Utility-optimal Partial Spectrum Leasing for. Future Wireless Services. Sang Yeob Jung, Seung Min Yu, and Seong-Lyun Kim. School of Electrical and Electronic ...
Utility-optimal Partial Spectrum Leasing for Future Wireless Services Sang Yeob Jung, Seung Min Yu, and Seong-Lyun Kim School of Electrical and Electronic Engineering, Yonsei University 50 Yonsei-Ro, Seodaemun-Gu, Seoul 120-749, Korea Email: {syjung, smyu, slkim}@ramo.yonsei.ac.kr
Abstract—One of the challenges facing the next-generation wireless networks is to cope with the expected demand for data. This calls for an efficient spectrum regulation that can enable mobile subscribers to support high quality of service (QoS) and mobile network operators (MNOs) to leverage their profit streams. In this paper, we present a new spectrum allocation policy in a monopoly situation. The problem is formulated as a Stackelberg game. We show that the conventional spectrum leasing contract may lead to the unprecedented scenario in which costs outweigh their revenues. On the other hand, our proposed spectrum leasing contract can not only maximize user welfare but also leverage MNO’s profit streams. We show that our spectrum leasing contract can increase user welfare and MNO’s profit up to 75% and 20%, respectively, relative to the conventional spectrum leasing contract. Thus, regulators must rewrite their spectrum allocation policy in order to maximize user welfare and leverage MNO’s profit streams.
I. I NTRODUCTION Due to the exploding popularity of all things wireless, the demand for wireless data traffic increases dramatically. According to a Cisco report, global mobile data traffic will increase 26-fold between 2011 and 2015 [1]. This growth drives mobile network operators (MNOs) to make excessive investment in a radio spectrum in order to cope with the expected demand for data. Although MNOs remain committed to deliver reliable mobile broadband services over their networks, a number of critical questions remain unanswered. How much revenue can MNOs generate with a new long-term evolution (LTE) network? What regulation should the regulator adopt to leverage their profit streams? Since their revenues begin to flatline, which is the result of market saturation allied to declining average revenue per users (ARPUs), they will possibly face an unprecedented scenario in which costs outweigh their revenues. The 700MHz band, which is currently used to deliver digital terrestrial TV (DTT), will be allocated by governments in Korea, Japan, and the United Kingdom, etc. They have been rewriting their spectrum allocation policies to preserve the ecosystem in wireless services market. In the conventional spectrum leasing contract1 , the regulator sets the leasing cost for a given bandwidth in a monopoly situation. Because the spectrum is a scarce resource, the leasing cost tends to 1 A contract is the binding agreement with specific terms between MNO and the regulator (e.g., FCC in USA, Ofcom in UK) in which there is a promise to offer network services to users in return for MNO’s profit.
increase. To guarantee the quality of service (QoS), MNO has tried to lease2 the full allocated bandwidth within the current spectrum policy. Due to the market saturation in next-generation wireless networks, however, MNO’s profit is expected to decrease [2]. There has been a substantial amount of work on the price competition of operators in wireless services markets [3]–[5]. However, limited attention has been paid to the spectrum regulation. For preserving the ecosystem, the spectrum regulation is also a significant factor in the fast-growing wireless communication market. In [6], various possible regulation methods are analyzed by the existence of neutral operators. The price and QoS subsidy schemes are introduced for guaranteeing user welfare in [7]. However, the existing literatures do not fully address MNO’s leasing strategies that are tightly related with pricing strategies and profit maximization. In this paper, we introduce a new spectrum leasing contract in a monopoly situation. In this contract, the regulator defines a rule that MNO can dynamically adjust the amount of leasing bandwidth in a given bandwidth subject to offering free services to all users in a leftover bandwidth. Thus, MNO should jointly consider leasing and pricing decisions to maximize its own profit. We show that there is an optimal spectrum price not only to maximize user welfare but also to leverage MNO’s profit streams. By taking proper account of the spectrum price, we can achieve the user welfare gain and the profit gain of the new spectrum policy to the conventional spectrum policy, which are 1.75 and 1.20, respectively. The rest of paper is organized as follows. Section II analyzes the conventional spectrum leasing contract and discusses the worst scenario where costs overtake MNO’s revenue. In Section III, we introduce our spectrum allocation contract for leveraging MNO’s profit streams and maximizing user welfare. Section IV looks at how efficient our spectrum leasing contract is compared to the conventional one and draws out some interesting insights. We conclude in Section V together with the future research direction. II. S YSTEM M ODEL Consider a regulator, an MNO, and users in a wireless services market. The interactions among the regulator, MNO, 2 MNO is permitted to hold exclusive rights to spectrum for a defined period of time and return it to regulator so we use the term ’lease’.
