te value of r max, te feasible region as two possible cases. Te optimal solution can only be eiter te interior points inside te feasible region or te extreme points on te boundary. As suc, we only need to ceck weter all te possible extreme points and te interior points satisfying KKT conditions are local optima. We skip te details (wic can be found in [23]) due to space limit, and summarize te optimal pricing results in te following proposition. Proposition 3: Te optimal pricing in time slot n under te strategy is summarized in Table IV, wic depends on te values of R n+1 R n+2 and k /k l. Te closed-form optimal pricing solutions in Table IV are given as follows, respectively, { r I0 l (n) = 1 2k ( l ) : r 1 4k + 1 (n) = l k + R n+1 R, n+2 E1 : { r E2 l (n) = 1 k : + R n+2 R n+1 r (n) = 1 k l (n) = ( R n+1 R n+2 )+ r r (n) = 2( R n+1 R n+2 )+ 2, and ( R n+1 R n+2 )2 + 3 k l k 3 ( R n+1 R n+2 )2 + 3 k l k 3 Te proof of Proposition 3 is given in Appendix E of [23]. In Table IV, I0, E1, and E2 represent te unique optimal solution in different cases (i.e., one interior point solution and two extreme point solutions). N/A represents te cases were te combinations of conditions are infeasible. For example, wen 4/3 k /k l < 3, it follows tat R n+1 R n+2 > (4k l 3k )/(4k k l ); wen k /k 3, we ave R n+1 R n+2 (2 1+k /k l )/k. Hence, te corresponding cell is labeled as N/A. Tables IV sows te optimal dynamic pricing in eac time slot n under te strategy. Given te demand elasticities k l and k, te solution will be uniquely given by one of te tree cases of R n+1 R n+2 regimes. In Subsection IV-C, we will propose an algoritm to compute R n+1 R n+2 iteratively for all time slots. 2) Optimal Pricing under Ligt-Priority Strategy: Given LP strategy cosen( in time slot n, we derive te expected total revenue Rn LP r LP l (n),r LP(n)) by setting a n = 2 and. a n = 1 in te last two terms of (9), respectively. Te database operator needs to solve te following problem. P5: Optimal Pricing for time slot n under LP maximize R LP n ( r LP l (n),r LP (n)) (33) subject to r LP n+2 rlp l (n)+ R n+1, (34) r LP (n)+ R n+2 R n+1, (35) 0 rl LP (n) rl max, (36) 0 r LP (n) r max, (37) variables rl LP (n),r LP (n). (38) Constraints (34) and (35) guarantee tat te ligt-priority strategy is optimal in time slot n. Te analysis for Problem P5 is similar to tat for Problem P4, due to te similar structures of te two problems. We tus ave Proposition 4 as follows. Proposition 4: Te optimal solution to Problem P5 can also be summarized in a table as in Table IV, only wit different conditions in te rows and te columns and expressions of I0 LP, E1 LP and E2 LP. Due to space limitation, te proof of Proposition 4 and te detailed solutions can be found in Appendix F of [23]. 3) Optimal Pricing under Ligt-Dominant Strategy: Given LD strategy cosen in time slot n, we derive te expected total revenue R LD n ( r LD l (n),r LD (n)) by setting a n = 0 and a n = 1 in te last two terms of (9), respectively. Te database operator needs to solve te following problem. P6: Optimal Pricing for time slot n under LD maximize Rn LD ( r LD l (n),r LD (39) subject to r LD (n)+ R n+2 0+ R n+1, (40) 0 rl LD (n) rl max, (41) 0 r LD (n) r max, (42) variables r LD l (n),r LD (n). (43) Unlike te and te LP cases, we can derive te optimal prices under LD in closed-form. Proposition 5: Te optimal prices in time slot n under te LD strategy are given by te interior point solution I0 LD : rl LD (n) = 1,r LD 2k (n) = min( R n+1 R n+2,rmax ). (44) l Te proof of Proposition 5 is given in Appendix G of [23]. We ave analyzed te price optimization under any cosen admission strategy. Next, we will compare te expected total revenuesrn, Rn LP, and Rn LD to pick te optimal pricingadmission strategy.

