[10pt]
Combining Spot and Futures Markets: A Hybrid Market Approach to Dynamic Spectrum Access
Lin Gao
School of Electronic and Information Engineering, Harbin Institute of Technology (Shenzhen), China,
and Department of Management Sciences, City University of Hong Kong, gaolin@hitsz.edu.cn
Biying Shou
Department of Management Sciences, City University of Hong Kong, biying.shou@cityu.edu.hk
YingJu Chen
School of Business and Management and School of Engineering, The Hong Kong University of Science and Technology, imchen@ust.hk
Jianwei Huang
Department of Information Engineering, The Chinese University of Hong Kong, jwhuang@ie.cuhk.edu.hk
Dynamic spectrum access is a new paradigm of secondary spectrum utilization and sharing. It allows unlicensed secondary users (SUs) to exploit opportunistically the underutilized licensed spectrum. Market mechanism is a widelyused promising means to regulate the consuming behaviours of users and, hence, achieve the efficient allocation and consumption of limited resources. In this paper, we propose and study a hybrid secondary spectrum market consisting of both the futures market and the spot market, in which SUs (buyers) purchase underutilized licensed spectrum from a spectrum regulator, either through predefined contracts via the futures market, or through spot transactions via the spot market. We focus on the optimal spectrum allocation among SUs in an exogenous hybrid market that maximizes the secondary spectrum utilization efficiency. The problem is challenging due to the stochasticity and asymmetry of network information. To solve this problem, we first derive an offline optimal allocation policy that maximizes the exante expected spectrum utilization efficiency based on the stochastic distribution of network information. We then propose an online Vickrey¨CClarke¨CGroves (VCG) auction that determines the realtime allocation and pricing of every spectrum based on the realized network information and the prederived offline policy. We further show that with the spatial frequency reuse, the proposed VCG auction is NPhard; hence, it is not suitable for online implementation, especially in a largescale market. To this end, we propose a heuristics approach based on an online VCGlike mechanism with polynomialtime complexity, and further characterize the corresponding performance loss bound analytically. We finally provide extensive numerical results to evaluate the performance of the proposed solutions.
Key words: Secondary Spectrum Market, Dynamic Spectrum Access, Asymmetric Network Information, Game Theory, Pricing
Subject classifications: Games/Group Decisions; Communications
Area of review: Games, Information, and Networks
History: Received October 2012; revisions received June 2014, May 2015; accepted March 2016
Radio electromagnetic spectrum is a limited resource for wireless communications, and is becoming more congested and scarce with the explosive development of wireless services and networks. Nonetheless, extensive studies (e.g., FCC (2002), McHenry et al. (2006)) have shown that many spectrum bands are heavily underutilized due to the significant variations in spectrum occupying time and space. This suggests that the traditional static licensing approach, i.e., granting exclusive spectrum usage right to a single commercial or government department over a long time and in a large area, is no longer efficient or suitable for future wireless networks. Dynamic spectrum access (DSA) has been recently recognized as a novel and promising approach to increase spectrum utilization efficiency and alleviate spectrum scarcity. The key idea is to enable unlicensed secondary users (SUs) to exploit the underutilized licensed spectrum in an opportunistic manner (see Haykin (2005), Akyildiz et al. (2006), Buddhikot (2007), Zhao & Sadler (2007)).^{}^{}endnote: This idea has been embraced by many government spectrum regulatory bodies (e.g., FCC in the U.S. and Ofcom in the UK), standardization organizations (e.g., IEEE and ECC), and major industry players (e.g., Google, Microsoft, Motorola, and Huawei). The most recent progress along this frontier is the finalization of the IEEE 802.22 Wireless Regional Area Network standard (see IEEE Standards Association (2011)) on the opening and secondary utilization of underutilized spectrum in the UHF band (between around 500MHz and 800MHz) licensed for broadcast TV. The longterm success of DSA requires many innovations in technology, economics, and policy. In particular, it is essential to design a secondary spectrum sharing mechanism that achieves a high secondary spectrum utilization efficiency, and meanwhile offers necessary incentives for licensees to open their licensed spectrum for secondary sharing and unlicensed access.^{}^{}endnote: For example, when FCC in the U.S. decided to open up the TV band for secondary unlicensed access, the National Association of the Broadcasters and Association for Maximum Service Television sued the FCC as they do not have incentives for such spectrum sharing (see NAB/MSTV (2009)).
Market mechanism is a widelyused promising means to regulate and coordinate the consuming behaviours of users and, hence, achieve the efficient allocation and consumption of limited resource. Therefore, marketdriven secondary spectrum trading is a natural choice to achieve the efficient spectrum utilization, and also a promising approach to address the incentive issue (see Hossain et al. (2009), Tonmukayakul & Weiss (2009)). With secondary spectrum trading, SUs temporarily purchase the underutilized licensed spectrum from a spectrum regulator (SR), such as an agent representing the Federal Communications Commission (FCC) in the United States or Ofcom in the United Kingdom.^{}^{}endnote: A realworld secondary spectrum sharing example (that involves the interactions of SR and unlicensed SUs) is the TV white space network (see FCC (2010)). In a TV white space network, SUs request and exploit the underutilized UHF/VHF radio spectrum licensed for broadcast television service (called TV white space) from a SR which is designated by the FCC in the United States to operate a white space database. The secondary spectrum market differs from conventional markets (e.g., paintings, bonds, and electricity markets) in the following two aspects. First, radio spectrum is spatially reusable. That is, a spectrum can be potentially used by multiple SUs simultaneously, as long as these SUs do not generate mutual interferences (see Section id1 for details). Second, the availability of spectrum is not deterministic. To protect the subscribed primary users (PUs) benefits, a spectrum is only available to SUs when it is not occupied by any PU (i.e., when it is idle). Due to the stochastic nature of PUs’ activities, the spectrum availability is often a random variable (see Section id1 for details). Based on the above two facts, the mechanism design in such a secondary spectrum market needs to overcome a different set of challenges than in a conventional market.
In this paper, we consider the secondary spectrum trading between one SR (seller) and multiple SUs (buyers) for all idle spectrum in a given time period which consists of multiple time slots.^{}^{}endnote: Detailed discussions regarding time period and time slot can be referred to Section id1. Moreover, we consider the shortterm spectrum trading, i.e., the idle spectrum is traded on a slotbyslot basis. Figure 1 illustrates such a timeslotted secondary spectrum market within a period of time slots. The blank and shadowed squares denote the idle spectrum and busy spectrum in each time slot, respectively. The edge connecting two SUs denotes the interference relationship between them. In other words, any two SUs without a connecting edge (e.g., SUs {4, 5} or SUs {6, 8}) are interferencefree (also called independent), and thus can use the same (idle) spectrum concurrently without any mutual interference. In this example, the idle spectrum at time is allocated to SUs {2, 6, 8} simultaneously, as these SUs are independent of each other.
