Carrier Aggregation Between Operators in Next

1 Carrier Aggregation between Operators in Next Generation Cellular Networks: A Stable Roommate Market Yong Xiao, Senior Member, IEEE, Zhu Han, Fellow, IEEE, Chau Yuen, Senior Member, IEEE, Luiz A. DaSilva, Senior Member, IEEE

Abstract—This paper studies carrier aggregation between next generation of mobile technology will rely on CA to multiple mobile network operators (MNO), referred to as achieve its promised peak data rates. In a typical network, inter-operator carrier aggregation (IO-CA). In IO-CA, each a mobile network operator (MNO) may aggregate MNO can transmit on its own licensed spectrum and aggregate the spectrum licensed to other MNOs. We focus frequency resource blocks contiguously or on the case that MNOs are distributedly partitioned into non-contiguously within a single frequency band, i.e. small groups, called IO-CA pairs, each of which consists of intra-band CA, or it may aggregate resources which are two MNOs that mutually agree to share their spectrum with located in separate frequency bands. While much of the each other. We model the IO-CA pairing problem between current work on CA investigates the aggregation of MNOs as a stable roommate market and derive a condition for which a stable matching structure among all MNOs exclusive blocks of spectrum from the perspective of exists. We propose an algorithm that achieves a stable typical macrocell topologies [3]–[5], we study carrier matching if it exists. Otherwise, the algorithm results in a aggregation (CA) for cellular networks from a cognitive stable partition. For each IO-CA pair, we derive the optimal radio (CR) network perspective. transmit power for each spectrum aggregator and establish In this paper, we investigate a framework that allows a Stackelberg game model to analyze the interaction between the licensed subscribers and aggregators in the multiple MNOs to access and aggregate each other’s spectrum of each MNO. We derive the Stackelberg licensed spectrum for the purpose of providing more equilibrium of our proposed game and then develop a joint spectrum for the low power elements of their network optimization algorithm that achieves the stable matching topology. As such, we propose a system that allows for structure among MNOs as well as the optimal transmit the dynamic aggregation of spectrum resources over both powers for the aggregators and prices for the subscribers of each MNO. the MNO’s own spectrum holdings and the spectrum holdings of other MNOs. In this scenario, an MNO may Index Terms—Carrier aggregation, cellular network, operate high power macrocells in its own exclusively cognitive radio, stable roommate, stable marriage, matching, graph, game theory. licensed spectrum and may dynamically aggregate additional sub-bands, for lower power use, in another network’s licensed spectrum. We refer to this type of I.INTRODUCTION carrier aggregation as inter-operator CA (IO-CA), i.e. a With the fast growing demand for mobile data service, heterogeneous mix of users, having different rights and it becomes more and more difficult to allocate a wide and employing different transmit powers, exploits the same contiguous frequency band to support high speed data frequencies over the same area. communication for each user equipment (UE). A new From a CR network perspective [6]–[8], each MNO technology proposed in LTE-Advanced (LTE-A) [2], and its corresponding subscriber UEs, who are spectrum referred to as carrier aggregation (CA), allows network license holders, are also referred to as the primary users operators to support high data rates over large bandwidths (PU). The subscriber UEs of each MNO have priority to by aggregating frequency resources that lie in different use their own licensed spectrum, but they can also tolerate frequency bands or which may not be contiguous. The a certain interference increase caused by subscriber UEs from other MNOs. These subscriber UEs from other Some of the results in this paper have been presented at the IEEE MNOs, also referred to as secondary users (SU), can Global Communications Conference (GLOBECOM), Anaheim, CA, USA, access the spectrum licensed to the MNO as long as the Dec., 2012 [1]. This work is partially supported by the National Science Foundation under Grants CNS-1443917, ECCS-1405121, CNS- resulting interference is lower than the tolerable level of 1265268, and CNS-0953377, the Science Foundation Ireland under Grant the spectrum license holders, i.e. the PUs. We propose 10/IN.1/I3007, and A*Star SERC Project Grant 142 02 00043. two IO-CA approaches: a direct extension of the Y. Xiao and Z. Han are with the Department of Electrical and Computer Engineering at University of Houston, TX, USA 77004 (e-mails: traditional CA in LTE Advanced into the inter-operator [email protected] and [email protected]). scenario, called regular IO-CA, and a spatial spectrum C. Yuen is with Singapore University of Technology and Design, sharing-based IO-CA framework for multi-operator Singapore 487372 (email: [email protected]) L. A. DaSilva is with CONNECT, Trinity College Dublin, Ireland cellular networks, called sharing IO-CA. In both Dublin 2 (e-mail: [email protected]). approaches, each MNO and its subscriber UEs are 2 regarded as PUs in its own spectrum. Some UEs from introduce two operations: deletion and addition, for each MNO can also access and aggregate the spectrum of cellular networks. Specifically, if an MNO observes other MNOs, where they will be regarded as the SUs and increasing traffic demands in its network and would should always control their access to keep the resulting like to seek extra network capacity by forming an interference under a given tolerable level. IO-CA pair with others, it will join the roommate There are several challenges to enable IO-CA between market by using the addition operation to decide its MNOs. Specifically, in the traditional CA within the IO-CA pairing partner and still maintain the stability spectrum of one MNO, the MNO’s infrastructure (e.g., of the existing partitions. Similarly, if the service eNB in LTE Advanced) controls and manages the demand for an MNO in an IO-CA pair decreases to spectrum aggregation behavior of its UEs in a centralized a level that can be satisfied without using IO-CA, the fashion. In cellular networks with multiple MNOs, delete operation will be applied to remove this MNO however, there is no central controller and, because of from the existing IO-CA pair. We observe that a privacy and business reasons, each MNO cannot disclose stable matching structure may not always exist. We its private information (such as the payoffs and preference derive a condition for which a stable matching of its UEs and the performance improvement brought by structure exists and propose an algorithm that can IO-CA) to other MNOs. In addition, since each MNO has detect whether this condition is satisfied and, if it is, already been licensed an exclusive spectrum band and achieves this matching structure. does not have to always rely on IO-CA to achieve the 2) Price Adjustment Problem: in this problem, each basic quality-of-service (QoS) for its subscriber UEs, each MNO charges a price to the subscriber UEs from MNO will only allow its spectrum to be aggregated by other MNOs that aggregate its spectrum. We observe others when it has an incentive to do so. that for each MNO, charging a high price to the One approach that has been proposed in the existing aggregators will deter the potential MNOs that are literature is for all the operators to merge their licensed willing to form an IO-CA pair. On the other hand, spectrum to form a common spectrum pool [9]–[12]. charging a low price will decrease the profit and However, the spectrum pooling system generally requires increase possible interference between pairing all operators to give up their exclusive use of spectrum. In MNOs. We hence propose a Stackelberg game-based addition, the coordination and competition for the hierarchical framework to investigate the case where spectrum usage among all the operators and their one MNO and its subscriber UEs are the leaders in corresponding UEs does not always lead to an efficient their own licensed spectrum, with the ability to set solution, especially when the size of the coverage area prices for secondary use of the spectrum, and and the number of operators and UEs becomes large [12]. subscriber UEs, also called the aggregators, of other A tradeoff between spectrum pooling and intra-operator MNOs are the followers who can optimize their CA can be achieved by partitioning all MNOs into small performance under the prices imposed by the leaders. groups, each of which consisting of a limited number of 3) Power Control Problem: in this problem, each MNOs that are willing to coordinate and share their subscriber UE using sharing IO-CA can optimize its spectrum with each other. However, as observed in [8], transmit power to further improve its performance there may not always exist a stable coalition formation without causing intolerably high interference to the structure in a distributed multi-agent system and even if it subscriber UEs of the primary operator. exists, there is still a lack of an effective algorithm to 4) Joint Optimization problem: we consider the joint allow all MNOs to distributedly negotiate and form this optimization of the above three problems and structure. That is, finding a coalition formation structure develop a distributed algorithm to jointly optimize in a distributed multi-agent system has been proved to be the resulting decisions. NP-hard [13], [14]. To the best of our knowledge, this is the first work that In this paper, we study the joint optimization for an IO- studies the optimization of IO-CA in cellular networks, CA-based cellular network with multiple MNOs. We focus adopting a framework that combines the stable roommate on solving four optimization problems: market problem and a Stackelberg game. 1) IO-CA Pairing Problem: in this problem, we focus We observe that not all network systems can support on the case that all MNOs can be partitioned into the optimization of all the above problems at the same different groups, called IO-CA pairs, each of which time. Therefore, in addition to presenting numerical consists of two MNOs. In our model, all MNOs can results for our joint optimization algorithm, we also establish a preference over each other and an IO-CA compare the performance improvement brought by each pair can only be established when two MNOs individual solution in our proposed optimization mutually agree to share spectrum with each other. framework. We model this problem as a stable roommate market The remainder of this paper is organized as follows. and seek a stable matching structure among all The background and related works are presented in MNOs such that no MNO or a pair of MNOs has the Section II. The network model and problem formulation intention to unilaterally deviate. We allow each MNO are established in Section III. The game theoretic analysis to dynamically join or leave an IO-CA pair and is presented in Section IV. We provide numerical results 3 and discussions in Section V and conclude the paper in Set of sub-bands for MNO M3 Set of sub-bands for MNO M4 ...... Section VI. M4 1 S4 ˆ1 M3 II.BACKGROUNDAND RELATED WORKS 2 S34 S 3 Most existing works in inter-operator spectrum sharing M1 3 M2 S1 focus on cases in which multiple operators share or 2 S 3 1 S1 Sˆ compete for a common pool of spectrum resources. For 2 21 example, the authors in [15] have studied power ...... allocation problem for multiple competitive operators Set of sub-bands for MNO M1 Set of sub-bands for MNO M2 coexisting at the same time in the same spectrum. The authors in [16] have applied the two-player non-zero-sum Fig. 1. Network model for an IO-CA-based cellular network with 4 games to study the spectrum allocation problem for MNOs. multiple operators that share a common spectrum. In [17], the authors have focused on the spectrum allocation have different meanings in different systems. For problem by assuming all the operators are centrally example, if each MNO corresponds to a cellular coordinated and studied a multi-carrier wave-form based telecommunication network operator, a UE is equivalent inter-operator spectrum sharing concept. In [12], spectrum to a cellular UE and the corresponding communication pooling has been modeled as a hierarchical game and a channel connecting itself and the infrastructure (e.g., base joint optimization framework has been proposed. Different station). If each MNO corresponds to a device-to-device from these existing works, in our paper, each operator has communication network, each UE then becomes the been licensed an exclusive portion of the spectrum and communication channel between a pair of can autonomously decide whether to share its licensed device-to-device source and destination. Let the set of all spectrum with others. UEs currently using the service of MNO M be S . It has We study the joint optimization problem for cellular i i been recently proposed in [32] that each UE should be networks with multiple MNOs from the game theoretic able to access more than one CC to support high data rate perspective. Game theory has been widely applied to transmission. In this paper, we use the term “sub-band” to study distributed optimization problems for spectrum denote the subset of CCs that can be allocated to each sharing-based CR networks. More specifically, the authors UE. Let the set of all sub-bands of MNO M be B . The in [18], [19] have introduced a non-cooperative game i i list of notation used in this paper is provided in Table I. theoretic model to study the sub-band competition among We consider an inter-operator carrier aggregation SUs in a CR network. Stackelberg game-based models (IO-CA) system in which a subset of sub-bands of each have been proposed in [8], [12], [20], [21] to study the MNO M , denoted as L for L ⊆ B , is aggregated by interaction between SUs and PUs. In [22], [23], the i i i i subscriber UEs from other MNOs and, in exchange, a authors have applied a coalitional game theoretic model to subset N ⊆ S of UEs from MNO M can aggregate the analyze the possible cooperations among different users i i i sub-bands licensed to other MNOs. We refer to the UEs who share the same spectrum. A detailed survey of game that subscribe to each MNO as the subscribers and those theory and its application into CR networks has been UEs from other MNOs aggregating the spectrum of an presented in [24]–[26]. MNO as the aggregators. Let Sk be the subscriber We model the pairing problem among MNOs as a i occupying the kth sub-band of MNO M . Let Sˆl be an stable roommate market. The roommate market and its i ij aggregator of MNO M that accesses the lth sub-band of variations have been extensively studied from both i MNO M for i ≠ j and l ∈ L . Each MNO needs to first theoretical and practical perspectives [27]–[31]. More j j obtain permission from other MNOs before aggregating specifically, the stable roommate market with ties and their spectrum. In most existing network systems, each incomplete lists has been analyzed in [27]. In [29], the MNO controls the usage of its sub-bands through its stable roommates market with parallel edges and multiple infrastructure (e.g., eNB in LTE systems). Once an MNO partners has been considered. A detailed survey for M agrees to allow other MNOs (e.g., MNO M ) to different variants of the stable marriage problem and j i aggregate its spectrum, it will allocate each aggregator a stable roommate problem has been presented in [30]. specific sub-band, i.e., there is a function mapping each k aggregator Sˆ to sub-band k of MNO Mj. This function III.NETWORK MODELAND PROBLEM FORMULATION ij can be centrally decided by the pairing MNOs or it can A. Network Model be the result of sub-band competition between aggregators Consider a cellular network consisting of a set of inside an IO-CA pair [33], [34]. The detailed analysis of closely located MNOs, labeled as K = {M1,M2,..., the sub-band cooperation or competition among MK } as shown in Figure 1. Each MNO Mi is licensed an aggregators is outside the scope of this paper. Interested exclusive frequency band consisting of a set of readers can see [8], [18], [19], [33]–[35] for the details. component carriers (CCs) each of which can be allocated In this paper, we consider the following two IO-CA to support the MNO’s UEs. Note that the term “UE” may approaches: 4