9 9 Algoritm 2: Optimal Dynamic Pricing and Admission Policy 1: Set n = N + 1, R N+1 = 0 2: Set r l (N),r (N) by (44) and R LD N by R N (r l (N),r (N)). 3: for n = N 1,,2,1 do 4: Derive r l (n),r (n), R n by Table IV. 5: Derive r LP l (n),r LP (n), R LP n by Prop. 4. 6: Derive r LD l (n),r LD (n) and R LD n by (44). 7: R n max{r n,r LP n,r LD n } and r l (n),r (n) argmax{r n,r LP n,r LD n }. 8: if r l (n),r (n) = r l (n),r (n) ten 9: Te eavy-priority strategy is optimal. 10: else if r l (n),r (n) = rlp l (n),r LP (n) ten 11: Te ligt-priority strategy is optimal. 12: else 13: Te ligt-dominant strategy is optimal. 14: end if 15: end for 16: return Pricing-Admission policy r and π. C. Optimal Dynamic Pricing and Admission Policies After deriving te optimal prices under eac admission strategy, we can now compare te corresponding revenues and coose te best admission strategy for time slot n. We need to do tis for eac of te N time slots. We sow tis process in Algoritm 2, wic involves te previous solutions (Table IV, Proposition 4, and Equation (44)). More specifically, te algoritm iteratively computes te prices and revenues under te tree admission strategies, respectively, and ten selects te optimal prices and te corresponding admission strategy wic lead to te largest revenue (lines 3 to 15). Te complexity of Algoritm 2 is low and in te order of te total time slots O(N), as it only needs to ceck te tables and Equation (44) we derived. We summarize te optimality result as follows. Teorem 2: Te dynamic prices r = {r (n), n N} and te dynamic admission policy π = {a n (S n,x n,y n ), n N} derived in Algoritm 2 are te unique optimal solution to Problem P2. Te proof of Teorem 2 is given in Appendix H of [23]. Note tat te optimal prices and admission policy form a contingency plan tat contains information about te optimal prices and admission decisions at all te possible system states (S n,x n,y n ) in any time slots n N. To implement te optimal policy from time slot 1 to N, te database operator needs to decide te actual admission actions according to te realizations of random demands and te transition of system states. More specifically, at te beginning of eac time slot n, te operator first announces prices r (n) according to r and cecks te actual demands (X n,y n ). Ten, te admission decisions are determined by cecking te optimal policy π and te state component S n is updated accordingly. V. EXTENSIONS Te analysis of te simplified case in Sections II to IV paves te way for te analysis of te general case of multiple types of SUs. Next, we will first consider te case of arbitrary spectrum occupancies of two SU types, and ten te general case of more tan two SU types. A. Extension to Arbitrary Spectrum Occupancies of Two SU Types In Sections II to IV, we ave assumed tat a eavy-traffic SU occupies 2 consecutive time slots. Now we proceed to consider te general case were a eavy-traffic SU occupies M consecutive time slots. Te cannel occupancy of a ligttraffic SU is still normalized to a unit time slot. Naturally, we ave2 M N. Following similar notations as in Section II, in order to caracterize te spectrum occupancy information over time, we define S n as te number of remaining occupied time slots before making te admission action a n in time slot n, were S n {0,1,,M 1}. At te beginning of time slot n, we first ceck te SU occupancy of te current time slot, i.e., { 1,,M 1, if time slot n is occupied, S n = (45) 0, if time slot n is idle. For example, if M = 3 and we start admitting a eavytraffic SU in time slot n, ten S n+1 = 2,S n+2 = 1, and S n+2 = 0. If we define te possible admission action as a n = 0 (admitting no SU), a n = 1 (admitting a ligttraffic SU), and a n = M (admitting a eavy-traffic SU), ten te dynamics of te system state in (2) still olds ere, i.e., S n+1 = (S n +a n (1 S n ) 1) +, n {1,,N 1}, and we define te wole system state in time slot n as (S n,x n,y n ) similarly as in Section III. Te problem formulation turns out to be te same as Problem P1. As a result, te optimal admission policy can also be computed similarly as Algoritm 1. 