More specifically, we consider a hybrid secondary spectrum market consisting of both the futures market and the spot market. In such a market, SUs can purchase spectrum either through predefined futures contracts via the futures market, or through spot transactions via the spot market. In the futures market, each SU reaches an agreement, called a futures contract or guaranteed contract, directly with the SR. The contract specifies not only the SU’s total demand and payment in the given period, but also the SR’s penalty if violating the contract.^{}^{}endnote: In this sense, the futures market in this paper can be seen as a generalized version of those in stock markets (see Kolb & Overdahl (2007)). In futures stock markets, the seller is obligated to deliver the commodities at an explicit future date; whereas, in our market model, the seller is allowed to deliver the commodities at any time before the due date, and even to violate the contract with certain penalty. In the spot market, SUs buy spectrum in a realtime and ondemand manner through spot transactions, and multiple SUs may compete with each other for limited spectrum resource (e.g., through an auction). In the example of Fig. 1, we denote the contract users (blue) by 1, 2, 3, and 4, each requesting a prespecific number of spectrum. Likewise, we denote the spot market users (red) by 5, 6, 7, and 8, each requesting and competing for spectrum in an ondemand manner. It is easy to see that the futures market insures buyers (sellers) against uncertainties of future supply (demand) through predefined contracts, while the spot market allows buyers to compete for limited resource based on their realtime demands and preferences.^{}^{}endnote: Guaranteed contract has been widely adopted by wireless service providers for different kinds of services, e.g., monthly subscription of data traffic (e.g., $25 for 2G wireless iPad data) or voice traffic (e.g., $35 for 180 minutes of voice communication in Hong Kong). On the other hand, with the explosive development of technological innovations, especially the emergence of DSA and cognitive radio, spot transaction, e.g., auction (see Huang et al. (2006), Gandhi et al. (2007), Li et al. (2008), Zhou et al. (2008), Zhou & Zheng (2009), and Wang et al. (2010)), has grown in prominence in secondary spectrum markets. Therefore, such a hybrid market has both the reliability of the futures market and the flexibility of the spot market; hence, it is highly desirable for Quality of Service (QoS) differentiations in secondary spectrum utilization. Specifically, an SU with elastic traffic (e.g., file transferring or FTP downloading) may be more interested in spot transactions to achieve a flexible resourceprice tradeoff; whereas, an SU with inelastic traffic requiring specific data rates (e.g., Netflix video streaming or VoIP) may prefer the certainty of contracts in the futures market.
We consider a quasistatic market scenario, where both the network topology and the hybrid market structure are exogenously given and fixed in each time period, but may change across different time periods.^{}^{}endnote: Here the network topology regards the interference relationships among all SUs, and the market structure regards the SUs in each market and the contract of each SU in the futures market. In particular, we focus on the optimal spectrum allocation in a particular time period with a particular exogenous and fixed hybrid market, i.e., with given contract users (in the futures market) and given spot transaction users (in the spot market).^{}^{}endnote: Namely, we do not consider the endogenous market formulation and evolution in this work. In a more general case, we can consider the problem of an endogenous hybrid spectrum market structure, in which an SU can choose which market(s) to join, and what contract to accept if he or she is in the futures market. Our study in this paper provides important insights into the exogenous market with fixed usermarket associations, and serves as an important first step for understanding the endogenous market. Namely, we want to answer the following question: how to optimally allocate spectrum among the given contract users and spot market users to maximize the secondary spectrum utilization efficiency in that time period. Here, the secondary spectrum utilization efficiency (also called spectrum efficiency) is defined as the total social welfare generated by all SUs from utilizing idle spectrum in the whole time period (see Section id1 for the detailed definition).
This problem is challenging due to the stochasticity and asymmetry of network information. Here, network information mainly refers to the benefit (utility) of each SU from using each (idle) spectrum in each time slot. Stochasticity means that no one can observe the network information in the future, as the network changes randomly over time. Asymmetry means that the currently realized information of an SU is the private information, and cannot be observed by other entities, such as the SR and other SUs. Nevertheless, in the hybrid spectrum market, the allocation of spectrum in the whole time period must be jointly optimized due to the timecoupling constraint on the contract user demand (e.g., each contract user requires a specific number of spectrum in the whole period). This implies that solving the optimal spectrum allocation problem directly would require the complete network information in all time slots. As mentioned above, however, in practice the SR only has partial knowledge (e.g., the stochastic distribution) about the future network information because of the stochasticity of information. Moreover, it cannot observe all of the realized current network information, especially the private information of SUs, due to the asymmetry of information. Therefore, the key research problem becomes the following:
Problem 1
How should the SR optimally allocate the idle spectrum in the given period among the contract users and spot market users to maximize the spectrum efficiency, taking into consideration the spatial spectrum reuse, information stochasticity, and information asymmetry?
To solve this problem, we first derive an offline optimal policy that maximizes the exante expected spectrum efficiency based on the stochastic distribution of network information. We then design an online Vickrey¨CClarke¨CGroves (VCG) auction that elicits SUs’ private information realized in every time slot. Based on the elicited network information and the derived offline policy, the VCG auction determines the realtime allocation and pricing of every spectrum. Such a solution technique (i.e., offline policy and online auction) allows us to optimally allocate every spectrum in an online manner under stochastic and asymmetric information. We further show that with the spatial spectrum reuse, the proposed VCG auction relies on solving the maximum weight independent set (MWIS) problem, which is wellknown to be NPhard. Thus, it is not suitable for online implementation, especially in a largescale market. This motivates us to further study lowcomplexity suboptimal solutions. To this end, we propose a heuristics approach based on an online VCGlike mechanism with polynomialtime complexity, and further characterize the corresponding performance loss bound analytically. Our numerical results indicate that the heuristics approach exhibits good and robust performance (e.g., reaches at least 70% of the optimal efficiency in our simulations).
Problem  Solution  Performance  Complexity  Section No. 

ESEM: Expected Spectrum  Policy + VCG  Optimal  Exponential  id1, id1 
Efficiency Maximization  Policy + VCGlike  Suboptimal  Polynomial  id1, id1, id1 
In summary, we list the key results and the corresponding section numbers in Table 1. It is important to note that the main contribution of this work is not the development of new auction theory, but rather the formulation of the hybrid spectrum market and the solution techniques (including the use of auction theory) to optimize the spectrum utilization in a given hybrid market. Specifically, the main contributions of this paper are as follows:

New modeling and solution technique: We propose and study a hybrid spectrum market, which has both the reliability of futures market and the flexibility of spot market. Hence, it is highly desirable for QoS differentiations in secondary spectrum utilization. To the best of our knowledge, this is the first paper to study such a hybrid spectrum market with spatial spectrum reuse.