978-1-4673-6337-2/13/$31.00 ©2013 IEEE
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Fig. 2. MNO’s equilibrium profits in two cases when B is 10: a conventional spectrum leasing contract in Section II and a new spectrum leasing contract in Section III.
Fig. 1. The conventional and new spectrum leasing contract in a monopoly situation
and users can be modeled as a three-stage Stackelberg game [8]. The regulator sets the leasing cost per unit bandwidth in Stage I. In Stage II, MNO jointly determines the service price and the amount of leasing bandwidth to maximize its profit. In Stage III, each user chooses the amount of bandwidth to maximize its net-utility. We solve this three-stage Stackelberg game by applying the concept of backward induction. We first look at the conventional spectrum leasing contract in Figure 1. The detailed analysis is as follows. A. Users’ Demand in Stage III Each user has a different utility for the network service and is characterized by a user type parameter v that is uniformly distributed in [0, 1]. We introduce a user type parameter to model different willingness to pay as in [9], [10]. The higher v users have more payoff for the same data-rate. The user type v obtains a utility u(v, b) (e.g., data rate) when using a bandwidth b [11], [12], u(v, b) = v · ln(1 + b).
pc
pc 1 3 − ln(pc ) − . = pc 1 − 4 2 4
The optimal bandwidth that maximizes the user’s net-utility is v if pc ≤ v, pc − 1, b∗ (v, pc ) = (2) 0, otherwise.
(4)
B. MNO’s Pricing and Leasing in Stage II In this stage, the pricing and leasing strategies of MNO should be investigated to maximize its profits. Given the users’ demand (2) in Stage III, the total bandwidth demand in the network is derived as follows: 1 v 1 pc − 1, (5) − 1 dv = + Q(pc ) = pc 2pc 2 pc which is decreasing in pc . Note that a user with a type-v will only access the network when pc ≤ v. If MNO has no other option but to lease a full bandwidth B, then lease it from the regulator unless its profit is negative. MNO chooses the optimal price p∗c to maximize its profit, i.e., max π M N O (pc )
0≤pc ≤1
MNO charges users a linear payment pc per unit bandwidth while guaranteeing quality of service (QoS). The user’s netutility with a type v is the difference of its utility and payment, i.e., (1) u(v, b, pc ) = v · ln(1 + b) − pc · b.
For a user with a type-v, the maximum net-utility is v · ln( pvc ) − v + pc , if pc ≤ v, ∗ u(v, b (v, pc ), pc ) = 0, otherwise,
which is always nonnegative. Assuming the price pc is normailized into the interval [0, 1], the total net utility of users served by MNO is defined as follows: 1 u(v, b∗ (v, pc ), pc )dv Uc =
subject to
= pc · Q(pc ) − c · B
(6)
Q(pc ) ≤ B
where c is the leasing cost per unit bandwidth. The unique optimal solution is described in the following proposition. Proposition 1: The equilibrium service price p∗c is: p∗c = 1 + B − (1 + B)2 − 1.
Proof: The objective function is a convex and monotonic decreasing function of pc and the feasible region is convex. Thus, MNO should decrease pc as much as possible to obtain its maximal profit. Due to the bandwidth constraint, however, (3) MNO can decrease pc until Q(pc ) = B.