TABLE I 1) Regular IO-CA: This approach directly extends the LISTOF NOTATION CA in existing LTE-Advanced into the multiple MNOs case where each MNO assigns each Symbol Definition aggregator a vacant sub-band that is unoccupied by K Set of MNOs Mi ith MNO its subscribers. Si Set of UEs for MNO Mi 2) Sharing IO-CA: In this approach, each MNO Mi Bi Set of sub-bands licensed to MNO Mi L allocates each aggregator to a sub-band that is i Subset of sub-bands of MNO Mi allowing aggregation from other MNOs currently occupied by a subscriber. To maintain the Ni Subset of UEs of MNO Mi that can aggregate the quality of service of the subscriber, Mi should be spectrum of other MNOs k able to limit the resulting interference caused by Si Subscriber occupying the kth sub-band of MNO Mi Sˆk Aggregator from MNO M aggregating the kth sub- each aggregator from other MNOs using a pricing ij i band of MNO Mj mechanism which will be described in detail in k k Di Corresponding destination of Si k Section IV-B. Bi Bandwidth of the kth subscriber of Mi hk Ratio of the channel gain between aggregator Sˆk In regular IO-CA, the transmission of each aggregator ij ij and subscriber Sk to the additive noise received by cannot affect the performance of subscribers to the primary j Sk ˆk j MNO, and hence the transmit power of each aggregator Sij k k wi Transmit power of subscriber Si does not need to consider interference constraints. The main k ˆk wîj Transmit power of aggregator Sij shortcoming of regular IO-CA is that the maximum number qj Maximum tolerable interference in each sub-band of aggregators cannot exceed the number of vacant sub- of MNO j βk Pricing coefficient charged to the aggregators in the bands. Sharing IO-CA, on the other hand, allows sub-band i kth sub-band of MNO Mi sharing between aggregators and subscribers and hence can ϖi(j) Part of the payoff of MNO Mi obtained by allowing provide further performance improvement even when there its spectrum aggregated by aggregators from MNO are no vacant sub-bands available. The main challenge for Mj ϖ(i)j Part of the payoff of MNO Mi obtained by sharing IO-CA is that the cross-interference will adversely aggregating the spectrum of MNO Mj affect the performance of both sub-band sharing aggregator ϖij Total payoff of MNO Mi obtained from an IO-CA and subscriber. Therefore, one of the most important issues pair between Mi and Mj for sharing IO-CA is interference control [6], [7], [12]. That ˆk is, each aggregator Sij should always maintain its resulting ′ k k ∈ Lj\Rij: interference to the sub-band sharing subscriber Sj below a ′ q tolerable level denoted as qj. ≤ k ≤ j 0 wîj k′ , (2) We assume that each IO-CA group can only be formed hij by two MNOs and refer to a group of MNOs that allow where q is the maximum tolerable interference of MNO each other to aggregate their sub-bands as an IO-CA pair. j M . This assumption is reasonable in practical j In a regular IO-CA system, each aggregator Sˆk can implementations to make it easy for each MNO to ij wˆk = qk manage interference of the aggregators in its licensed transmit at the maximum power ij i in its assigned ˆk spectrum. More specifically, whenever an MNO detects sub-band k and we assume the price paid by Sij to MNO intolerable interference caused by an aggregator in its Mj is a linear function of its transmit power denoted by k k k 1 sub-band, it will inform the other paired MNO that it ζ(i)j = βj wîj . We can hence write the payoff of Mi obtained from regular IO-CA as should perform interference control or even stop [ ( ) ] aggregating the sub-band in question. We also assume ∑ hk wˆk R k ii ij − k each sub-band (occupied or unoccupied by a subscriber) ϖij = Bj log 1 + k ζ(i)j , (3) ϱi can be aggregated by at most one aggregator. k∈Rij Let the channel gain between source and destination of Bk k aggregator Sˆk be hk for i ≠ j and k ∈ L . Let R be the where i is the bandwidth of the th sub-band of MNO ij ij j ij k ˆk Mi and ϱi is the additive noise received by Sij in subset of sub-bands of MNO Mj that aggregators from MNO band k. Note that, in regular IO-CA, each aggregator can Mi will access using regular IO-CA, with Rij ⊆ Lj. We consider the following power constraint of each sub-band only access the vacant sub-band and hence there is no cross interference between the aggregators and subscribers. k for k ∈ Rij: In a sharing IO-CA system, the subscribers always have ≤ k ≤ k 0 wîj q˜ij, (1) priority to access the sub-bands of their MNOs. More k ∈ Li specifically, a subscriber Si in sub-band k can where q˜k is the maximum transmit power that can be k ij choose its transmit power wi without considering the ˆk k supported by aggregator Sij. wîj is the transmit power of settings or parameters of the potential aggregators in its ˆk aggregator Sij in the kth sub-band of MNO Mj. Suppose 1The linear pricing function is motivated by the fact that many the rest of the sub-bands in Lj are aggregated by existing telecommunication mobile systems charge UEs according to their aggregators from another MNO Mi using sharing IO-CA. communication data rates, which are monotonically increasing functions We can write the power constraints of sub-band of their transmit powers. 5