Optimal solution under stochastic and asymmetric information: We analyze the optimal spectrum allocation in an exogenous hybrid market (in a particular time period) under stochastic and asymmetric information systematically. Our proposed solution consists of two parts: (i) an offline allocation policy that maximizes the exante expected spectrum efficiency based on the stochastic network information; and (ii) an online VCG auction that determines the realtime allocation of every (idle) spectrum based on the realized network information and the prederived policy. Such a solution technique allows us to optimally allocate every spectrum in an online manner.

Heuristic solution with polynomialtime complexity: We propose a heuristic approach based on an online VCGlike mechanism with polynomialtime complexity, and further characterize the corresponding performance loss bound analytically. This polynomialtime solution is particularly useful for achieving the efficient spectrum utilization in a largescale network.

Performance evaluation: We provide extensive numerical results to evaluate the performance of the proposed solutions. Our numerical results show that: (i) the proposed optimal allocation significantly outperforms the traditional greedy allocations, e.g., with an average increase of 20% in terms of the expected spectrum efficiency; and (ii) the proposed heuristics approach exhibits good and robust performance, e.g., reaching at least 70% of the optimal efficiency in our simulations.
The rest of this paper is organized as follows. After reviewing the literature in Section id1, we describe the system model in Section id1, and present the problem formulation in Section id1. Then we derive the offline optimal policy in Section id1, and design the online VCG mechanisms in Section id1. In Section id1, we analyze the performance loss in the lowcomplexity heuristic solution. in Section id1, we provide the detailed simulation results. We finally conclude in Section id1.
A major motivation of this work is to establish economic incentives and improve spectrum utilization efficiency in dynamic spectrum access (DSA) and cognitive radio networks (CRNs). There are several comprehensive surveys on the technical aspects of DSA and CRNs (see Haykin (2005), Akyildiz et al. (2006), Buddhikot (2007), Zhao & Sadler (2007)).
Recent years have witnessed a growing body of literature on the economic analysis (in particular the incentive issues) of DSA (see, e.g., Hossain et al. (2009), Tonmukayakul & Weiss (2009)). Marketdriven secondary spectrum trading is a promising paradigm to address the incentive issue and achieve high spectrum efficiency in DSA. The literature on secondary spectrum trading often considers pricing, contract, and auction. Pricing and Contract are generally adopted by the futures market, wherein sellers and buyers enter into certain agreements, specifying the price, quality and demand in advance. In particular, Pricing is often used under information symmetry, where sellers and buyers possess the same market information (e.g., utility and quality) or hold the same degree of uncertainty about the market information (see, e.g., Kloeck et al. (2005), Wang et al. (2008), Niyato & Hossain (2008)). Contract has been widely used in supply chain models, with a main focus on the supply chain coordination (see de Kok & Graves (2003) for a comprehensive survey). Contract is effective in a market under information asymmetry, where sellers and buyers possess different information. The seller (or buyer) can offer a selling (purchasing) contract to extract the buyer surplus (seller surplus) as much as possible (see, e.g., Gao et al. (2010), Duan et al. (2011), Kalathil & Jain (2010)). In contrast to Pricing and Contracts, Auctions are one of the most common mechanisms used by the spot market, wherein buyers compete to obtain commodities by offering different bids. Auctions are particularly suitable for the information asymmetry scenario. With a welldesigned auction, buyers (bidders) have the incentive to bid for the resource in a truthful manner, and the seller can allocate the resource efficiently without knowing the buyers’ private information in advance (see, e.g., Huang et al. (2006), Li et al. (2008), Gandhi et al. (2007), Wang et al. (2010), Zhou et al. (2008), Zhou & Zheng (2009)). However, the above work considers the secondary spectrum trading in a pure spectrum market (i.e., either spot or futures market). Instead, our setup incorporates the shortterm secondary spectrum trading in a hybrid spectrum market.
Kasbekar et al. (2010) and Muthusamy et al. (2011) considered the secondary spectrum trading in a hybrid market. In their settings, primary sellers offer two types of contracts: the guaranteedbandwidth contract and the opportunisticaccess contract. The main difference between these two prior papers and our paper lies in the formulation of the guaranteed contract. Specifically, in Kasbekar et al. (2010) and Muthusamy et al. (2011), the guaranteedbandwidth contract provides guaranteed access to a certain amount of bandwidth at every time slot. In our model, the guaranteeddelivery contract provides guaranteed access to a total amount of bandwidth in one time period; nevertheless, the bandwidth delivery at every time slot can be different, depending on the PUs’ own demand. The main advantage of our approach is its flexibility in shifting secondary demand across time slots (to comply with the PUs’ random demand). That is, it enables opportunistic delivery of a small (or large) bandwidth to SUs in those time slots that the PUs’ own demand is high (or low). Additionally, our model is also more practically relevant to a wide range of applications, which do not require fixed data delivery per time slot, but demand a guaranteed average data rate over each time period. Furthermore, the underlying market models are also different. Kasbekar et al. (2010) and Muthusamy et al. (2011) assumed that the demand (supply) markets have infinite liquidity. That is, any bandwidth amount supplied by the seller can be sold out (any bandwidth amount demanded by the buyer can be bought from the market) at an “outside fixed price”. In this sense, their market models are closely related to the ideal competitive market. We assume that the market price is endogenously determined by the associated seller and buyer (through, for example, an VCG mechanism). Thus, we essentially consider the monopoly market.
Market Type  Related Work 

Futures Market  Pricing: Kloeck et al. (2005), Wang et al. (2008), Niyato & Hossain (2008). 
Contract: Gao et al. (2010), Duan et al. (2011), Kalathil & Jain (2010).  
Spot Market  Auction: Huang et al. (2006), Gandhi et al. (2007), Li et al. (2008), Wang et al. (2010), 
Zhou et al. (2008), Zhou & Zheng (2009).  
Hybrid Market  Kasbekar et al. (2010), Abhishek et al. (2012), Muthusamy et al. (2011), Gao et al. (2012a) 
Abhishek et al. (2012) also considered a hybrid market, in which a cloud service provider sells its service to users via two different pricing schemes: payasyougo (PAYG) and spot pricing. Under the PAYG, users are charged a fixed price per unit time. Under the spot pricing, users compete for services via using an auction. They focused on the optimal market formulation, that is, the service provider selects different PAYG prices such that different users will choose different pricing schemes or market types; consequently, different hybrid markets will be formulated. In our work, we focused on the optimal spectrum allocation in a given hybrid market. In other words, each SU is associated and fixed in a particular market (based on his or her application type), and the SR determines the optimal spectrum allocation among the given contract users and spot market users.