1) Stationary Admission Policy under Static Pricing: Wen we analyze te static pricing for tis general case, a new callenge is to understand tat under wic combination of system parameters te stationary admission policies are optimal, wic is different from tose in Subsection III-C. Next we take te Heavy-Priority Admission Policy as an example, and derive te condition of te parameters p l, p, r l, and r, under wic te stationary admission policy is optimal under static pricing. Proposition 6: Te optimal policy for solving te revenue maximization Problem P1 degenerates to te eavy-priority stationary admission policy wen price ratio between te eavy-traffic SU and te ligt-traffic SU is larger tan a tresold θ t (p l,p ), i.e., r /r l > θ t (p l,p ), (46) were te tresold ratio θ t (p l,p ) can be determined by solving te following: r + R n+m = r l+ R n+1, n {1,2,,N M+1}. (47) Te proof of Proposition 6 is given in Appendix I of [23]. We give te proof sketc as follows. First, we derive te expected revenue R n as a function of r l,r,p l,p, given te eavy-priority stationary admission policy. Second, we determine r /r l in terms of p l, p, and n, by plugging R n+1 and R n+m into te condition (47), i.e., r /r l = f(p l,p,n). Tird, we denote f(p l,p,n) as θt (p l,p,n), and derive te final tresold θt (p l,p ) by optimizing θt (p l,p,n) over n {1,2,,N M + 1}. It tus follows tat te

10 10 eavy-priority stationary admission policy is optimal to solve te operator s revenue maximization problem if (46) olds. Proposition 6 sows tat our analysis in Section III also applies to te general case. We can also derive te tresold condition for te ligt-priority admission policy by considering R n+1 r + R n+m r l + R n+1, and te ligt-dominant admission policy by considering r + R n+m < R n+1 similarly. Te related analysis are similar to Teorem 1. We skip te detailed analysis due to space constraints. 2) Dynamic Pricing and Performance Evaluation: Te analysis under dynamic pricing is also similar to tat in Section IV, were we decompose te problem into tree subproblems in eac time slot. We sow te main result in te following proposition, by focusing on te eavy-priority strategy for te illustration purpose. Proposition 7: Given an arbitrary value of spectrum occupancy M, te optimal dynamic pricing under te eavypriority strategy is te same as tat in Proposition 3 and Table IV, once we replace R n+1 R n+2 by R n+1 R n+m. Te proof of te proposition is given in Appendix J of [23]. Proposition 7 sows tat previous analysis for dynamic pricing can be directly extended to te arbitrary occupancy case. B. Extension to Multiple Types of SUs In tis subsection, we furter extend te analysis in Sections III to IV and Subsection V-A to te case wit a total of I types of SUs seeking for spectrum access, including one type of ligt-traffic SUs and I 1 types of eavy-traffic SUs wo occupy 2,3,,I consecutive time slots, respectively. 1) Problem Formulation: We use I = {1,2,,I} to denote te set of SU types. To analyze te stationary admission policy, we need to compare a total of I+1 admission coices (including no admission) as in te analysis in Section III and Subsection V-A. Te difference is tat tere are two revenue constraints for eac policy in Section III and Subsection V (e.g., (10) and (11)), wile tere are I+1 revenue constraints ere. We continue te procedure and derive te associated tresolds, ten determine te stationary admission policy by comparing te price relations wit tose tresolds. More specifically, we define te prices carged to all types of SUs as R = {r i, i I}, were r i is te price carged to a type-i SU for using te spectrum resource. Let te demand probabilities of all types of SUs be P = {p i, i I}, and te realizations of all types of SUs demands in time slot n be X n (i), i I,n N. Given r i R and p i P, te expected total revenue in time slot n, i.e., R n (S n,x n (1),,X n (I),a n ), is te summation of te immediate revenue (as a result of te immediate action a n ) and te expected future revenue R n+1 (S n+1) (if a n = 0 wit no admission) or R n+i (S n+i) (if a n = i, admitting a type-i SU), considering all possible SU demands (X n (1),,X n (I) ) in time slot n. Te detailed expression is similar to (9), and can be found in [23]. At te beginning of time slot n, we determine te optimal admission decision by comparing te total revenue of admitting a particular type of SU, wic involves bot te immediate revenue r i and te maximum expected future