sub-band. The price charged by each MNO to aggregators Each MNO Mi also needs to pay a price when its from other MNOs can be in real currency, such as the aggregators access the sub-bands of other MNOs. We ˆm ∈ N spectrum rental fee charged by an MNO [36], or it can be hence define the cost of each aggregator Sij i of in a virtual currency used by each MNO to manage or MNO( Mi accessing) the mth sub-band of MNO Mj as m m m m m m regulate the interference and accessibility of the ζ(i)j βj , wîj = βj hij wîj . aggregators [7], [37]. The second part of the payoff function of MNO Mi We assume that each MNO can only control the price in a sharing IO-CA pair can then be written as ( ) of its own sub-bands and should always follow the prices ′ ϖ β , wj, wˆ ij decided by other MNOs when it aggregates their spectrum. (i)j j ∑ ( ) m m m m In this way, the utility function of each MNO contains two = ϖ(i)j βj , wj , wîj , (7) m∈N ∈L \R parts: Si i,m j ij 1) The first part consists of the payoff obtained from its ( ) where ϖm βm, wm, wˆm = own spectrum. We assume each MNO tries to ( ) (i)j j( j )ij πˆm wm, wˆm − ζm βm, wˆm . maximize its transmission rates and follow the (i)j j ij (i)j j ij Note that if there is no subscriber currently occupying widely adopted revenue function [8], [12], [35], [38] k k M sub-band k,( i.e., wi = 0 )and hij = 1, the resulting to define the revenue of MNO i obtained from its m m m m k ∈ L payoff ϖ(i)j βj , wj , wîj will become equivalent to subscriber Si which shares sub-band k i with an Sˆk the payoff of regular∑ IO-CA( in (3), i.e., we) can also aggregator ji as R m m m m ( ) write ϖij = ϖ(i)j βj , wj = 0, wîj . ( ) k k k∈Rij k k k k hiiwi πi(j) wi , wˆji = αiBi log 1 + k k k . (4) By combining (5) and (7), we can write the total payoff ϱi + hjiwˆji of MNO Mi when it forms an IO-CA pair with MNO Mj where αi is the unit price charged by MNO Mi from as its subscriber for sending each bit per second of data. ( ) ϖ β , β , w , wˆ , w , wˆ Each MNO can also obtain revenue by charging the ij i j i ji j ij ( ) aggregator in each of its sub-bands. Let the revenue = ϖi(j) (βi, wi, wˆ ji) + ϖ(i)j βj, wj, wˆ ij . (8) ˆk of MNO Mi obtained by charging aggregator Sji for where k ∑ ( ) causing interference( ) to its subscriber Si in sub-band ϖ (β , w , wˆ ) = ϖk wk, wˆk = 0, βk + i(j) i i ji ( k∈Rij i()j) i ji i k be πˆk βk, wˆk = βkhk wˆk where βk is the ∑ i(j) i ji i ji ji i ϖk wk, wˆk ≠ 0, βk and k∈L(ij \Rij i(j) ) i ji i pricing coefficient of MNO Mi for the aggregator in ϖ(i)j ∑βj, wj, wˆ ij ( ) = the kth sub-band for causing each unit of m m m m m k ϖ(i)j βj , wj = 0, hij = 1, wîj + interference on subscriber Si . m∈N ∈R Si i,m∑ ij ( ) We hence can write the first part of the payoff function m m m m ϖ(i)j βj , wj , wîj . m∈N ∈L \R of MNO Mi in a sharing IO-CA pair formed by Mi Si i,m j ij and Mj for Mi ≠ Mj as follows: Note that if an MNO Mi decides to use neither regular ′ nor sharing IO-CA, the resulting payoff is only related to ϖ (β , wi, wˆ ji) i(j) i ∑ ( ) the transmit power of each subscriber, i.e., we have k k k k = ϖi(j) wi , wˆji, βi , (5) ϖii (ϖi) = ϖij (wi, wˆ ji = 0, wj = 0, wˆ ij = 0) . (9) k∈L \R i ( ji ) k k k k It can be observed that sharing IO-CA is particularly where( ) ϖi(j) w(i , wˆji, β)i { } = k k k k k k k useful in heterogeneous networks with multi-tiers in π wi , wˆji +π ˆ βi , wˆji , βi = βi ∈L , i(j) { } i(j) { } k i which the network of each MNO consists of both k k wi = w and wˆ ji = wˆ . Note that if i k∈Li ji k∈Li macro-cells with high-power infrastructure and low there is no aggregator sharing sub-band k which is powered operator- or user-deployed small-cells or k currently occupied by subscriber Si , we can write femto-cells. In this case, allowing the high-power the payoff of MNO Mi obtained( from its subscriber) infrastructure of an MNO to share the same spectrum as k k k k in the kth sub-band as ϖi(j) wi , wˆji = 0, βi . the small-cell infrastructure of the same MNO will have 2) The second part of the payoff function of MNO Mi the potential to cause large interference to both the consists of the payoff obtained by aggregating sub- macro-cell and small-cell users. On the other hand, bands of others. We can write the revenue of MNO allowing the low power infrastructure of one MNO to ˆm Mi obtained from aggregator Sij using the mth sub- aggregate the spectrum used by the high power band of Mj as ( ) infrastructures of another MNO can alleviate the cross-tier m m m interference. We will provide a more detailed discussion πˆ(i)j wj , wîj ( ) of this scenario using simulation results in Section V. m m ′ m hii wîj = αiBj log 1 + m m m , (6) ϱj + hji wj B. Problem Formulation ′ where αi is the price charged by Mi to its subscriber In cellular network systems, different MNOs have for aggregating sub-band m of MNO Mj. different infrastructure and spectrum resources and always 6 have the incentive to maximize their performance by K MNOs taking full advantage of the aggregated spectrum of other A Joint Optimization Framework MNOs. This makes it natural to study the IO-CA from the game theoretic perspective. In this paper, we assume A Stable Roommate Market each MNO is selfish and can strategically decide its Pairing Problem parameters to optimize its performance and will only seek Γ cooperation with other MNOs when this cooperation can IO-CA Pairs (w*, β*) provide mutual benefits. In this paper, we focus on the following problems, A Stackelberg Game 1) Pairing Problem: In an IO-CA-based cellular network, each MNO is selfish and can always make Pricing Adjustment Problem for Subscribers autonomous decisions about its IO-CA partner. More specifically, each MNO Mi needs to send a request β w to another MNO (e.g., Mj) and an IO-CA pair Power Control Problem between Mi and Mj can only be established when for Aggregators the request is accepted by Mj. To attract other MNOs to aggregate its spectrum, each MNO should also reveal some information such as the set of Fig. 2. Relationship among different problems and game models in an sub-bands allowed for aggregation and the prices IO-CA-base cellular network. charged to each aggregator, for other MNOs to evaluate the achievable performance in an IO-CA problems. More specifically, we model the pairing pair. If MNO Mj in an IO-CA pair can further improve its payoff by pairing with a different MNO problem as a stable roommate market and seek a stable matching structure among all the MNOs. We (e.g., Mk for k ≠ i, j) which will also accept the then establish a Stackelberg game-based hierarchical request from Mi, the IO-CA pair between Mi and framework [8] within each IO-CA pair in which, in Mj will not be stable. Therefore, it is important to decide a stable IO-CA pairing structure in which the licensed spectrum of each MNO Mi, Mi and its every MNO sticks to its IO-CA pairing partner and corresponding subscribers are the leaders (or primary has no intention to unilaterally deviate. Note that the user, seller, etc.) and the aggregators from the other ̸ pairing request signals sent by each MNO only need pairing MNO Mj for j = i are the followers (also to include the identity information of each MNO and called secondary user, buyer, etc.). Using this therefore, the communication overhead caused by framework, we propose an algorithm to jointly sending and responding to the pairing request optimize the transmit power, prices, and pairing between two pairing MNOs can be regarded as a partner of each MNO and its corresponding small constant which is neglected in this paper. subscribers and aggregators. 2) Pricing Adjustment Problem for Each MNO in Its Licensed Spectrum: Each MNO can control the price IV. A JOINT OPTIMIZATION FRAMEWORKFORAN charged to the spectrum aggregators in its licensed IO-CASYSTEM spectrum. Therefore, the pricing adjustment problem In this section, we discuss the solutions of the problems for each MNO Mi in an IO-CA pair formed between described at the end of Section III. We first study the power Mi and Mj can be written as control problem for an IO-CA pair in Section IV-A. We ( ) ∗ then discuss Stackelberg game modeling and the joint price max ϖij βi, βj , wi, wˆ ji, wj, wˆ ij . (10) βi and transmit power optimization problem in Section IV-B. ∗ Finally, we establish a stable roommate market to study the where βj is the optimal price of MNO Mj. 3) Power Control Problem for Each MNO in the IO-CA pairing problem for a network with three or more Spectrum of Others: When accessing the spectrum of MNOs in Section IV-C. The relationship between different other MNOs, each MNO can use power control models in our joint optimization framework is illustrated in methods to further improve its performance given the Figure 2. prices imposed by the spectrum license holders. The power optimization problem for each aggregator of A. Optimal Power Control for each MNO M an MNO i can be formulated as Once an IO-CA pair has been established and a sub-band ( ∗ ) max ϖij βi, βj, wi, wˆ ji, wj, wˆ ij . (11) has been aggregated by an aggregator, it is important for wˆ ij each aggregator to maintain the resulting interference below ∗ where wji are the optimal transmit powers of the the tolerable levels defined in (1) and (2). In this subsection, aggregators from MNO Mj. we assume an IO-CA pair has already been formed between 4) Joint Optimization Problem: In this paper, we MNOs Mi and Mj for i ≠ j and Mi,Mj ∈ K and pricing consider the joint optimization of the above three coefficients βi and βj for both MNOs are constants. Let 7