In Gao et al. (2012a), we studied the secondary spectrum trading in the same hybrid market. However, Gao et al. (2012a) did not consider spatial spectrum reuse, which is the key contribution of this paper. With spatial spectrum reuse, the same spectrum can be potentially used by multiple SUs simultaneously; thus, the total spectrum efficiency can be greatly improved. This new coupling in the spatial dimension creates many challenges (e.g., solving MWIS problems) in the optimal mechanism design, and makes the problem significantly different from those without spatial spectrum reuse. Precise mathematical modeling and understanding of the spatial coupling are often lacking in the wireless literature. One of the main contributions of this paper is to propose a lowcomplexity heuristic algorithm to tackle this issue, and to quantify the performance bound of the proposed algorithm. For convenience, we summarize the key literature in Table 2.
The theoretic model used in our work is related to the dual sourcing problem in supply chain management. Specifically, with dual sourcing, a firm can procure a single commodity from a supplier via a longterm contract and/or from a spot market via shortterm purchases (see Kleindorfer & Wu (2003)). In such a context, the longterm contract and shortterm purchase in the supply chain model corresponds to the guaranteed contract and spot transaction in our model, respectively. Note that the firm in the supply chain model buys commodities through dual sourcing, while the SR in our model sells commodities through guaranteed contracts and spot transactions. The longterm/guaranteed contract ensures consistency over time availability of commodities with guaranteed quality at a predetermined price; whereas, the shortterm purchase/spot transaction provides high inventory flexibility, allowing firms to buy and sell commodities at any quantity with zero lead time, but at a random market price. The main advantage of dual sourcing is to hedge the future uncertainties in the supplier’s commodity supply and the end users’ demand, so as to hedgy the financial risk in the supply chain (see Kleindorfer (2008, 2010)).
The integration of longterm contracts and shortterm purchases is of particular interest to the firm in a supply chain, due to the following two reasons: (i) sourcing competition can keep purchasing prices under control by the firm; and (ii) a wider supply base can mitigate the risk induced by the uncertainty at one supplier. Lee & Whang (2002) were the first to integrate, after sales, the spot market considerations within a newsvendor ordering framework. Peleg et al. (2002) studied the longterm and shortterm integrated sourcing using a stylized two period model. Yi & SchellerWolf (2003) studied the nature of the optimal inventory policy when such a dual sourcing is used, in the presence of a fixed cost for the spot market participation. Wu & Kleindorfer (2005) proposed a general framework with integrated longterm and shortterm contract decisions for nonstorable commodities. Kouvelis et al. (2013) studied the problem of dual sourcing with financial hedging for storable commodities.
Our work differs from the above work in the following aspects. First, the spectrum in our model can be potentially used by multiple SUs simultaneously, while a traditional commodity can usually be used by one user only. Second, the availability of spectrum is stochastic, while the availability of a traditional commodity is usually deterministic. Third, we focus on the spectrum utilization efficiency maximization, rather than the financial risk hedging. Fourth, instead of designing the optimal contracts, we treat contracts as exogenously given, and focus on the problem of how to fulfill these contracts and cope with additional demand from spot markets. Finally, we consider stochastic and asymmetric network information.
We consider a DSA system with one SR and multiple SUs. The SR has certain licensed frequency band, which is divided into orthogonal channels using channelization methods such as frequency division and code division. In Fig. 1, for example, we have . The frequency band is licensed to a set of subscribed users (here we call primary users, PUs), who access channels with the slotted transmission protocol which is widelyused in today’s wireless communication systems (e.g., GSM, WCDMA, and LTE). That is, the total time is divided into fixedtime intervals, called time slots, and each PU transmits over one or multiple channel(s) according to a synchronous time slot structure. Depending on the activities of PUs, some channels may be not used by any PU (i.e., idle) in some time slots, which can be potentially assigned for the secondary utilization of SUs.
We consider the marketdriven dynamic spectrum access, also called secondary spectrum trading, in which SUs temporarily purchase the idle channels from the SR. More specifically, we consider the shortterm secondary spectrum trading, in which the idle channel is traded on a slotbyslot basis. Namely, the basic resource unit for trading is “a particular channel at a particular time slot”, referred to as a spectrum opportunity or spectrum. The main motivation for considering the shortterm spectrum trading is as follows. The spectrum availability changes frequently and randomly over time due to the stochasticity of PUs’ activities; thus, a channel that is idle in a particular time slot may not always be idle in the future.
Let denote the spectrum on the th channel at time slot . Let denote the state (availability) of spectrum , with indicating that the spectrum is not used by any PU and thus is available for SUs, and otherwise. We consider the operations in a given period of time slots. The total amount of spectrums in the period is referred to as the SR’s spectrum supply, denoted by The size of total supply is denoted by . The states of all spectrums in are referred to as the spectrum availability, denoted by
Due to the uncertainty of PUs’ activities, the spectrum availability changes randomly across both time and frequency. Fig. 1 illustrates an example of such a system, where , , and . At the time slot , the spectrum is idle, while the spectrum and are busy, i.e., and . Note that in practical wireless communication systems, is usually very large. This is mainly due to two reasons. First, the length of each time slot in a wireless system is often quite small, e.g., in milliseconds or even in microseconds, which corresponds to the typical length of frame in many wireless communication systems (such as 4G LTE and WiFi). In fact, the physical limit of choosing the time sloth length is the so called coherence time, which is the time within which the channel condition does not change. In wireless communications, such coherence time is usually very small due to fast small scale multipath fading. Second, the time scale of each time period is relatively large, e.g., in minutes or even hours, which corresponds to the validity period of contracts (to be defined in Section id1). Hence, the number of slots in each time period (i.e., ) is very large, e.g., when the length of slot is 1 millisecond and the length of period is 10 minutes.
Following are our assumptions on the spectrum availability and spectrum usage.
Assumption 1
The spectrum availability is independently and identically distributed (i.i.d.) across time and frequency.^{}^{}endnote: We will show in Gao et al. (2012b), that our results can be generalized to the noni.i.d. spectrum availability case with minor modifications. Moreover, the spectrum availability is ergodic.
Assumption 2
Each SU can transmit over multiple channels simultaneously.^{}^{}endnote: This is supported by most physical layer access technologies (e.g., OFDMA) even with just one transmitter antenna per SU, and is possible with other technologies when each SU has multiple transceivers.