C. Regulator’s Pricing in Stage I
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Next, we consider the regulator’s leasing cost of the unit bandwidth c in Stage I. In a monopoly case, the regulator charges MNO the particular leasing cost3 by allocating a bandwidth B. Since the amount of bandwidth is fixed, there is no incentive for MNO to change its equilibrium service price. Thus, the total net utility of users is same in the conventional spectrum leasing contract. As the motivation of this paper, we look at how MNO’s profit changes as c increases. Figure 2 shows that MNO’s profit comes to 0 as c is close to p∗c , which is 0.046. This motivates us to introduce a new leasing contract not only to leverage its profit streams but also to maximize user welfare. III. A NEW SPECTRUM LEASING CONTRACT In this section, we consider how the regulator can leverage MNO’s profit streams and maximize user welfare. In our spectrum leasing contract, MNO can dynamically adjust the amount of leasing bandwidth in a given bandwidth B. However, it should offer free services to all users in a leftover bandwidth as described in Figure 1. The leftover bandwidth is the difference between the given bandwidth B and the amount of bandwidth leased from the spectrum regulator. Thus, MNO should jointly consider leasing and pricing strategies to maximize its own profit. We also solve this three-stage Stackelberg game by backward induction. A. Users’ Demand in Stage III The analysis of this stage is similar to that of Section II-A. This means that the user type-v achieves the net-utility i.e., u(v, b, pn (c)) = v · ln(1 + b) − pn (c) · b, where pn (c) is a function of the bandwidth c. The optimal demand is v pn (c) − 1, b∗ (v, pn (c)) = 0,
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leasing cost of the unit for a user with a type v if pn (c) ≤ v, otherwise.
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The maximum net-utility for a user type-v is v ∗ u(v, b (v, pn (c)), pn (c)) = v · ln − v + pn (c), (9) pn (c) where pn (c) ≤ v. Let us define the total net-utility as follows: 1 Ut = u(v, b∗ (v, pn (c)), pn (c))dv pn (c) pn (c) 3 1 = pn (c) 1 − − ln(pn (c)) − . (10) 4 2 4 The key difference here is that MNO should offer free services to all users in a leftover bandwidth. Because users are normalized into the interval [0, 1], the average user surplus is defined as follows: 1 (11) Usurplus (Blef tover ) = ln(1 + Blef tover ), 2 3 A regulator sets the spectrum leasing cost by reflecting the economic value of it.
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MNO’s equilibrium leased bandwidth and free bandwidth when B
where Blef tover is the leftover bandwidth after MNO decides optimal amount of leasing bandwidth in a given bandwidth B. Let us define user welfare as follows: Un = Ut + Usurplus .
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B. MNO’s Pricing and Leasing in Stage II We consider MNO’s pricing and leasing strategies in this stage. Since MNO’s leasing strategies are tightly related with its pricing strategies, MNO needs to take proper account of the tradeoff between the revenue earned by network services and the cost of leasing bandwidth. To maximize its own profit, MNO chooses pn (c), i.e., max
0≤pn (c)≤1
π M N O (pn (c)) = (pn (c) − c) · Q(pn (c)) (13)
subject to
Q(pn (c)) ≤ B.
Let us assume that the leasing cost of the unit bandwidth c is normalized into the interval [pn (c)3 , p∗c ]. We only consider the case in which the maximum value of c is p∗c because p∗c makes MNO’s profit 0 in the conventional spectrum leasing contract. Further, c is usually higher than pn (c)3 since spectrum is a valuable and scarce resource. Thus, the optimal solution of (13) is described in the following proposition. Proposition 2: The equilibrium service price p∗n (c) on c ∈ (pn (c)3 , p∗c ] is: c θ + 4π 1 p∗n (c) = max p∗c , 1+ 1 + 2 cos 3 2 3 √
3 c(ε− 4c ) 2 1 + 2c . , ε = 27 where θ = tan−1 ε− c 2
Proof: Appendix. Proposition 2 states that MNO should jointly consider leasing and pricing strategies to leverage its profit streams. The tradeoff between the revenue earned by network services and the cost of leasing bandwidth is an important factor to
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obtain the maximal profit in our proposed spectrum leasing contract. Figure 2 shows MNO’s optimal profit as a function of c in a given bandwidth B. Compared with the conventional spectrum leasing contract, its profit streams can leverage up to 0.25 when c is 0.046. Figure 3 shows the change of leased bandwidth from the regulator and free bandwidth. As the leasing cost increases, MNO dynamically adjust the amount of leasing bandwidth to maximize its own profit in a given cost c. Thus, MNO needs to take proper account of the tradeoff between the revenue earned by network services and the cost of leasing bandwidth to leverage its profit streams. C. Regulator’s Pricing in Stage I Next we consider how the regulator chooses the optimal spectrum prices in Stage I. Although many significant objectives should be realized by use of spectrum and spectrum pricing, however, the regulator mainly focuses on the economic benefits for the public. In other words, the spectrum regulator determines a spectrum price c per unit bandwidth to maximize user welfare i.e., maximize
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Fig. 4. User welfare in two cases when B is 10: a new leasing contract in Section III and a conventional leasing contract in Section II.