ˆk k∗ ˆk us consider the power control of an aggregator Sij from wij = 0, it means that Sij cannot aggregate the kth MNO Mi aggregating the kth sub-band of MNO Mj. The sub-band of MNO Mj. optimization of the pricing coefficients and pairing partners for MNOs will be discussed in the next two subsections. B. Optimal Price and Stackelberg Game for each MNO Following the same line as Section III, we can rewrite the Let us consider the joint optimization of the transmit optimization of the power control problem for MNO Mi as follows: powers and pricing coefficients for an IO-CA pair formed ( ) between MNOs Mi and Mj for Mi ≠ Mj and ∗ ∈ K max ϖij βi, βj, wi, wˆ ji, wj, wˆ ij Mi,Mj . By substituting the( optimal transmit powers) wˆ ij ∗ ∗ ∗ ∗ wˆ and wˆ in (13) into ϖij wi, wj, β , β , wˆ , wˆ , k k ≤ l l∗ ≤ k ≤ k ij ji i j ij ji s.t. hijwîj qj, hjiwˆji qi, wîj q˜j we can observe that the payoff of each MNO depends on k ≤ k ∀ ∈ L ∈ L and wˆji q˜i k j, l i the pricing coefficients of both paired MNOs. This means l k k l∗ that the pricing optimization problems for both MNOs are w , w , wˆ , wˆ ≥ 0, ∗ i j ij ji correlated. Specifically, the optimal β decided by M ∀ ∈ L ∈ L l ∈ N k ∈ N i i k j, l i,Si i,Sj j. (12) affects the optimal transmit powers of the aggregators from MNO M , which also determines the pricing As the transmissions in different sub-bands are j coefficients and the incentive of M to form an IO-CA independent( with) each other and the payoff function j k k k k ˆk pair with MNO Mi. However, each MNO can only ϖ(i)j wj , wîj, βj of aggregator Sij from MNO Mi is k k k control the price of its own spectrum. Recall from Section concave in wîj for a given βj and wj , we can then derive k∗ III, the payoff of each MNO(Mi consists) of two parts: the following optimal transmit power wîj of each ϖi(j) (β , wi, wˆ ji) and ϖ(i)j β , wj, wˆ ij . The pricing ∂ϖk (wk,wˆk ) i j ˆk (i)j j ij aggregator Sij of MNO Mi by setting k = 0 coefficient βi decided by Mi can only affect the first part ∂wîj ∗ k ϖi(j) (βi, wi, wˆ ji)), and the second part where wîj is given by, ϖ(i)j βj, wj, wˆ ij depends on the pricing coefficient βj k∗ wˆ = (13) controlled by Mj. ij ( {( ) }) + As each MNO has the autonomy to decide and manage Bk ϱk + hk wk j − i ji j k qj the spectrum usage in its own sub-bands, it can determine min k k k , q˜j , k , βj hij hii hij the price charged to each aggregator in each of its sub-bands considering that all aggregators will use the where (·)+ = max{0, ·}. We can write the optimal{ transmit} ∗ k∗ optimal transmit powers discussed in Section IV-A. The power of all aggregators of MNO Mi as wˆ ij = wîj k∈Lj interactions between users that must decide what actions k∗ where wîj is given in (13). to take in a sequential manner make it natural to model ˆk It can be observed that aggregator Sij can only the above pricing and transmit power optimization calculate the optimal transmit power by knowing the problem as a Stackelberg game as follows: in its own k pricing coefficient βj , the channel gain and transmit licensed spectrum, each MNO is a leader and its action is k power wj . In a practical system, each MNO Mi will rely to select the pricing coefficient when the strategic on the other paired MNO Mj to provide this information. aggregators access its spectrum. Each aggregator is a ˆk If Mj refuses to disclose such information to Mi, Sij follower and its action is to optimize the transmit power cannot determine the optimal transmit power but has to according to the prices imposed by the MNOs. We seek a send signals using a pre-defined fixed power. In Section Stackelberg equilibrium solution for our proposed game V, we compare the performance of an IO-CA-based which is formally defined as follows. cellular network with and without using optimal power Definition 1: [39, Definition 3.26-3.28] Suppose MNOs control. Mi and Mj form an IO-CA pair. In the spectrum of MNO Another observation from (13) is that the optimal Mi, Mi is the leader and aggregators( from) MNO Mj are k∗ ˆk ∗ ∗ transmit power wîj of aggregator Sij decreases with the the followers. An action pair βi , wˆ ji is a Stackelberg k ∗ pricing coefficient βj . In other words, each MNO Mj can equilibrium if wˆ ji satisfies ( ) control the interference level of the aggregator Sˆk in each ∗ ∗ ∗ ∗ ij ϖ β , β , wˆ , wˆ , w , w (14) of its sub-bands by adjusting the corresponding pricing ij i j( ji ij i j ) ∗ ∗ ∗ |Li| k ≥ ϖij β , β , wˆ ji, wˆ , wi, wj , ∀wˆ ji ∈ R , coefficient βj . We will provide a more detailed discussion i j ij on how each MNO decides the optimal pricing coefficient ∗ where ( βi ) satisfies in the next subsection. ∗ ∗ ∗ ∗ β = arg max ϖij β , β , wˆ , wˆ , wi, wj . i |L | i j ji ij ∈R j We can prove the following result: βi ∃ ∈ L k∗ Proposition 1: If k j, wîj > 0, then ϖ(i)j > 0. We can prove the following results about the Stackelberg Proof: See Appendix A. equilibrium for our proposed game. The above proposition says that if the optimal transmit Theorem 1:( For each) IO-CA pair formed by MNOs ∗ ∗ power of at least one of aggregators from MNO Mi is Mi and Mj, βi , wˆ ji is a Stackelberg equilibrium in the k∗ positive, MNO Mi can always obtain benefits by licensed spectrum{ } of Mi where wˆji is given in (13) and ∗ ∗ ∗ aggregating the kth sub-band of MNO M . Note that if β = βk for βk is given as follows: If j i i k∈Li i 8

( ) ( ) − k k 2 k k − k k∗ k− k− 2 2ρi + θi < θi θi ρi , βi = βi where βi [43], in which K students (or, in our model, MNOs) will ⌈ K ⌉ is given by try to be assigned into 2 rooms (or, in our model, IO-CA pairs) each of which accommodates two students. hk Bk βk− = ii i . (15) Let us first define the roommate market as follows. i k k k 2 k∗ hiiqi + hii + hji wˆji Definition 2: [44, Chapter 4.1] A roommate market is K hk 1+hk wk ( ) specified by a set of K students and a preference list Pk k ji( ij i ) k k k − k k 2 ≥ and ρi = k , θi = hiiwi . If 2 2ρi + θi ∈ K ( ) hjj for each student k for k .A preference relation for the k k − k ⟨K ⟩ θi θi ρi , then roommate market is a tuple R = , P where P is the { ( )} ⟨ ⟩ k∗ k k∗ k∗ preference table of all students defined as P = Pk k∈K. βi = arg max ϖi βi , wˆji , (16) k∗∈{ k1∗ k2∗ k− k+} We define the IO-CA pairing problem for a cellular βi βi ,βi ,βi ,βi ∗∈ k− k+ network as a stable roommate market, referred to as the βi [βi ,βi ] IO-CA market, in which the students are modeled as k k+ B k1∗ i MNOs and each MNO Mn has a preference over all the where βi = ρk and βi is given by i other MNOs that can improve its payoff by forming a ∗ βk1 = IO-CA pair, i.e., we use P (M ) to denote the rank of i √ ( ) n i ( )2 ( ) MNO Mi in the preference list of Mn and Bk ρkθk 2 − 2ρk + θk + θk ρk − θk i i i i i i i i Pn(Mi) < Pn(Mj) means ( )( ) ∗ ∗ ∗ ∗ k k − k − − k ϖni (wn, wi, βn, βi , wˆ ni, wˆ in)) > 2ρi ρi 1 ρi 1 θi ∗ ∗ ∗ ∗ ( ) ϖnj wn, wj, β , β , wˆ , wˆ for Mi,Mn,Mj ∈ K. Bkρk 2 − 2ρk + θk n j nj jn − (i i )( i i ), (17) Note that IO-CA cannot always improve the payoff for k k − k − − k 2ρi ρi 1 ρi 1 θi both paired MNOs and, if an MNO Mi cannot improve k2∗ its payoff by forming IO-CA with any MNOs in the βi = ( ) Bkρk 2 − 2ρk + θk market, it will not share its spectrum with others but only (i i )( i i ) (18) use its own exclusive spectrum to support services for its 2ρk 1 − ρk ρk − 1 − θk i √ i i i subscribers, i.e., if M occupies the lth position in the (( ) ( )) n k k k k k 2 k k k preference list of itself, it means that B ρ θ 2 − 2ρ + θ + θ ρ − θ ∗ ∗ ∗ ∗ i i i i i i i i ϖ (w , w , β , β , wˆ , wˆ ) > ϖ (w ) for all − ( )( ) . ni n i n i ni in nn n k k − k − − k ∈ K 2ρi ρi 1 ρi 1 θi Mi satisfying 0 < Pn(Mi) < l. Different MNOs generally have different peak hours. Proof: See Appendix B. ( ) ∗ ∗ We hence can assume, in a cellular network, MNOs Similarly, we can also observe that β , wˆ is the j ij sequentially join or leave the IO-CA market2. If an MNO Stackelberg equilibrium solution in the spectrum licensed ∗ ∗ M decides to join the market, it will send a message to to MNO M where β and wˆ can be obtained by i j j ij inform all the MNOs in the current IO-CA market that swapping i and j in Theorem 1 and equation (13). the spectrum of M will be available to share. MNOs in Note that the optimal pricing coefficients Theorem 1 i the market can use the message sent by M to evaluate are calculated by assuming all aggregators use the optimal i the performance of M and insert M into the proper power control methods derived in (13). If the aggregators i i positions in their preference lists. All MNOs will also use constant power to send signals, both MNOs should feedback a confirmation message to M which can be charge the highest prices they can in each of their i used by M to establish the preference list over all the βk → βk+ i sub-bands to maximize their revenue, i.e., i i P ˜ k MNOs. Let i be the preference of Mi. Let Mi be the ∀k ∈ Li. ( ) ∗ ∗ kth most preferred MNO in the preference list of MNO The optimal pricing coefficient pair β , β for both i j M . paired MNOs M and M derived in the above theorem i i j One of the main solution concepts for the roommate also corresponds to the Bertrand equilibrium solution if we market is the matching which is defined as follows. model the price competition between the pairing MNOs Definition 3: [44, Chapter 4.1] A (one-sided) matching as an oligopoly market where two market dominant firms Γ for a roommate market is a function from sets K to K compete with each other with different prices [38], [40], such that Γ(M ) ∈ K, Γ(M ) ∈ K, and Γ(M ) = M ⇔ [41]. i j i j Γ(Mj) = Mi for every Mi,Mj ∈ K. Note that Γ(Mi) = Mi means that Mi cannot form an C. A Stable Roommate Market for the IO-CA Pairing IO-CA pair with any of the other MNOs. It can be observed Problem K/2 ( ) ∏ − that there are K 2i / K ! number of possible matchings Let us consider the pairing problem for a cellular 2 2 ( ) i=1 network with three or more MNOs. In our model, an m where n is the number of m combinations from a set of IO-CA pair can only be formed when two MNOs mutually agree to share their spectrum with each other. 2If multiple MNOs decide to join the market simultaneously, a random This makes it natural to model the interaction among duration of delay can be introduced for these MNOs. That is, if an MNO M decides to join the IO-CA market, it will delay for η amount of time MNOs as a roommate market, also known as one-sided i i before sending the joining request where ηi is a bounded random variable. matching market [42] or non-bipartite matching market We have included this random delay in Algorithm 1. 9