For convenience, we denote the availability of spectrum as a random variable , and the probability mass function of as .^{}^{}endnote: We will use and to denote the probability distribution function (PDF) and cumulative distribution function (CDF) of a continuous random variable or vector . For notational convenience, we use the same notation and to denote the probability mass function and CDF of a discrete random variable . Suppose is the idle probability of spectrum . Then, if , and if .
Wireless applications can be broadly categorized as either elastic or inelastic depending on QualityofService (QoS) requirements.^{}^{}endnote: Note the meaning of “elastic/inelastic demand” in wireless networks (in this paper) is a bit different from that in microeconomics, where it is mainly used to characterize whether a consumer’s demand changes with the market price. In wireless networks, however, it is mainly used to characterize an SU’s inherent demand for spectrums. With elastic traffic, SUs have elastic demands for spectrum, in the sense that the tasks are not urgent in terms of time; thus, the QoS will not be significantly affected, even if the spectrum resource is limited and the transmission rate is low for a substantial amount of time. With inelastic traffic, SUs have inelastic demands for spectrum. In other words, the tasks can only function well when the data rate is guaranteed to be above certain thresholds such that the delay requirements are met; otherwise they will suffer significant performance loss. It is important to note that the information of the application type is usually explicitly represented in the headers of data packets, and can be easily extracted by the network operator through deep packet inspection. Hence, it is reasonable to assume that an SU cannot fake his or her application type arbitrarily. Examples of elastic traffic include FTP downloading, data backup, and cloud synchronization, and examples of inelastic traffic include VoIP, video streaming, and realtime data collection.
To accommodate the various requirements of elastic traffic and inelastic traffic (so as to achieve desirable QoS differentiations), we propose a hybrid spectrum market combining both the futures market and the spot market. Next we define the futures market and spot market formally.
Futures Market. In the futures market, each SU enters into an agreement, called a futures contract or guaranteed contract, directly with the SR. The contract specifies the key elements in trading, e.g., the SU’s total demand and payment in the given period. Once the SR accepts a contract, it is committed to deliver the specified number of spectrum to the SU. If the SR fails to do so, it needs to pay certain penalty for compensating the SU’s potential welfare loss. This penalty can be a unit price paid for every undelivered spectrum (called a soft contract) or simply a total payment for the violation of contract (called a hard contract).
Spot Market. In the spot market, each SU purchases spectrums in a realtime and ondemand manner through spot transactions. That is, an SU initiates a purchasing request only when he or she needs spectrum, and multiple SUs requesting the same spectrum would compete with each other for the spectrum, e.g., through an auction. The spectrum is delivered immediately to the winner at a realtime market price, which depends on both the SUs’ preferences and competitions. The winner’s payment is also dependent on the SUs’ preferences and competitions.
Although, in practice, it is more desirable to allow SUs to have the flexibility to choose which market to join, in this work we assume that the SUs with inelastic traffic always choose the futures market, and the SUs with elastic traffic always choose the spot market.^{}^{}endnote: As mentioned previously, one key motivation for this assumption is that SUs cannot fake their applications arbitrarily. In Gao et al. (2012b), we will further show that even if an SU has the capability of faking his or her application type, he or she does not have the incentive to do so, if the hybrid market is properly designed. In other words, we will leave out the SU’s market selection problem, and instead focus exclusively on the spectrum allocation in the given hybrid markets in each time period. This setup helps us to concentrate on the technical challenges brought by spatial spectrum reuse and information stochasticity & asymmetry.
Next we provide the key assumptions on the hybrid market structure.
Assumption 3
We assume a quasistatic market scenario, where the hybrid spectrum market is exogenously given and fixed in each time period, but may change across different time periods.^{}^{}endnote: This means that the usermarket association is fixed, and the contract of each futures market user is also fixed during the whole period of interest. The changes of hybrid market in different time periods can be caused by the changes of user parameters (e.g., user service types) as well as the changes of the contract selection of the SR at the beginning of each period.
Assumption 4
We assume that each SU has one application, and all SUs with inelastic traffic (elastic traffic) are in the futures market (spot market).^{}^{}endnote: Note that the assumption can be easily generalized: For an SU having multiple applications with different QoS requirements, we can simply divide the SU into multiple virtual SUs, each associated with one application.
Based on these assumptions, we can divide SUs into two disjoint sets, each associated with one market. Let and denote the sets of SUs in the futures market and the spot market, respectively. For convenience, we will use the notation “” to denote a contract user, and “” to denote a spot market user. When needed, we will use the superscripts “c” and “s” to indicate variables related to the futures market and spot market, respectively.
For analytical convenience, we further assume that the size of futures market is much smaller than that of the spot market, i.e., . This assumption is used to facilitate the computation of independent contract user sets in the later analysis.^{}^{}endnote: Although this is an assumption, it can be justified by the fact that the SR can actually control the size of the futures market, via intelligently choosing a limited set of contract users at the beginning of each period. Namely, to balance the performance and the complexity, the SR can intelligently choose a proper set of contract users to serve in a particular time period. Hence, from the system perspective, the size of the futures market is controllable, whereas the size of spot market is generally random and uncontrollable.
A contract could be quite complicated, depending on specific requirements of a contract user’s application. In this work, we focus on a basic contract form, which consists of two parts: (i) the SU’s demand and payment for spectrum in each period; and (ii) the SR’s penalty when not delivering the demanded spectrums to the SU. Formally, we write the contract of an SU as:
(1) 
where is the SU’s payment, is the SU’s demand in one period, and is the SR’s penalty scheme. Here, indicates the contract type (soft or hard), is the unit penalty for a soft contract (), and is the total penalty for a hard contract (). Thus, the actual payment of contract user (denoted by ) depends on the number of spectrums he or she actually obtains (denoted by ), that is:
(2) 
where is the SR’s penalty if violating the contract, given by:
(3) 
where and . That is, if , and and if . For convenience, we denote the set of all contracts in the futures market by . As each contract user is guaranteed to obtain the desired number of spectrum (otherwise he can get the SR’s penalty as compensation), we refer to such a contract as the guaranteeddelivery contract.
The important assumptions on the guaranteed contract are listed below.
Assumption 5
We assume that contract users care about the expected number of spectrums they obtain (hence the SR’s penalty is based on the expected number of allocated spectrums).