revenue R n+i (S n+i). Given SUs demands in time slot n, if te optimal decision is no admission (a n = 0) due to a more profitable type of SU in te next time slot, te total revenue in time slot n is 0+ R n+1 (S n+1). To summarize, te optimal decision in time slot n is a n { =arg max 0+ R n+1 (S n+1 ), a n I {0} (r i + R n+i (S n+i)) 1 (i) {X, i I}. (48) n =1} Te above argument reveals a backward induction algoritm of determining te optimal admission decision in eac time slot, wic is similar to Algoritm 1. We are interested in te optimality of te stationary admission policies as discussed in Subsection III-C. 2) Stationary Admission Policies under Static Pricing: We first consider a type-i and a type-j SU (i > j > 1) wo seek to occupy arbitrarily consecutive time slots i and j, respectively. In tis case, te priority of admitting a particular type of SUs depends on te values of r i + R n+i, r j + R n+j, and 0 + R n+1. For a particular time slot n, for example, if r i + R n+i > r j + R n+j > 0 + R n+1, we prefer to serve te type-i SU type rater tan te type-j SU (i.e., te admission priority follows Λ(i) > Λ(j) > Λ(0)). By specifying te values ofa n according to tis admission priority in R n (S n,x n (1),,X n (I),a n ), we determine te differences R n+j R n+i and R n+1 R n+j similarly as Teorem 1 and Proposition 6. Te tresold tat guarantees te condition r i + R n+i > r j + R n+j > 0 + R n+1 can be derived by solving tis condition. Furter, by optimizing te derived tresold over all time slots n N, we derive te final tresold tat guarantees te optimality of te admission priority Λ(i) > Λ(j) > Λ(0) for all time slots. Hence, tis admission priority becomes one of te stationary admission policies. Similarly, te tresolds for te oter five admission priorities can be determined by solving te corresponding revenue conditions. Te above discussions can be generalized to te case of multiple types of SUs as follows. For a particular time slot n, for example, if te revenue conditions satisfy r I + R n+i > r I 1 + R n+i 1 > > r 1+ R n+1 > 0+ R n+1, te admission priority follows Λ(I) > Λ(I 1) > > Λ(1) > Λ(0). By specifying te values ofa n according to tis admission priority in R n (S n,x n (1),,X n (I),a n ), we determine te difference R n+j R n+i, j {1,,I 1} similarly as Teorem 1 and Proposition 6, respectively. We ten proceed to derive te tresolds suc tat te revenue conditions old for all time slots. Tese tresolds guarantee tat te admission priority Λ(I) > Λ(I 1) > > Λ(1) > Λ(0) is optimal for all time slots, and ence it becomes a stationary admission policy. Proposition 8: Given te set I of I types of SUs, tere are (I + 1)! admission priorities. For eac admission priority, tere exist tresolds of te price ratios suc tat te optimal admission priority for a time slot is optimal for all time slots (corresponding to an optimal stationary admission policy). Te proof of Proposition 8 is given in Appendix K of [23]. Proposition 8 sows tat te tresold-based stationary policy still olds in te general scenario, and tere exist (I + 1)!

11 11 Heavy-traffic demand elasticity k Ligt-Dominant Admission Policy Ligt-traffic demand elasticity k l Ligt-Priority Admission Policy Fig. 4. Optimal coices of admission policies for different values of elasticity parameters k l and k. Te yellow (Ligt-Dominant Admission Policy), cyan (Ligt-Priority Admission Policy), and blue (Heavy-Priority Admission Policy) regimes represent tree stationary admission policies, i.e., I, II, and IV regimes in Fig. 2, respectively. Te brown (Nonstationary Policy) regime requires Algoritm 1 to compute te optimal policy. Nonstationary Policy Heavy-Priority Admission Policy Heavy-traffic demand elasticity k LD Policy and Dynamic Pricing Ligt-traffic demand elasticity k l LP Policy and Dynamic Pricing Fig. 5. Optimal admission policies under dynamic pricing over N = 100 time slots. Te yellow (LD Policy and Dynamic Pricing), cyan (LP Policy and Dynamic Pricing), and blue ( Policy and Dynamic Pricing) regimes are te tree stationary admission policies, i.e., I, II, and IV regimes in Fig. 2, respectively. Te brown (Nonstationary Policy) regime requires Algoritm 2 to compute te optimal policy. Nonstationary Policy Policy and Dynamic Pricing tresolds 6 for all types of SUs I, wic are completely determined by te values of {0+ R n+1,r i + R n+i, i I} in eac time slot. Recall tat in Subsection III-C, we sould ave (2 + 1)! stationary admission policies. However, due to te fact r 1 + R n+1 > 0 + R n+1, finally we ave a total of (2 + 1)!