of n elements. In this paper, we seek a matching structure ii) Mi then sends the IO-CA pairing request to its most preferred K that is stable which is defined as follows. MNO in . If the request sent by Mi is rejected, Mi sends a request to the next most preferred MNO in its preference list. Definition 4: A stable matching is a partition of set K This process is repeated until Mi has been matched with an ⌈ K ⌉ ∈ K K into 2 disjoint IO-CA pairs such that no two MNOs who MNO Mj or been rejected by all MNOs in . There are are not in the same IO-CA pair but each of whom prefers four possible results of the above process: the other to its partner in the matching. a) If the request sent by Mi has been rejected by all MNOs, then Mi will not form any matching pair with MNOs in It has already been observed in [44] that a stable K, matching for the stable roommate market may not always b) If the request sent by Mi has been accepted by an MNO Mj and Mj has not sent the IO-CA pairing request to exist. This is because the preference of each MNO over another MNO before, it will have the following two results: each other may form a cyclic sequence. For example, it b-i) Mj is currently not matched with any other MNO, then can be easily shown that if the preferences of four MNOs Mi and Mj will form a matching pair with each other, b-ii) If the request sent by Mi has been accepted by an MNO M1, M2, M3 and M4 are given by P1 = ⟨M2,M3,M4⟩, ⟨ ⟩ ⟨ ⟩ Mj that is currently matched with another MNO Mk, P2 = M3,M1,M4 , P3 = M1,M2,M4 , and then Mi and Mj will form a matching pair with each P4 = ⟨M1,M2,M3⟩, respectively, it is impossible for find other. Mk will repeat the same Step ii) as Mi, c) If the request sent by M has been accepted by an MNO a stable matching, e.g., any MNO, for instance Mi, that is i M who has sent an IO-CA pairing request to another matched with M will be able to find another more j 4 MNO before, it means that both Mi and Mj belong to a preferable MNO which also prefers Mi to its current cycle sequence and there is no stable matching for these matching MNO. two MNOs. Another concept called stable partition, which can be Let us consider the case that the MNOs sequentially regarded as a generalization of the stable matching, was join the IO-CA market using the above operation. More first proposed in [45], [46]. It has already been proved in specifically, at the beginning of the IO-CA market, there [45] that a stable partition always exists in any instance of is only one MNO (e.g., Mi) in the market. When the the roommate market. Let us present the formal definitions second MNO Mj joins the market for Mj ≠ Mi, its as follows. preference list only consists of two elements Mi and Mj Definition 5: [46, Section 2] A stable partition P for and if Pj(Mi) < Pj(Mj), Mj will send an IO-CA a roommate market is a permutation Π of the set K such request to MNO Mi and a stable matching pair can only −1 that (i) for every Mi ∈ K, either Π(Mi) = Π (Mi) or be formed when Mi also observes Pi(Mj) < Pi(Mi). If a −1 Mi prefers Π(Mi) to Π (Mi), (ii) if Mi prefers Mj to third MNO Mk tries to join the market for −1 −1 ∈ { } Π (Mi) then Mj prefers Π (Mj) to Mi. We refer to Mk / Mi,Mj , it will sequentially send pairing requests −1 Π(Mi) and Π (Mi) as the successor and predecessor of to the MNOs in its preference list. If Mi is the first MNO Mi, respectively, relative to Π. We refer to a cycle in Π of that accepts the request of Mk, it means that Mk is more odd (or even) length as an odd (or even) party. preferred by Mi than Mj, i.e., Note that the stable partition is in fact a permutation Pi(Mk) < Pi(Mj) < Pi(Mi). Since MNO Mk sends instead of a matching structure that is stable [47], [48]. requests to MNOs according to its preference list, Mi is In the rest of this paper, we will first establish the also the most preferred MNO that accepts the request of condition for which a stable matching exists in our Mk. In this case, Mi and Mk will be paired with each IO-CA market. We will then develop an algorithm that other and Mj will be left without any IO-CA pairing achieves a stable matching structure if it exists. partner. When a fourth MNO Ml tries to join the market Otherwise, the proposed algorithm results in a stable for Ml ∈/ {Mi,Mj,Mk}, Ml will send a pairing request partition among MNOs. according its preference list which will result in the As mentioned previously, in practical systems, each following possible cases: 1) if the request of Ml has been MNO can decide to join or leave the IO-CA market under rejected by all three MNOs in the market, Ml will not be different situations. If an MNO that is not in the current paired with any MNO, 2) if MNO Mj is the first MNO IO-CA market applies to join the market due to the that accepts the request of Ml, then an IO-CA pair will be increasing of the traffic in its network, it needs to go formed by Ml and Mj, 3) if MNO Mi (or Mk) accepts through a procedure, referred to as addition operation, to its request, an IO-CA pair will be formed by Ml and Mi decide its potential IO-CA pairing partner before it starts (or Mk) and Mk (or Mi) will again restart the requesting to share the spectrum. process by sending an IO-CA request to its next preferred Let us present the detailed operation as follows. MNO. If during the requesting process of Mk, Mj is the first MNO to accept the request, then the resulting IO-CA Operation 1: Addition market will consist of two stable matching pairs ⟨Ml,Mi⟩ M ∈/ K Suppose an MNO i tries to join the IO-CA market. and ⟨Mj,Mk⟩. However, it is also possible that Ml will i) M broadcasts the pairing request and price information to i be the first MNO that accepts the request of Mk (For MNOs in K. Each MNO M ∈ K then evaluates the j example, during the previous requesting process of M , resulting payoff when forming an IO-CA pair with Mi. Each l Mj ∈ K inserts Mi into its own preference list and then Ml prefers Mk to Mi. However, Mk rejects the request feedbacks a confirmation message to M . M can use this i i sent by Ml because Mk prefers Mi to Ml.). This means received feedback message to evaluate the performance and establish its preference over all MNOs in K. All MNOs that Ml,Mk and Mi form a cyclic sequence and the update K = K ∪ {Mi}. requesting process will be infinitely repeated among 10

MNOs Ml,Mk and Mi. In this case, no stable matching 4) Each MNO Mi obtains the resulting payoff ϖij and updates R R ∪ { } exists. We therefore can have the following results. i = i Mj . Mi also updates the preference list by ranking all MNOs in the updated Ri from the highest to the Proposition 2: Suppose all MNOs sequentially join the lowest payoffs. IO-CA market following the procedure described in the ENDWHILE addition operation. The resulting structure is a stable Phase II — IO-CA Pairing ∃ ∈ K matching if the IO-CA market consists of a stable WHILE Mi who did not receive any pairing request or did not send any pairing request to other MNOs, matching. Otherwise, the resulting structure will be a 5) Each MNO Mi ∈ K waits for a bounded random amount of stable partition. time before using the addition operation to join the IO-CA Proof: See Appendix C. market. 6) Whenever an MNO Ml wishes to join or leave the market, We can prove the following complexity results about the it uses the addition or deletion operation to join or leave the addition operation. market. Proposition 3: The complexity of the addition operation is O(K2) in the worst case, where K is the number of Proposition 5: Algorithm 1 either reports no stable MNOs3. matching exists and achieves a stable partition or Proof: See Appendix D. generates a stable matching for the IO-CA system. For M M Similarly, if an MNO that is currently in the market any IO-CA pair between MNOs i and j for M ≠ M decides to retrieve its exclusive spectrum and quit the i j, the transmit power of each aggregator achieves IO-CA market due to the decrease of the trafﬁc in its the optimal transmit power derived in Section IV-A. The β wˆ exclusive spectrum, it also needs to inform all other pricing coefﬁcient i and the transmit power ji in the M MNOs that its spectrum will no longer be available. Let spectrum of each MNO i achieve the Stackelberg us present the deletion operation as follows. equilibrium. Proof: The second part of the above theorem directly Operation 2: Deletion comes from (13) and the results of Theorem 1. In the

Suppose an MNO Mi ∈ K decides to leave the IO-CA market. Phase II of Algorithm 1, each MNO enters or leaves the Then we have IO-CA market by using the addition and deletion K\{ } i) Mi broadcasts a leaving message to MNOs in Mi and operators introduced in Operations 1 and 2, respectively. ∈ K\{ } then each MNO Mj Mi will remove Mi from its Following the results in Propositions 2 and 4, the ﬁrst preference list and update K = K\{Mi}. ii) If Mi is currently in an IO-CA pair with Mj = Γ(Mi) and part of Proposition 5 can be proved. ̸ Mi = Mj , Mj will send requests to the remainder of the From the above proposition, if Algorithm 1 reports a MNOs in K following the exactly the same line as steps ii) in the addition operation. stable matching structure, we can claim the existence of at least one stable matching structure. However, if a stable Following the same line as Proposition 2, we can prove matching does not exist, then Algorithm 1 will result in a the following results. stable partition. Note that a stable partition is actually a Proposition 4: Suppose an MNOs has been deleted permutation and is not necessarily stable because of the from the IO-CA market following the procedure described existence of odd parties. It has been proved in [49] that in deletion operation. The resulting structure is a stable for each odd party, if an MNO can be removed from this partition. odd party, the rest of the MNOs can form a stable Proof: See Appendix E. matching with each other. In other words, a possible Let us now present the following algorithm that can solution to reach a stable structure among MNOs can be jointly optimize the transmit powers, pricing coefﬁcients and pairing of MNOs. obtained by choosing an MNO from each odd party and forcing it to quit the market. However, which MNO Algorithm 1: A Joint Optimization Algorithm should quit and how to design a distributed mechanism to

Initialization: Let Pi be the preference list of Mi and Ri be the incentivize the quitting process of these MNOs is out of P domain of i. the scope of the current paper. Phase I — Price Adjustment and Power Control WHILE ∃Mi ∈ K, |Ri| ≤ K − 1, It can be observed from Algorithm 1 that the resulting 1) Each MNO Mi randomly chooses another MNO Mj ∈/ Ri to matching structure among MNOs is closely related to the form an IO-CA pair. preference relation of each MNO, which also depends on 2) Once an IO-CA has been formed, the pairing MNOs inform the resulting payoff, by forming different IO-CA pairs each other regarding their sets of sub-bands allowing aggregation. Each MNO also sends a short training signal in with each other. In addition, from the discussion of each of these sub-bands for the other pairing MNO to Section IV-A, we can observe that the resulting payoffs as estimate the channel gain between the licensed subscribers well as the preference list of each MNO, are directly and aggregators as well as the transmit powers of the subscribers in each of these sub-bands. determined by their transmit powers and pricing 3) Both of the pairing MNOs inform each other of their optimal coefficients. By using the optimal transmit power in (13) pricing coefficients calculated by Theorem 1 and each and optimal pricing coefficients derived in Theorem 1, aggregator transmits using the transmit power calculated by (13). each MNO can obtain the highest payoff when forming an IO-CA with another MNO in the market and hence each 3In this paper, we follow Bachmann-Landau notations: f = O(g) if MNO cannot further improve its payoff by unilaterally lim f(n) < +∞. n→∞ g(n) changing its price, transmit power or pairing partner. 11

140

2 120 S2 1 S .. 1 . 100 10 S2 80 Macro−cell and small−cell Small−cell

M1 1.5 km M2 60 Macro−cell

2 S1 ˆ1 S21 40 ˆ1 1 S12 S 2 ... .. m . 200 ˆ10 10 S21 20 ˆ d M’2 M’

1 Average Transmission Rate (mbps) S12 M1 M ' 2 km 2 0 ... 0 0.0002 0.0004 0.0006 0.0008 0.001 10 Maximum Tolerable Interference S1

Fig. 6. Average transmission rate for each subscriber with different maximum tolerable interference of small-cell subscribers. Fig. 3. Simulation setup with two MNOs and multiple UEs.