The main practical motivation for such an assumption is that most wireless applications in practice require an expected/average data rate during a certain time period. For example, video streaming concerns the average downloading rate in every minute, and VoIP concerns the average downloading/uploading rate in every second. When the actual data rate is occasionally less than the average rate, various coding and error concealment technologies can be employed, so that the SUs will not feel significant performance degradation. Thus, with a proper choice of the length of allocation period (e.g., one minute for video streaming or one second for VoIP), such a contract with the expected demand is suitable for most wireless applications. In fact, with the uncertainties of spectrum availability and the stochasticity of SU utility on each spectrum, it is practically impossible to guarantee a strict spectrum supply for each contract user. To the best of our knowledge, no such guarantee has been provided even in the latest communication standard. To make our analysis more complete, we also provide detailed theoretical analysis and numerical evaluation for the impact of this assumption on the performance in Gao et al. (2012b).
As mentioned previously, an important characteristic of a spectrum market is that radio spectrum is spatially reusable subject to certain spatial interference constraints (see Tse & Viswanath (2005)). For example, a spectrum can be used by multiple SUs simultaneously if they are far enough apart and do not interfere with each other. This leads to a fundamental difference between the spectrum market and conventional markets: the same spectrum can be potentially sold to multiple SUs.
In this work, we adopt a widelyused physical model to capture the interference relationships between SUs, namely, the protocol interference model (see Gupta & Kumar (2000)). Under the protocol interference model, multiple SUs can use the same spectrum simultaneously without mutual interference, if and only if none of the SUs falls inside the interference ranges of others (e.g., they are sufficiently far away from each other). The network under a protocol interference model can usually be represented by a conflict graph, with each vertex denoting an SU, and each edge indicating that two associated SUs fall inside the interference range of each other. Thus, any two SUs not connected by an edge (also called independent users) can use the same spectrum simultaneously, while any two connecting SUs cannot.
Fig. 2 illustrates the conflict graph representation for a network with 6 SUs, in which each dash circle denotes the interference range of the associated SU. In this example, SUs 1 and 3 are independent of each other (and therefore can use the same spectrum simultaneously), while SUs 1 and 2 are not. Fig. 1 also illustrates a similar conflict graph, where SUs 2, 6, and 8 are independent of each other, whereas SUs 1, 2, and 3 are not.
Let denote the graph consisting of all SUs (and the associated edges, similarly hereafter), where is the set of vertices (SUs), and is the set of edges. Let denote the spot market subgraph consisting of all spot market users, and denote the futures market subgraph consisting of all contract users, that is, , , , and . For presentation convenience, we further introduce the concept of side market.
Definition 1 (Side Market)
The side market (subgraph) of a contract user set , denoted by , is a subgraph of the spot market, consisting of all spot market users except the neighbors of contract users .
Here, the “neighbors” of refer to spot market users connecting to at least one contract user in the set . It is easy to see that for any , since a neighbor of must be a neighbor of if . In Fig. 1, we have , , and examples of side markets include and . Obviously, .
Let and denote the sets of all independent sets and all cliques of a graph G, respectively.^{}^{}endnote: The detailed definitions for independent set and clique can be referred to a standard textbook, e.g., West (2001). Then, the spatial interference constraint can be defined as follows.
Definition 2 (Spatial Interference Constraint – SIC)
Any two SUs in a clique of G cannot use the same spectrum simultaneously; on the other hand, all SUs in an independent set of G can use the same spectrum simultaneously.
Finally, we list the key assumption on the network topology or graph G below.
Assumption 6
We assume that the network topology G is exogenously given, and remains unchanged during the whole period of interest.
This corresponds to a static or quasistatic network scenario, in which SUs do not move or move slowly. With this assumption, it is reasonable to assume that the network topology is public information to the SR.
The valuation of an SU over a spectrum reflects the SU’s preference for the transmission rate (capacity) achieved on spectrum . Specifically, it is determined by two factors: (i) the transmission rate of SU over spectrum (denoted by ), reflecting how efficiently SU utilizes spectrum ; and (ii) the user preference of SU (denoted by ), reflecting how eagerly SU desires for spectrum . To avoid confusion, we use notation and to denote the valuations of spot market user and contract user over spectrum , respectively. Formally, we can write the valuations and as generic functions:
(4) 
where is the generic valuation function of SU , which generally increases in both the transmission rate and the user preference. Typical examples of valuation function include: (a) linear functions, e.g., ; (b) sigmoid functions, e.g., ; and (c) step functions, e.g., if exceeds a certain threshold , and 0 otherwise.
The user preference is a subjective feeling of SU for spectrum , and is related to factors such as traffic state, urgency, and importance. The transmission rate is an objective attribute of spectrum for SU , and is related to a spectrumspecific quality and a userspecific channel coefficient . The is common for all SUs, regarding factors such as channel bandwidth, noise level, and power constraint. The is usually independent between different SUs, regarding factors such as path loss, shadow fading, and small scale fading (or multiplepath effect). Typically, is given by the ShannonHartley theorem (see Shannon (2001)):
(5) 
where is the channel bandwidth, is the transmission power, is the noise level over spectrum (captured by the common spectrum quality ), and is the channel gain of SU over spectrum (captured by the userspecific channel coefficient ).
By (id1) and (id1), an SU’s valuation (over a spectrum) is essentially a joint function of the common spectrum quality, userspecific channel coefficient, and user preference. For notational convenience, we will write it as and . In general, the spectrum quality, userspecific channel coefficient, and user preference change randomly across time and frequency. For simplicity, the following assumptions are used in our work.
Assumption 7
The spectrum quality, userspecific channel coefficient (of every SU), and userpreference (of every SU) are all i.i.d. across time and channels.
Based on the above assumption, we can see that the valuation of each SU is also i.i.d. across time and channels.^{}^{}endnote: Note that in practice, an SU’s valuation may not be i.i.d. across time and channels, due to the noni.i.d. of spectrum quality, userspecific channel coefficient, and/or user preference. We will show in Gao et al. (2012b), that our work can be easily applied to the scenario with noni.i.d. valuation across time or channels. Let random variable denote the spectrum quality, and denote the userspecific channel coefficient and preference of SU . Let random variables and denote the valuations of spot market user and contract user , respectively. We can further see that the valuations of different SUs (over the same spectrum) are partially correlated by the common spectrum quality . That is, the valuation of each spectrum is partially correlated across users. Such a valuation formulation generalizes both the case of independent user valuation (by removing the uncertainty of ) and the case of completely correlated user valuation (by removing the uncertainty of and , ).
Let random vector denote the valuations (vector) of all SUs, and denote a realization of on spectrum . As will be shown later, in the spot market, the valuation uniquely reflects the SU ’s maximum willingnesstopay (or utility) for the spectrum . In the futures market, however, the valuation only partially reflects the SU ’s maximum willingnesstopay (or utility) for the spectrum , which depends also on the total number of spectrums that he obtains. Since each spectrum is fully characterized by the utility vector and its availability , we also refer to as the information of spectrum . Formally,
Definition 3 (Network Information)
The information of each spectrum consists of its availability and the valuations of all SUs over that spectrum, i.e., . The information of all spectrums in is referred to as the network information, i.e., .