/2! = 3 stationary admission policies. 3) Optimal Dynamic Pricing and Dynamic Admission: In te dynamic pricing setting, te joint pricing and admission problem in time slot n can be formulated similarly as Problem P3 in Section IV, by canging te objective function to R n (S n,x n (1),,X n (I),a n ). Since tere are I + 1 possible revenues in (48) and we need to determine teir value orders, tere are (I + 1)! admission strategies (admission priorities) as in Proposition 8. We follow te same pricing-admission decomposition procedure to transform te joint problem into (I+1)! subproblems corresponding to te (I+1)! admission strategies in tis time slot. As suc, we can also derive te optimal pricing for maximizing te revenue in eac time slot by solving tose subproblems as we did in Section IV, and ten coose te admission strategy tat leads to te largest revenue as sown in Algoritm 2. Te analysis procedure is identical wit tat in te previous scenario. Te only difference is tat tere are I rater tan two constraints (revenue conditions) in eac optimization problem wen assuming a particular admission strategy, ence it will be more complicated to optimize te prices in eac subproblem. VI. SIMULATION RESULTS In tis section, we provide te simulation results to illustrate our key insigts regarding te performances of te dynamic admission control under bot static pricing and dynamic pricing. We first illustrate te stationary admission policies for te dynamic admission control under static pricing and dynamic pricing, respectively. We ten compare te revenue improvement of dynamic pricing over static pricing under a wide range of system parameters. 6 To determine te specific admission strategy (priority) in eac time slot, we need to sort te I+1 revenues in (48) to te corresponding order. Hence, we ave a I + 1 permutation of I + 1, wic involves (I + 1)! admission strategies (priorities). A. Optimal Static Pricing and Stationary Admission Policy In Subsection III-D, we derived te optimal static pricing by first assuming tat one of te stationary admission policies is optimal. Recall tat te tree conditions in Teorem 1 are caracterized by te price ratio r /r l. Given any demand elasticities k l and k (ence any r /r l relation wit respect to p l andp ), it is natural to ask weter te optimal static pricing satisfies one of te conditions in Teorem 1, so tat it is indeed optimal to coose a stationary admission policy after we optimize te static prices. Fig. 4 illustrates te corresponding result, sowing wen a stationary admission control policy is optimal under te optimal static prices for particular system parameters k l and k. As we can see, except te small brown (Nonstationary Policy) regime wic corresponds to regime III in Fig. 2, te stationary policies are optimal in most cases. B. Optimal Dynamic Pricing and Stationary Admission Policy In Subsection IV-C, we ave sown tat in te general case of dynamic pricing and dynamic admission control, te optimal admission strategies in different time slots may be different. On te oter and, it would be interesting to study under wat system parameters te optimal admission decisions of different time slots (under dynamic pricing) will coincide wit one of te stationary admission policies defined in Table II. Recall tat in our system model, as long as we adopt te linear demand functions, te system only as two parameters k l and k, and te oter parameters (e.g., probabilities p l and p ) are determined by k l and k. Fig. 5 illustrates te optimal admission and pricing decisions under dynamic pricing. We can see tat te optimal admission strategies in Algoritm 2 degenerate to stationary admission policies in most cases, and it is only optimal to switc between different admission strategies (, LD, and LP) in a small regime (te brown regime in Fig. 5). Observation 1: Under a wide range of system parameters k l and k, te optimal admission decisions developed in Algoritm 2 (wit te optimized optimal dynamic prices) degenerate to stationary admission policies over all time slots.