Payoff of M with IO−CA and power control 14 1

120 Payoff of M with IO−CA and power control 2 Without IO−CA 12 Payoff of M with IO−CA With sharing IO−CA 1 Payoff of M with IO−CA 100 With regular IO−CA 10 2 Payoff of M without IO−CA 1 8 Payoff of M without IO−CA 2

80 Payoff 6

4 60

Average Transmission Rate (mbps) 40 0 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 7 8 9 10 11 β Percentage of occupied small−cell sub−bands 1

Fig. 4. Average transmission rate for each subscriber with regular IO-CA, Fig. 7. Payoffs of MNOs with different pricing coefficients: Each MNO sharing IO-CA and without IO-CA. consists of 10 subscribers and 10 aggregators and applies the same pricing coefficient to all sub-bands. Pricing coefficient β2 of MNO M2 has been fixed to β2 = 6. V. DISCUSSIONSAND NUMERICAL RESULTS

Payoff of M with IO−CA, power control and optimal pric e 1 18 Payoff of M with IO−CA, power control and optimal pric e Before presenting the simulation results of our 2 Payoff of M with IO−CA and power control proposed joint optimization framework, let us ﬁrst verify 1 16 Payoff of M with IO−CA and power control 2 Payoff of M with IO−CA the performance improvement measured by transmission 1 14 Payoff of M with IO−CA 2 rate brought by the IO-CA in a two-tier heterogeneous Payoff of M without IO−CA 12 1 Payoff of M without IO−CA network with two closely located MNOs M1 and M2, i.e., 2 k 10 we apply the utility function given in (8) with αj = 1 and k ∀ ∈ { } ∈ L Payoff 8 βj = 0 k 1, 2,...,K and j . We assume the network of each MNO consists of a macro-cell overlaid 6 with a small-cell, and that the macro-cells and small-cells 4 associated with the same MNO operate in different 2 spectrum. We consider the downlink transmission and 0 1 2 3 4 5 6 7 8 9 10 11 assume only the macro-cell of each MNO can share the Distance between two MNOs spectrum licensed to the small-cell of the other MNO. With sight abuse of notation, we denote the macro-cell Fig. 8. Payoffs of MNOs under different d , measured in meters. ′ M1M2 BS and small-cell BS for each MNO Mi by Mi and Mi , respectively for i ∈ {1, 2}. We assume the locations of the

Optimal pricing coefficient of M 1 0.198 Optimal pricing coefficient of M 2 0.196 120 0.194 Without IO−CA 110 0.192 With IO−CA 0.19 100 0.188 90 0.186

Pricing Coefficient 0.184 80 0.182 70 0.18 0.178 60

Average Transmission Rate (mbps) 2 4 6 8 10 50 0 250 500 750 1000 1250 1500 Distance between two MNOs Distance (meter) 1 1 Fig. 9. Optimal pricing coefﬁcients for two subscribers S1 and S2 with Fig. 5. Average transmission rate for each subscriber with different different dM1M2 , measured in meters. distances between the macro-cell base station and small-cell base station. 12

Random Pairing

12 IO−CA Optimal transmit power w 10 21 IO−CA with power control Optimal transmit power w IO−CA with power control and optimal price 12 9 10 8

7 8

6 Transmit Power 5 6

Number of Matchings 4 3 2 4 6 8 10 Distance between two MNOs 2 ˆ1 ˆ1 Fig. 10. Optimal transmit powers of two aggregators S21 and S12 with 0 different dM1M2 , measured in meters. 10 15 20 25 30 35 40 45 50 Length of the Side of Coverage Area

Average payoff of MNOs with IO−CA, power control and optimal price 1.8 Average payoff of MNOs with IO−CA and power control Fig. 14. Number of IO-CA pairs with different lengths of the side of 1.6 Average payoff of MNOs with IO−CA coverage area. Average payoff of MNOs with random pairing 1.4 Average payoff of MNOs with global optimal solution

1.2 1 macro-cell BSs and small-cell BSs for the two MNOs are 0.8 symmetric as shown in Figure 3. We assume each (macro- 0.6 or small-) BS has been associated with the same number Average payoff of MNOs 0.4 of subscribers which are uniformly randomly distributed 0.2 within the circular coverage area of its BS with radius of 0 2 4 6 8 10 12 14 16 18 20 2 km and 200 m for macro- and small-cell, respectively. Number of MNOs We set the maximum transmit powers for macro-cell BS and micro-cell BS as 40 and 20 dBm, respectively [50], Fig. 11. Average payoff of MNOs with different numbers of MNOs. and assume each BS can adjust its optimal transmit power using (13) when possible. 10 Random Pairing We ﬁrst present the average transmission rates of each 9 IO−CA IO−CA with power control subscriber achieved by regular IO-CA and sharing IO-CA 8 IO−CA with power control and optimal price and compare it with the system without IO-CA in Figure 7 4. It can be observed that if all the sub-bands in the 6 small-cell are vacant, sharing IO-CA and regular IO-CA 5 will result in the same performance. However, if all the 4 sub-bands are fully occupied by small-cell subscribers, Number of Matchings 3 regular IO-CA cannot provide any performance 2 improvement compared to the system without IO-CA. We 1 can also observe that the sharing IO-CA can always 0 2 4 6 8 10 12 14 16 18 20 improve the transmission rate of the subscribers even Number of MNOs when there is no vacant small-cell sub-band available. In Fig. 12. Number of IO-CA pairs with different numbers of MNOs. Figure 5, we compare the average transmission rate of the subscriber for each MNO with and without sharing IO-CA under different distances d ′ . It can be easily MiMj 0.9 observed that when d ′ approaches zero, our sharing MiMj 0.8 IO-CA can be regarded as special case of the traditional 0.7 carrier aggregation between a small-cell and macro-cell of

0.6 the same MNO sharing the same spectrum. We can observe from Figure 5 that if d ′ is close to zero, 0.5 MiMj IO-CA cannot improve the transmission rate for its 0.4 Average payoff of MNOs with CA, power control and optimal price subscriber compared to the case without IO-CA. This is 0.3 Average payoff of MNOs with CA and power control Average payoff of MNOs because the high-power macro-cell subscribers and the Average payoff of MNOs with CA 0.2 Average payoff of MNOs with random pairing low-power small-cell subscribers can cause large Average payoff of MNOs with global optimal solution 0.1 cross-interference when they are closely located within

0 the same macro-cell. However, with the increasing of the 10 15 20 25 30 35 40 45 50 Length of the Side of Coverage Area distance between the macro-cell BS of one MNO and small-cell BS of the other MNO, the transmission rate for Fig. 13. Average payoff of MNOs with different lengths of the side of each subscriber can be significantly improved. This coverage area. confirms our previous observation that IO-CA has the 13 potential to significantly improve the performance of Phase-II of Algorithm 1 to decide its IO-CA pairing cellular networks compared to carrier aggregation within a partner. single MNO’s network. In sharing IO-CA, the aggregators 4) IO-CA with power control and optimal price: each from one MNO should always control their transmit MNO uses Algorithm 1 to decide its transmit power, powers to avoid intolerable interference to the subscribers pricing coefficient and the pairing partner. in the small cell of the other MNO. Therefore, in Figure From the discussion in Section III, we can observe that 6, we assume the small cell subscribers have the same in regular IO-CA, there is no incentive for each MNO to maximum tolerable interference levels and each control the transmit powers of the aggregators by macro-cell subscriber adapts its transmit power to the optimizing its price. In addition, regular IO-CA can be maximum tolerable interference level of its sub-band regarded as a special case for sharing IO-CA when the sharing small-cell subscriber. We compare the average interference caused by the sub-band sharing aggregator is transmission rate for our simulated network system under lower than the maximum tolerable interference of the different tolerable interference levels. We can observe that subscriber even when the aggregator uses its maximum k ≤ qj the transmission rates of the small-cell subscribers transmit power, i.e., q˜ij k . Therefore, in this section, hij decrease with the maximum tolerable levels. On the other we mainly focus on the sharing IO-CA. We first simulate hand, the increasing of the transmission rate for the IO-CA-based cellular network with two MNOs, each of aggregators from the other MNO can compensate the which corresponds to a cellular network with a base performance degradation of the small-cell subscribers and station located at the center of the coverage area. Each the total average transmission rate for each subscriber can MNO also contains a set of subscribers and a set of increase with the maximum tolerable interference level. aggregators uniformly and randomly located in the In Figures 4-6, we assume all subscribers are uniformly overlapped coverage area of both MNOs as is shown at randomly located within a fixed coverage area and the top of Figure 3. Each subscriber or aggregator compare the performance of cellular networks with and corresponds to the uplink communication channel from without IO-CA. The performance of each subscriber is each UE to the base station. Assume that the channel gain also closely related to its relative distance to the hˆk k k ij ∈ { } ˆk hij is given by hij = k ξ for i, j 1, 2 where hij is corresponding BS. We will provide a more detailed dij k discussion about this in examining Figures 13 and 14 at the average channel fading coefficient, dij is the distance ˆk the end of this section. between aggregator Sij and the base station of Mj and ξ In this paper, we consider the joint optimization of is the fading exponent. We also use dM1M2 to denote the three problems: pairing problem, pricing adjustment distances between base stations of M1 and M2. Let us problem and power control problem. We derive solutions focus on the performance of both source-to-destination for each of these problems and propose a joint pairs with different values of dM1M2 . optimization algorithm that simultaneously achieves all We first consider the effects of the changing pricing these solutions. Our algorithm is general in the sense that coefficients on the payoff of the MNOs. In Figure 7, we each separate part of our algorithm can be individually assume each MNO applies the same pricing coefficient to applied to optimize IO-CA-based cellular networks under all of its sub-bands. We then fix the pricing coefficient of different situations. For example, if each aggregator one MNO and compare the payoffs of MNOs when the cannot keep track of the channel gains between itself and other MNO changes its pricing coefficient. It is observed the subscribers, it will fix its transmit power. However, that the payoff of both MNOs will be affected even when MNOs can still use Phase-II of Algorithm 1 to decide the price of only one MNO changes. This is because in their IO-CA pairing partners. In the rest of section, we our model each MNO can use the price to control the present numerical results to access the performance of our payoff obtained from its own spectrum as well as that proposed optimization algorithms. We mainly compare the obtained from aggregating the spectrum of the other following approaches for our IO-CA-based cellular MNO. Another observation is that the IO-CA with power networks, control significantly increases the payoff of both MNOs, 1) Random pairing: all K MNOs are randomly and more importantly, it also reduces the payoff ⌈ K ⌉ partitioned into 2 groups each of which consists difference between the MNOs caused by the price change. of two MNOs. If both of MNOs in a group can The payoffs of both MNOs with different optimization

improve their payoffs using IO-CA with predeﬁned algorithms under different values of dM1M2 are compared ﬁxed powers and prices, they will form an IO-CA in Figure 8. It is observed that the payoffs of both MNOs

pair. Otherwise, both MNOs will only use their own decrease with the distance dM1M2 . This is because when licensed spectrum to transmit signals without the distance between MNOs becomes large, each MNO aggregating the spectrum of each other. will decrease its pricing coefficient to attract more 2) IO-CA: each MNO fixes the powers and pricing aggregators from the other MNO which also reduces the coefficient and only uses Phase-II of Algorithm 1 to revenue obtained from the other MNO. We can also decide its IO-CA pairing partner. observe that the payoff improvement brought by IO-CA 3) IO-CA with power control: each MNO uses the with power control and/or optimal price is larger than optimal transmit power calculated from (13) and those brought by other two approaches. 14