For convenience, we will refer to a spectrum with information as spectrum . Note that the private information of a contract user (or spot market user ) is defined as his or her valuation (or ) for each spectrum . That is, we consider all other information of a contract user (e.g., the spectrum demand) as public information. It is important to note that in a broad sense, we can generalize our current analytical framework by allowing multidimensional private information, as we adopt a general VCG framework in the later analysis.
When the SR leases an idle licensed spectrum to an SU for secondary utilization, the SU achieves a certain benefit, called SUsurplus, which equals the difference between the total utility achieved from using these spectrums and the payment transferred to the related PU; the PU also achieves a certain benefit, called PUsurplus, which equals the difference between the payment from the SU and the cost of holding spectrum and sharing spectrum with unlicensed SUs.^{}^{}endnote: Such a cost may include the spectrum license fee, management fee, maintenance fee, etc. The social surplus generated by such a process is the sum of the SUsurplus and the PUsurplus, which equals the difference between the (SU) utility generated and the (PU) cost involved. That is:
Notice that the PU’s cost is a constant (sunk cost) independent of the secondary spectrum allocation. Without loss of generality, we normalize it to zero. As a result, the social surplus is equivalent to the utility generated by the SU. Next we define the utility of each SU formally.
Utility of Spot Market User. In the spot market, based on the characteristics of elastic traffic, an SU achieves an immediate utility from any spectrum allocated, and such a utility is directly related to his or her valuation of the spectrum. Thus, the total utility of a spot market user (denoted by ) can be directly defined as the total valuation achieved: ^{}^{}endnote: Our method can be directly applied to a general case, in which the SU’s utility is a generic function of his valuation. In that case, we have , where is the valuation function of SU .
(6) 
where indicates whether to allocate a spectrum to SU . The factor means that only the idle spectrums can be used by SUs.
Utility of Contract User. In the futures market, however, an SU’s utility is not only related to the valuation that he or she achieves, but also related to the total number of spectrums that he or she obtains. First, depending on the requirement of inelastic traffic, an SU achieves a fixed total utility if the number of allocated spectrums meets (or exceeds) the demand; otherwise the SU suffers certain utility loss. Such a utility is usually referred to as the demandrelated utility, which coincides with the SU’s payment defined in the contract, i.e., Eq. (id1).^{}^{}endnote: It is worth noting that we are not contending that the contract user’s actual payment in (id1) or the contract details in (id1) determines the user’s (demandrelated) utility definition; on the contrary, the user’s (demandrelated) utility determines the type of contract that he or she will accept and the payment that he or she is willing to pay. For convenience, we will use the notation , , , and to denote the spectrum demand, the fixed total utility when the demand is satisfied, the unit utility loss (for soft contract) and the fixed total utility loss (for hard contract) when the demand is not satisfied, respectively. Since we do not address the problem of contract design in this work, we simply consider that each contract coincides perfectly with the contract user’s utility definition, i.e., , , , and . Thus, the total demandrelated utility of a contract user (denoted by ) has exactly the same form as his actual payment in the contract, that is:
(7) 
where indicates whether to allocate a spectrum to SU , is the total number of idle spectrums allocated to the SU , and is the potential utility loss of the SU caused by the unmet demand, that is:
(8) 
which has the same form as the SR’s potential penalty defined in (id1).
Second, similar as spot market users, a contract user’s utility is also related to the valuation that he or she can achieve on the allocated spectrum (or equivalently, the quality of the allocated spectrum). The reason is following: to achieve a desirable qualityofservice (QoS), a contract user needs to pay different levels of efforts (e.g., transmission powers) on spectrums with different qualities. For example, he may achieve the desirable QoS with a lower effort on a spectrum with a higher quality, hence lead to a higher utility. Such a utility is referred to as the qualitybased utility. Similar as that of spot market users, the total qualitybased utility of a contract user (denoted by ) can be defined as the total valuation achieved:
(9) 
To capture the above two factors in a contract user’s utility, we define the utility of contract user as a weighted sum of the demandrelated utility and the qualitybased utility , i.e.,
where is the factor weighting and . Such a utility definition is flexible in capturing contract users’ different preferences regarding spectrum demand and spectrum quality.
The spectrum secondary utilization efficiency (or spectrum efficiency) is the aggregate utility of all SUs, also called social welfare. Specifically, we have the following:
Definition 4 (Spectrum Efficiency)
The spectrum efficiency (or social welfare) is
(10) 
given any spectrum allocation that is feasible.^{}^{}endnote: A spectrum allocation is feasible if it satisfies the spatial interference constraint (SIC) in Definition 2.
Based on the above, the spectrum efficiency maximization (SEM) problem can be defined as follows: finding a feasible spectrum allocation , such that the spectrum efficiency defined in (10) is maximized.
We first formulate the spectrum efficiency maximization (SEM) problem under deterministic (complete) network information (Section id1), which is essentially a matching problem, and can be solved by many existing algorithms. Then we formulate the expected spectrum efficiency maximization (ESEM) problem under stochastic network information (Section id1), which is an infinitedimensional optimization problem. We will focus on solving the ESEM problem in Section id1.
With complete and deterministic network information, i.e., , the spectrum efficiency maximization (SEM) problem can be formulated as a matching between all spectrums and all SUs, and the optimal solution explicitly specifies the allocation of every spectrum to a particular set of (independent) SUs. Let (or ) indicate whether to allocate a spectrum to spot market user (or contract user ). Denote:
as the allocation (vector) of spectrum , which contains the allocation of spectrum to every SU. An allocation strategy consists of the allocations of all spectrums in the whole time period, denoted by . Intuitively, the allocation strategy is a mapping from every particular spectrum to an allocation vector .
By the Spatial Interference Constraint (SIC) given in Definition 2, an allocation strategy is feasible, if and only if the allocation for every spectrum satisfies:
(11) 
where is the th clique of graph G, and and denote the spot market user set and contract user set in , respectively. It is easy to see that , if (i.e., is not idle). Intuitively, (11) states that (i) only idle spectrums can be used by SUs, and (ii) every idle spectrum can be allocated to, at most, one SU in a clique.