12 12 Revenue Improvement < 10% Revenue Improvement 30% Revenue Improvement < 10% Revenue Improvement 30% Heavy-traffic demand elasticity k Ligt-traffic demand elasticity k l Fig. 6. Te revenue improvement of dynamic pricing over static pricing for different k l and k distributions. 10% Revenue Improvement < 30% Heavy-traffic demand elasticity k Ligt-traffic demand elasticity k l Fig. 7. Te revenue improvement of dynamic pricing over static pricing for different k l and k distributions. Here, te spectrum occupancy M = 3, wile M = 2 in Fig % Revenue Improvement < 30% Wen te stationary admission policy is optimal, we ave te following claims. If ligt-traffic SUs are muc more price-sensitive tan eavy-traffic SUs (i.e., k l is significantly larger tan k ), te optimal dynamic pricing degenerates to te eavypriority admission policy wic is stationary over time. If eavy-traffic SUs are muc more sensitive to prices tan ligt-traffic SUs (k l is significantly less tan k ), te optimal dynamic pricing degenerates to te ligtdominant admission policy wic is stationary over time. If bot ligt- and eavy-traffic SUs sensitivities k l and k are comparable, te optimal dynamic pricing degenerates to te ligt-priority admission policy wic is stationary over time. C. Performance Comparison of Optimal Dynamic Pricing wit Optimal Static Pricing In addition to te optimal pricing and admission policies, it is also important to compare te performance of dynamic pricing wit tat of static pricing. Te key benefit of static pricing is tat it does not cange over time. Unlike static pricing, te advantage of dynamic pricing is to acieve te maximum operator revenue. However, dynamic pricing as a iger implementational complexity. Next, we compare te optimal revenue of optimal dynamic pricing obtained in Teorem 2 wit tat of optimal static pricing obtained in Subsection III-D. Fig. 6 sows te revenue improvement of dynamic pricing over static pricing under different demand elasticity values (k l and k ). Here, we set te total time slots N = 100, so tat te time orizon is long enoug to approximate te time-average performance. Observation 2: As sown in Fig. 6, dynamic pricing outperforms static pricing by more tan 30% wen bot types of SUs are sensitive to prices (i.e., bot k l and k are ig). Wen bot types of SUs are not price-sensitive (i.e., k l and k are low), dynamic pricing only leads to limited revenue improvement (less tan 10%) tan static pricing, and it is better to adopt static pricing due to its low complexity. Te above comparison is based on te assumption tat eavy-traffic SUs request two consecutive time slots. In Section V, we ave extended te model to arbitrary spectrum occupancies. Hence, it is also interesting to sow te comparison wit more spectrum occupancies. Fig. 7 sows te revenue improvement of dynamic pricing over static pricing wit tree consecutive time slots occupancy of eavy-traffic SUs (M = 3). We can see tat dynamic pricing significantly outperforms static pricing wen SUs demands are igly elastic, wic is similar to Observation 2. Comparing wit Fig. 6 wit M = 2, te difference ere is tat a larger value of M reduces te benefit of dynamic pricing. For example, wen k l (90,120) and k (60,70), te revenue improvement of dynamic pricing over static pricing is more tan30% in Fig. 6, but is only around 10% in Fig. 7. Te intuition is tat a larger spectrum occupancy reduces te flexibility of dynamic pricing, since more slots will be occupied and cannot be dynamically allocated to new demands. Consider te extreme case M = N, ten all slots will be occupied wen admitting a eavy-traffic SU initially and dynamic pricing degenerates to static pricing. Tis implies tat as te cannel occupancy gap between te two SU types increases, it becomes increasingly attractive for te operator to coose te simple static pricing approac in order to acieve a close-to-optimal revenue. VII. CONCLUSION In tis paper, we consider a spectrum database operator s revenue maximization problem troug joint spectrum pricing and admission control. We incorporate te eterogeneity of SUs spectrum occupancy and demand uncertainty into te model, and consider bot te static and te dynamic pricing scemes. In static pricing, we sow tat stationary admission policies can acieve optimality in most cases. In dynamic pricing, we compute optimal pricing troug a proper pricingand-admission decomposition in eac time slot. Furtermore, we sow tat dynamic pricing significantly improves revenue over static pricing wen SUs are sensitive to prices cange. Finally, we sow tat wen te gap of te cannel occupation lengt between two types of SUs increases, te gap between static pricing and dynamic pricing srinks. REFERENCES [1] C. Jiang, L. Duan, and J. Huang, Joint Spectrum Pricing and Admission Control for Heterogeneous Secondary Users, Proc. IEEE WiOpt, 2014.