To study the payoff obtained by MNOs from each MNOs to aggregate their spectrum at a low price. We subscriber, we present the optimal pricing coefficient of refer to the solution that can maximize the total payoff each MNO in one of its sub-bands occupied by sum of all MNOs without the constraint of individual 2 1 subscribers S1 and S2 under different dM1M2 in Figure 9. rationality as the global optimal solution which is also We observe that the optimal pricing coefficients charged presented in Figure 12. As can be observed from Figure in both sub-bands of MNOs M1 and M2 decrease with 11, although the global optimal solution is significantly the distance dM1M2 . This is because when two MNOs are better than our proposed distributed optimization further away, each MNO will need to provide more approach, it cannot guarantee stableness and the incentive such as reducing its price to attract the other performance for each individual MNO, and hence cannot MNO to aggregate its spectrum. always incentivize IO-CA among MNOs. In Figure 10, we compare the optimal transmit powers In Figure 12, we compare the numbers of IO-CA pairs of two aggregators randomly chosen for both MNOs formed under different number of MNOs. It can be under different distances between the base stations. We observed that the number of IO-CA pairs between MNOs can observe that with the increasing of dM1M2 , the achieved by random pairing does not vary much as the aggregator should always increase its transmit power to number of MNOs increases. However, if the power further improve the payoff of its corresponding MNO control and/or optimal prices have been applied, the because the cross-interference for each sub-band sharing chance for each MNO to find another MNO to form an subscriber and aggregator decreases with dM1M2 . Note IO-CA pair will increase with the number of MNOs. In that the price decreasing process illustrated in Figure 9 addition, when the number of MNOs is large enough affects the revenue for MNOs at a much faster rate, which (e.g., exceeds 16 in Figure 12), IO-CA with power control eventually lowers the payoffs of the MNOs as observed in can achieve the maximum number of IO-CA pairs among Figure 8. MNOs. In other words, if the main objective for each We now simulate an IO-CA-based cellular network MNO that adopt IO-CA is to maximize the total number with more than two MNOs by considering a of spectrum sharing pairs, IO-CA with power control and square-shaped coverage area in which each MNO has a IO-CA with power control and optimal price achieve the fixed number of subscribers and aggregators uniformly same results if the density of MNOs in the coverage area randomly located in the coverage area. We follow the exceeds a certain threshold. same settings as the two MNO case introduced in the In Figure 13, we compare the payoffs of the MNOs beginning of this section. under different sized coverage areas. We observe that the In Figure 11, we fix the size of the coverage area and average payoff of MNOs decreases when the size of the compare the average payoff obtained by all MNOs under coverage area becomes large. This verifies our previous different total numbers of MNOs. We can observe that observation that our proposed optimization algorithm can random pairing cannot provide any payoff improvement to provide high performance improvement when the density MNOs because the chance for each MNO to pick up a of the MNOs is high. We also observe that the average high cross interfering pairing partner (e.g., another MNO payoff obtained only by IO-CA approaches that obtained that is close-by) increases with the density of the MNOs by random pairing when the length of the coverage area in the coverage area. However, the average payoff of becomes large. However, IO-CA with optimal price MNOs increases with the number of MNOs when IO-CA and/or power control can still provide significant payoff is allowed. This is because all MNOs are randomly improvement compared to the random pairing. located in the area and with the increasing of the number In Figure 14, we compare the number of IO-CA pairs of MNOs, each MNO will have more choice of its IO-CA between MNOs under different sizes of the coverage area. pairing partner using the Phase II of Algorithm 1. We can We observe that if the size of the network is small, there also observe that as the coverage area becomes more and are always some MNOs that cannot find a pairing partner more crowded, the payoff improvement brought by our to form an IO-CA pair. However, when applying IO-CA proposed IO-CA with power control and optimal price with optimal price and power control, the number of IO- becomes more significant. In other words, our proposed CA pairs between MNOs will reach the maximum number joint optimization algorithm is more useful in a high K 2 when the size of the network becomes large. population/MNO density area such as city center or during the peak hours of the data service demand. Note VI.CONCLUSION that, in our model, we assume MNOs are selfish and we focus on the distributed optimization for cellular networks This paper considers CA between MNOs in a cellular with multiple MNOs. In our setting, an IO-CA pair can network. In this network, an MNO can not only access its only be formed if both pairing MNOs can further improve own licensed spectrum, but can also aggregate the their performance by allowing their spectrum to be spectrum licensed to other MNOs by paying a certain aggregated by each other. This condition is referred to as price. We establish a stable roommate market to study the individual rationality in game theory. It can be observed pairing problem among the MNOs. We derive a condition that the payoff sum of the MNOs can be further increased for which a stable matching structure exists. We propose if some MNOs sacrifice their performance and allow other an algorithm to approach a stable matching structure if it 15

k k k k∗ k∗ exists. Otherwise, the algorithm results in a stable θi = hiiwi . Substituting wîj and wˆji in (13) into (8), partition. We then establish a Stackelberg game-based we have model to study the interaction between the subscribers ( ) ϖk wk, wˆk∗, βk and aggregators in the spectrum of each MNO. We derive i(j) i ji( i ) the optimal transmit power for each aggregator and the βkθk = Bk log 1 + i (i ) Stackelberg equilibrium for each MNO. We propose a i k k − k Bi + βi 1 ρi joint optimization algorithm that can achieve a stable ( ) k − k k + matching structure among MNOs if it exists as well as the + Bi βi ρi (20) optimal transmit powers and prices for each MNO. We k k present numerical results to verify the performance To find the optimal value of βi that can maximize ϖi(j), we have improvement brought by each of these optimization ( ) methods under different situations. k k k∗ k ∂ϖi(j) wi , wˆji , βi k = 0 ∂βi APPENDIX A ( ) Bk 1 + θk − ρk PROOFOF PROPOSITION 1 ⇒ i ( i i ) k k k − k ˆk Bi + βi 1 + θi ρi Let us consider the payoff of subscriber Sij of MNO Mi 1 − ρk obtained by aggregating the kth sub-band of MNO Mj as − ( i ) − k ρi = 0 follows, Bk + βk 1 − ρk ( ) i i i ( ) k k k − k k k k βi Bi ρi 2 2ρi + θi hiiwîj ⇒ ( ( )) ( ( )) ϖk = Bk log 1 + − βkhk wˆk . (19) k k − k k k − k (i)j j k k j ij ij Bi + βi 1 + θi ρi Bi + βi 1 ρi 1 + hjiwj ( ) ( )( ) k2 k − k − k2 k − k k − k Bi θi ρi βi ρi 1 ρi 1 + θi ρi It is observed that wˆk increases (or decreases) with ϖk + ( ( )) ( ( )) ij (i)j Bk + βk 1 + θk − ρk Bk + βk 1 − ρk k k∗ k ≥ k∗ k∗ i i i i i i i when wîj < wîj (or wîj wîj ) where wij is the optimal k k k = 0. solution of ϖ(i)j given in (13). Because ϖ(i)j = 0 if wîj = k ≥ k∗ ≥ (21) 0, we hence have ϖ(i)j 0 when wîj 0. From the above equation, it is observed that if ( )2 ( ) APPENDIX B − k k k k − k 2 2ρi + θi < θi θi ρi , there is no solution for PROOFOF THEOREM 1 ∂ϖk (wk,wˆk∗,βk) ∂ϖk (wk,wˆk∗,βk) i(j) i ji i = 0. In this case, i(j) i ji i < 0 ∂βk ( ) ∂βk Let us consider the optimization of the pricing i ∗ i which means that ϖk wk, wˆk , βk always decreases coefficients for the MNOs. Before the derivation of the i(j) i ji i with βk. Therefore, MNO M should choose the lowest optimal price, we first need to calculate the range of βk. i i i value of βk in the kth sub-band to maximize the payoff It is observed that the value of βk is limited by two ( i ) i ϖk wk, wˆk∗, βk , i.e., βk∗ = βk−. However, if constraints. The first one is the power constraints in (2). ( i(j) i ji) i ( i ) i − k k 2 ≥ k k − k The other one is the fact that the transmit power of each 2 2ρi + θi θi θi ρi , there exist two solutions aggregator should be a positive value. Otherwise Mi for the equation (21) which are given in (17) and (18), k βk cannot obtain any benefits by optimizing βi in the kth respectively. These solutions can be the( value of ) i that k∗ k∗ ϖk wk, wˆk∗, βk sub-band, i.e., wˆji > 0. Substituting wˆji in (13) into (2), either maximizes or minimizes i(j) i ji i . Note k− k k we can obtain the lower bound βi for βi which is given that in (16), we also consider the boundary values of βi . k∗ k∗ in (15). Similarly, applying wˆ > 0, we can calculate the This is because( the solution) βi that maximizes the value ji ∗ k+ k of ϖk wk, wˆk , βk may not always within the range upper bound βi of βi which is presented in the results i(j) i ji i k∗ k of Theorem 1. In other words, if the value of βi is less of βi . In this case, we need to choose the lowest or the k− ˆk k than that of βi , the transmit power of aggregator Sji highest value of βi to improve the payoff of MNO Mi in will exceed the maximum tolerable interference level of the kth sub-band. k∗ k+ ˆk Mi. While if βi is greater than βi , Sji will not Note that the pricing coefficients of MNOs control the aggregate the kth sub-band of Mi by transmitting with a optimal transmit powers of MNOs. We hence can claim that k∗ positive power and hence MNO Mi cannot obtain any the optimal pricing coefficient βi and the corresponding k∗ revenue from the kth sub-band of MNO Mj. transmit powers wˆji achieve an Stackelberg equilibrium k∗ If we substitute wîj in (13) into the( payoff function) of for the Stackelberg game of the leader (subscribers) and k k k∗ k the follower (aggregators). This concludes our proof. (8),( we can) find that ϖi(j) wi , wˆji , βi (or k k k∗ k k∗ k∗ ϖ(i)j wj , wîj , βj ) is only related to wˆji (or wîj ) k k which is controlled by βi (or βj ). Since MNO Mi can APPENDIX C k k only decide the value of βi , we assume βj has already PROOFOF PROPOSITION 2 been chosen by MNO Mj and hence MNO Mi only k∗ The proof of the above proposition follows directly from needs to focus on the optimization of βi to maximize ( ) k k k the results proved in [27], [45]–[47], [51], [52]. We list k k k∗ k k hji(1+hij wi ) ϖi(j) wi , wˆji , βi . Let us denote ρi = k and hjj these results as follows: 16