pn (c)3 ≤c≤p∗ (c)
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Unfortunately, the structure of (23) is so complex to obtain closed-form solutions. By computing numerically, we can illustrate some interesting insights. Figure 4 shows equilibrium user welfare as a function of c. An interesting observation is that there is the optimal spectrum price c∗ =0.015 to maximize user welfare. As c increases, MNO leases less bandwidth and charges users more to leverage its own profit streams. Thus, the total net utility of users served by MNO will be decreased. On the other hand, the average user surplus is going to increase as c increases due to the increase of the leftover bandwidth. The tradeoff between the total net utility and the average user surplus is an important factor to optimize user welfare. If the spectrum regulator focuses on the
Fig. 5.
User welfare gain and profit gain when B is 10.
economic benefits for the public, the spectrum price should be 0.015 to obtain maximal user welfare Un∗ , which is 1.452. IV. C OMPARISON OF TWO LEASING CONTRACTS In this section, we analyze how efficient our proposed leasing contract is compared to the conventional leasing contract. In order to see it, let us define the user welfare gain and the profit gain as follows: Ugain =
Un , Uc
Pgain =
π M N O (pn (c)) . π M N O (pc )
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Notice that user welfare in the conventional spectrum leasing contract is equivalent to the total net utility of users because there is no average user surplus. Figure 5 shows the user welfare gain and the profit gain as a function of c. Notice that the user welfare gain is maximized when c is 0.015. However, the profit gain increases as c increases. The maximal user welfare gain is 1.7512 and the following profit gain is 1.2. V. C ONCLUSION In this paper, we have presented a new spectrum leasing contract in a monopoly situation. In the conventional spectrum leasing contract, MNO will possibly face an unprecedented scenario in which costs overtake their revenues. With a new spectrum leasing contract, however, we show that the regulator cannot only maximize user welfare but also leverage MNO’s profit streams by setting an appropriate cost of leasing bandwidth. The general situation of multiple MNOs, a regulator, and users will be an interesting topic, where our current research is heading. A PPENDIX A. Proof of Proposition 2 To find the optimal price p∗n (c), we verify that the objective function of (13) should be a concave function of pn (c). The second order derivative of (13) is ∂ 2 πM N O = 1 − c · pn (c)−3 < 0. ⇔ pn (c)3 < c. ∂pn (c)2
(16)
Thus, the objective function of (13) is a concave function of pn (c) in c ∈ (pn (c)3 , p∗c ). The optimal price can be obtained by solving the following equation: ∂π M N O c 1 = pn (c) − 1 − =0 1− ∂pn (c) 2 pn (c)2 (17) = 2pn (c)3 − (2 + c)pn (c)2 + c = 0.
If we substitute pn (c) with x + 13 1 + 2c , then the above equation (15) is c 2 c 3 c 1 2 1+ 1+ (18) x− + = 0. x3 − 3 2 27 2 2
2
3 2 1 + 2c + 2c . Let us define τ = − 19 1 + 2c and σ = − 27 Then, the above equation (16) is reduced to: 3
x + 3τ x + σ = 0.
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If we substitute x with μ+ν and put it in (17), then μ3 + ν 3 + σ + 3(μν + τ )(μ + ν) = 0.
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Choose μ and ν to satisfy μ3 + ν 3 = −σ and μν = −τ to find x. If we let μ3 = α, ν 3 = β, then α + β = −σ and αβ = −τ 3 . Without loss of generality, α, β is the solution of a quadratic equation t2 + σt − τ 3 = 0. It is clear that α = μ3 =
√ −σ+ σ 2 +4τ 3 , 2
β = ν3 =
√ −σ− σ 2 +4τ 3 . 2
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To identify the solutions of equation (17) include complex roots, we should determine the sign of D = σ 2 + 4τ 3 c 3 c2 2