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D. Yao, Te Stocastic Knapsack Revisited: Switc-Over Policies and Dynamic Pricing, Operations Researc, vol. 56, no. 4, pp , July-August [23] C. Jiang, L. Duan, and J. Huang, Optimal Pricing and Admission Control for Heterogeneous Secondary Users, ttp://arxiv.org/abs/ Cangkun Jiang (S 14) is working towards te P.D. in Information Engineering at Te Cinese University of Hong Kong. His researc interests include dynamic pricing and revenue management in communication networks, game teory and incentive mecanisms design in network economics, and network optimization. During May 2013-Aug. 2013, e was a visiting P.D. student in Engineering Systems and Design at Singapore University of Tecnology and Design. He is a student member of IEEE. Lingjie Duan (S 10-M 12) is an Assistant Professor in Engineering Systems and Design at Singapore University of Tecnology and Design. He received te P.D. degree from Cinese University of Hong Kong in 2012 and was a Visiting Scolar at University of California at Berkeley in He as publised igly cited papers in top engineering and business journals. He as received te 10t IEEE ComSoc Asia-Pacific Outstanding Young Researcer Award in 2015, and e was te Finalist of Hong Kong Young Scientist Award. He now serves as an Editor of IEEE Communications Surveys and Tutorials, and e is also a SWAT team member of te Editorial Board of IEEE Transactions on Veicular Tecnology. He is also a Guest Editor in IEEE Wireless Communications magazine for a special issue about green networking and computing for 5G. Moreover, e is a TPC member of multiple top conferences (e.g., MobiHoc, SECON, and NetEcon). Jianwei Huang (S 01-M 06-SM 11-F 16) is an Associate Professor and Director of te Network Communications and Economics Lab (ncel.ie.cuk.edu.k), in te Department of Information Engineering at te Cinese University of Hong Kong. He received P.D. from Nortwestern University in 2005, and worked as a Postdoc Researc Associate at Princeton University during His main researc interests are in te area of network economics and games, wit applications in wireless communications, networking, and smart grid. He is a Fellow of IEEE, and a Distinguised Lecturer of IEEE Communications Society ( ). Dr. Huang is te co-recipient of 8 Best Paper Awards, including IEEE Marconi Prize Paper Award in Wireless Communications in 2011, and Best (Student) Paper Awards from IEEE WiOpt 2015, IEEE WiOpt 2014, IEEE WiOpt 2013, IEEE SmartGridComm 2012, WiCON 2011, IEEE GLOBECOM 2010, and APCC He as co-autored four books: Wireless Network Pricing, Monotonic Optimization in Communication and Networking Systems, Cognitive Mobile Virtual Network Operator Games, and Social Cognitive Radio Networks. He is a co-autor of six ESI Higly Cited Papers. He received te CUHK Young Researcer Award in 2014 and IEEE ComSoc Asia-Pacific Outstanding Young Researcer Award in Dr. Huang as served as an Editor of IEEE Transactions on Cognitive Communications and Networking (2015-), Editor of IEEE Transactions on Wireless Communications ( ), Editor of IEEE Journal on Selected Areas in Communications - Cognitive Radio Series ( ), Editor and Associate Editor-in-Cief of IEEE Communications Society Tecnology News ( ). He as served as a Guest Editor of IEEE Transactions on Smart Grid special issue on Big Data Analytics for Grid Modernization (2016), IEEE Network special issue on Smart Data Pricing (2016), IEEE Journal on Selected Areas in Communications special issues on Game Teory for Networks (2016), Economics of Communication Networks and Systems (2012), and Game Teory in Communication Systems (2008), and IEEE Communications Magazine feature topic on Communications Network Economics (2012). Dr. Huang as served as Vice Cair ( ) of IEEE Communications Society Cognitive Network Tecnical Committee, Cair ( ) and Vice Cair ( ) of IEEE Communications Society Multimedia Communications Tecnical Committee, a Steering Committee Member of IEEE Transactions on Multimedia ( ) and IEEE International Conference on Multimedia & Expo ( ). He as served as te TPC Co-Cair of IEEE WiOpt 2017 and 2012, IEEE SDP 2016 and 2015, IEEE ICCC 2015 (Wireless Communications System Symposium) and 2012 (Communication Teory and Security Symposium), NetGCoop 2014, IEEE SmartGridComm 2014 (Demand Response and Dynamic Pricing Symposium), IEEE GLOBECOM 2013 (Selected Areas of Communications Symposium) and 2010 (Wireless Communications Symposium), IWCMC 2010 (Mobile Computing Symposium), and GameNets He will serve as a General Co-Cair of IEEE WiOpt He is a frequent TPC member of leading networking conferences suc as INFOCOM and MobiHoc. He is te recipient of IEEE ComSoc Multimedia Communications Tecnical Committee Distinguised Service Award in 2015 and IEEE GLOBECOM Outstanding Service Award in 2010.