R1) [45, Theorem 6.6] For any stable roommate market, previously sent a request, it means that there is a there exists at least one stable partition and any two cycle sequence and the MNOs involved in the stable partitions have the same odd parties. sequence of processes will repeatedly send requests R2) [45], [46, Theorem 6.7] There is no stable matching to each other, forming a pair and separating from available for a roommate market if and only if there their matching partner and it is easy to verify that exists a stable partition with a set that has an odd party, the sequence of MNOs in the sequence of processes R3) Each even party can be broken into pairs of mutually forms an odd party. This results in the cases in Step agreed MNOs preserving stability [45], ii-c). Using the result R2), we can claim that there R4) If a stable roommate market contains at least one will be no stable matching for the MNOs in the stable matching structure, the set of all stable sequence of processes. Also using result R1), we can partitions and the set of all the stable matching claim that the addition operation always results in the coincide [47]. same odd parties. Finally, using result R4), we can claim that if there is no odd parity, the final result of Let us briefly describe how to prove Proposition 2 using the addition operation is always a stable matching. the above results. From Step ii) in the addition operation, Based on the above analysis, we hence can claim that if we can observe that if a new MNO Mi sequentially sends pairing requests to other MNOs in the market from the most at least one stable matching exists, the addition operation preferred MNO to the least preferred one, we can claim that will achieve it. If no stable matching exists for the IO- CA market, the addition operation will result in a stable for each MNO Mj that rejects the request of Mi, there must partition. This concludes the proof. exist another MNO Mk that is strictly preferred by Mj. In addition, any other MNO Ml which is more preferred APPENDIX D by Mi than Mj (e.g., satisfying Pi(Ml) < Pi(Mj)) has PROOFOF PROPOSITION 3 already rejected the requests of Mi. The sequential requests of Mi will have the following possible results: As can be easily observed from the addition operation, the worst case happens when the new MNO Mi joins the 1) If M has been rejected by all the MNOs in the i IO-CA market and finds a pairing partner (e.g., MNO Mj) market that are more preferred by M than matching i which is in an transposition with another MNO Mn. In with itself, it means that either MNO cannot find any this case, MNO Mn will repeat steps ii) again. If the same MNO that can improve its payoff by forming an situation happened repeatedly for each of the other MNOs IO-CA pair, or every MNO that can provide in K, this will cause all K MNOs to send requests to each performance improvement for Mi by forming an of the other K −1 MNOs and hence results in a complexity IO-CA pair has already been matched with another of O(K2) in the worst case. MNO that is more preferred than Mi. In this case, there is no stable matching for the IO-CA market APPENDIX E and Mi will not form an IO-CA pair with any other PROOFOF PROPOSITION 4 MNO in the market. This results in the cases in Step Let us use the following result and Proposition 2 to prove ii-a). Proposition E. 2) If there is one MNO M accepting the request of M j i R5) Suppose Π is a stable partition in a stable roommate for M ≠ M and P (M ) < P (M ), it means that j i i j i i market and C = ⟨a , a , . . . , a ⟩ is an odd party M prefers M to its current matching partner M i1 i2 i2k+1 j i l in Π for k ≥ 1. Then and all the other MNOs that are more preferred by ′ \C ∪ ⟨ ⟩⟨ ⟨ ⟩⟩ Π = (Π ) ai1 , ai2 ai3 , ai4 ,..., ai2k−1 , ai2k Mi than Mj prefer their current matching partner to \⟨ ⟩ is a stable partition of Π ai2k+1 [45], [49]. Mi. Therefore, both Mi and Mj have the incentive to form an IO-CA pair. Once an IO-CA pair has From the above result, we can claim that if MNO Mi has been deleted from the IO-CA market and Mi belongs to an been formed between Mi and Mj, Ml will start the odd party, each of the remaining MNOs in the same odd sequential requesting process as Mi. The same results as described in 1) and 2) will also apply for party as Mi will be able to find its IO-CA pairing partner and form a stable matching pair. Mi. It is possible for Ml to also find anther MNO If Mi has already been matched to another MNO Mj Mm that accepts its request and form a matching for Mj ≠ Mi, then Mj will separate with Mi and find its pair with Ml by separating from its current IO-CA pairing partner. If this process continues, it will result parting partner using the same procedure as the addition in a sequence of matching, separating and sequential operation. Therefore, we can use Proposition C to prove requesting processes of a set of MNOs. There are that the resulting matching will be a stable partition. This two possible results. If this sequence of processes concludes the proof. will result in a new stable matching structure in REFERENCES which all MNOs have found their new IO-CA pairing partners, it means that the final matching is [1] Y. Xiao, T. Forde, I. 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[48] P. Biro´ and G. Norman, “Analysis of stochastic matching markets,” Chau Yuen received the B.Eng. International Journal of Game Theory, vol. 42, no. 4, pp. 1021– and Ph.D. degrees from Nanyang Technological 1040, Nov. 2013. University, Singapore, in 2000 and 2004, [49] J. J. Tan, “Stable matchings and stable partitions,” International respectively. In 2005, he was a Postdoctoral Journal of Computer Mathematics, vol. 39, no. 1-2, pp. 11–20, 1991. Fellow with Bell Labs, Alcatel-Lucent, [50] T. Q. Quek, G. de la Roche, I. Guvenc¸,¨ and M. Kountouris, Small cell Murray Hill, NJ, USA. From 2006 to networks: Deployment, PHY techniques, and resource management. 2010, he was a Senior Research Engineer with Cambridge University Press, 2013. the Institute for Infocomm Research, Singapore, [51] B. G. Pittel and R. W. Irving, “An upper bound for the solvability where he was involved in an industrial project probability of a random stable roommates instance,” Random on developing an 802.11n wireless local area Structures & Algorithms, vol. 5, no. 3, pp. 465–486, Jul. 1994. network system and participated actively in the [52] D. J. Abraham, B. Pter, and D. Manlove, “almost stable?matchings in Third-Generation Partnership Project Long-Term Evolution and LTE the roommates problem,” in Approximation and Online Algorithms Advanced standardization. In 2008, he was a Visiting Assistant Professor (T. Erlebach and G. Persinao, eds.), vol. 3879 of Lecture Notes in with The Hong Kong Polytechnic University, Kowloon, Hong Kong. Computer Science, Springer Berlin Heidelberg, 2006. Since June 2010, he has been an Assistant Professor with the Department of Engineering Product Development, Singapore University of Technology and Design, Singapore. Dr. Yuen serves as an Associate Editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. He received the Lee Kuan Yew Gold Medal, the Institution of Electrical Engineers Book Prize, the Institute of Engineering of Singapore Gold Medal, the Merck Sharp & Dohme Gold Medal, the Hewlett Packard Prize (twice), and the 2012 IEEE Asia-Paciﬁc Outstanding Young Researcher Award. Yong Xiao (S’09-M’13-SM’15) received his B.S. degree in electrical engineering from China University of Geosciences, Wuhan, China in 2002, M.Sc. degree in telecommunication from Hong Kong University of Science and Technology in 2006, and his Ph. D degree in electrical and electronic engineering from Nanyang Technological University, Singapore in 2012. From August 2010 to April 2011, he was a research associate in school of electrical and electronic engineering, Nanyang Technological University, Singapore. From May 2011 to October 2012, he was a research fellow at CTVR, school of computer science and statistics, Trinity College Dublin, Ireland. From November 2012 to December 2013, he was a postdoctoral fellow at Massachusetts Institute of Technology. From December 2013 to November 2014, he was a MIT-SUTD postdoctoral fellow with Singapore University of Technology and Design and Massachusetts Institute of Technology. Currently, he is a postdoctoral fellow II at Department of Electrical and Computer Engineering at University of Houston. His research interests include machine learning, game theory and their applications in Luiz A. DaSilva (SM) is the Professor communication networks. He is a Senior Member of IEEE. of Telecommunications at Trinity College Dublin. He also holds a research professor appointment in the Bradley Department of Electrical and Computer Engineering at Virginia Tech, USA. His research focuses on distributed and adaptive resource management in wireless networks, and in particular wireless resource sharing, dynamic spectrum access, and the application of game theory to Zhu Han (S’01-M’04-SM’09) wireless networks. He is currently a Principal received the B.S. degree in electronic Investigator on research projects funded by the National Science engineering from Tsinghua University, in 1997, Foundation in the United States, the Science Foundation Ireland, and the and the M.S. and Ph.D. degrees in electrical European Commission under Horizon 2020 and Framework Programme engineering from the University of Maryland, 7. He is a Co-principal Investigator of CONNECT, the College Park, in 1999 and 2003, respectively. Telecommunications Research Centre in Ireland. Prof DaSilva is an From 2000 to 2002, he was an R&D IEEE Communications Society Distinguished Lecturer. Engineer of JDSU, Germantown, Maryland. From 2003 to 2006, he was a Research Associate at the University of Maryland. From 2006 to 2008, he was an assistant professor in Boise State University, Idaho. Currently, he is an Associate Professor in Electrical and Computer Engineering Department at the University of Houston, Texas. His research interests include wireless resource allocation and management, wireless communications and networking, game theory, wireless multimedia, security, and smart grid communication. Dr. Han is an Associate Editor of IEEE Transactions on Wireless Communications since 2010. Dr. Han is the winner of IEEE Fred W. Ellersick Prize 2011. Dr. Han is an NSF CAREER award recipient 2010.