According to the definition of spectrum efficiency in Definition 4 and the SIC in (11), the spectrum efficiency maximization (SEM) problem can be formally defined as follows:
(12)  
where is the number of spectrums allocated to contract user . The last constraint in (12) states that none of the contract users will get spectrums more than his or her demand. The reason for this is that overallocation brings trivial welfare gain for contract users. It is easy to see that the above problem is an 01 integer linear programming (also called matching problem), and many algorithms, e.g., branchandbound algorithm and KuhnMunkres algorithm (see Munkres (1957)), have been developed to solve a matching problem effectively. Due to space limitations, we omit the details here.
In practice, the assumption of complete and deterministic information is often unreasonable, as the network changes randomly over time. Hence, in this section we consider the case in which the network information is stochastic and the SR only knows its distribution, but not realization. In this case, we formulate the expected spectrum efficiency maximization (ESEM) problem based on the stochastic distribution (e.g., PDF or CDF) of network information.^{}^{}endnote: This stochastic network information can be obtained empirically after counting the realized network information over a sufficiently long time period. Note that in Gao et al. (2012b), we also study the problem when this stochastic distribution information is unavailable. In that case, we propose a learning mechanism, which converges to the optimal solution without stochastic information. The solution specifies the allocation of any spectrum under every possible information realization (rather than the explicit allocation of every particular spectrum). In this sense, it essentially defines a contingency plan (also called a policy) for allocating every spectrum under every possible information realization.
Based on the above, we can define the allocation strategy as a mapping from every possible information realization (rather than every particular spectrum, as under deterministic information) to an allocation vector consisting of the allocation probability to every SU. However, as shown in Gao et al. (2012b), such a primitive definition is not appropriate due to the high complexity in determining the SIC constraint.
To this end, we propose a new definition for the allocation strategy. The basic idea is to consider each independent SU set as a virtual player, and decide the spectrum allocation among the virtual players. That is, we transform the primitive allocation strategy, regarding the allocation probability (of every information realization) to every SU, into an equivalent strategy, regarding the allocation probability to every independent SU set.
Denote as the th independent set of the graph G, and as the total number of independent sets in the graph G. Let denote the allocation probability of a spectrum to independent SU set . The allocation probability vector of spectrum is:
and the new allocation strategy can be formally defined as follows.
Definition 5 (Allocation Strategy)
An allocation strategy is a mapping from every information realization to a vector consisting of the allocation probability of a spectrum to every independent SU set, denoted by .
Since the SUs in an independent set can use the same spectrum simultaneously, we can easily find that strategy is feasible, if the allocation for every spectrum satisfies:
(13) 
That is, the condition in (13) is sufficient for a feasible allocation strategy. The next question is whether it is able to represent all feasible allocations. The answer is YES, as shown below.
Proposition 1
For detailed proof, please refer to Gao et al. (2012b). Proposition 1 implies that the condition in (13) is an effective SIC constraint. From Proposition 1, we can see that the main advantage of such a transformation is that we can systematically identify the necessary and sufficient conditions of a feasible allocation strategy (i.e., the SIC constraint), which is essential for the ESEM problem formulation. Certainly, this will bring some challenges, among which the most critical one is that the problem of finding all independent sets is NPhard. This motivates our later study on the approximate algorithm and the associated performance loss analysis.
Next we can provide the ESEM problem formulation based on the new allocation strategy. Let (or ) indicate whether a contract user (or spot market user ) belongs to the th independent set, i.e., if , and otherwise. Then, the total allocation probability of spectrum to contract users or spot market user can be written as:
(14) 
Given a feasible allocation strategy , the expected welfare generated by a spot market is:
(15)  
where is the spot market user ’s utility (i.e., the th element) in the utility vector , and is the joint PDF of , i.e., . The second equation follows because and are i.i.d. across time and channels. The third equation follows because (i) if and (ii) if . Note that we write as and as for convenience.
The expected number of spectrums allocated to a contract user is:^{}^{}endnote: Here we impliedly use the timeaverage value to approximate the expected value. This is available due to the ergodicity of network information and allocation strategy.
(16)  
Then, the expected demandrelated utility generated by a contract user is:
(17) 
where is defined in (8). Note that the above utility formula is based on the expected number of spectrums that the contract user obtains, rather than the explicit number as in (7). The major motivation for this assumption is discussed in Section id1.
The expected qualitybased utility generated by a contract user is:
(18)  
where denotes the contract user ’s valuation in the vector .
The expected spectrum efficiency (expected social welfare) is the expected aggregate utility generated by all SUs, that is:
Therefore, we have the following expected spectrum efficiency maximization (ESEM) problem:
(19)  
Obviously, the ESEM problem (19) is an infinitedimensional optimization problem (see Fattorini (1999)), in which the solution (or strategy) is not a number or a vector, but rather a continuous quantity. By the second constraint (SIC), we immediately have: if , . Therefore, in the rest of this paper, we will focus on solving , i.e., the allocation of idle spectrums, denoted by . For notational convenience, we will write the tuple as , and thus denote .
In this section, we solve the ESEM problem given in (19). We first reduce the strategy space by removing some irrelevant strategies, so as to reduce the computational complexity.^{}^{}endnote: The “irrelevant strategies” denote those that never emerge in an optimal solution. Therefore, ignoring the irrelevant strategies does not affect the optimality of the solution. We will show this in Proposition 2. Then we derive the optimal solution systemically using the primaldual method. Due to space limits, we present all proofs in Gao et al. (2012b).
Now we simplify the ESEM problem (19) by restricting our attention to a particular subset of the strategy space. Define the weight of each vertex in the graph G as the utility of the associated SU. Notice that the welfare generated by a spot market user is exactly his or her utility, and thus we have: if an idle spectrum (or spectrum ) is allocated to an independent contract user set , it must be allocated to an MWIS of the side market at the same time. This observation helps us to eliminate some strategies without affecting the optimality of the solution.
Let and denote the spot market user set and contract user set in (i.e., the th independent set of graph G), respectively. Formally, we have:
Proposition 2
Proposition 2 implies that in an optimal solution, an idle spectrum will never be allocated to an independent set where is not an MWIS of the graph . We refer to such an independent set as an irrelevant independent set. Obviously, we can ignore those allocation strategies that allocate spectrums to the irrelevant independent sets, and focus only on the allocation among the “relevant” independent sets (where is an MWIS of the graph ). In effect, we can restrict our attention to the independent sets of the futures market graph , instead of the independent sets of the whole graph G: if a spectrum is allocated to an independent contract user set (say ) of , it must be allocated to an MWIS of the side market graph at the same time. Consider the example in Fig. 1. Any idle spectrum will not be allocated to the independent set , since is not an MWIS of ; otherwise, we can immediately increase the total welfare by reallocating the spectrum to another independent set .
The above result allows us to simplify the ESEM problem. Denote as the th independent contract user set of the futures market graph , and as the total number of independent sets of the graph . Let