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Journal of Computer Networks and Communications
Volume 2012, Article ID 794572, 14 pages
Research Article

Optimal Pricing of Spectrum Resources in Wireless Opportunistic Access

Chair of Wireless Communications, Poznan University of Technology, ul. Polanka 3, 60-965 Poznan, Poland

Received 8 February 2012; Accepted 23 April 2012

Academic Editor: Luca Ronga

Copyright © 2012 Hanna Bogucka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider opportunistic access to spectrum resources in cognitive wireless networks. The users equipment, or the network nodes in general are able to sense the spectrum and adopt a subset of available resources (the spectrum and the power) individually and independently in a distributed manner, that is, based on their local channel quality information and not knowing the Channel State Information (CSI) of the other nodes' links in the considered network area. In such a network scenery, the competition of nodes for available resources is observed, which can be modeled as a game. To obtain spectrally efficient and fair spectrum allocation in this competitive environment with the nodes having no information on the other players, taxation of resources is applied to coerce desired behavior of the competitors. In the paper, we present mathematical formulation of the problem of finding the optimal taxation rate (common for all nodes) and propose a reduced-complexity algorithm for this optimization. Simulation results for these derived optimal values in various scenarios are also provided.

1. Introduction

Opportunistic spectrum access and flexible and efficient spectrum allocation procedures as well are considered as measures to increase the utilization of the scarce radio resources in future wireless communication networks. Apart from the spectral efficiency, the Quality of Experience (QoE), and the associated fairness in resources distribution are in the focus of research towards the cognitive, opportunistic, and dynamic spectrum access. The spectrum allocation procedures are usually centralized, require the Channel State Information (CSI) of all links in the network, and involve the overhead traffic, which in turn occupies the scarce radio resources. For the future communication concepts, such as cognitive or opportunistic radio, the nodes are expected to take intelligent decisions on the amount of resources to be utilized in a distributed way, thus minimizing or eliminating the overhead traffic.

In this paper, we consider opportunistic acquisition of orthogonal frequency channels by the network nodes. An example of the multiple-access technique using such orthogonal channels is the well-known Orthogonal Frequency Division Multiple Access (OFDMA). In the opportunistic OFDMA, the network nodes are able to adopt a subset of accessible subcarriers (SCs) individually, as well as the transmission rate and power allocated to these SCs [1]. Below, we consider a more general scenario of the opportunistic access to frequency channels of any bandwidth, limited centralized management, and very limited control traffic, that is, there is no central frequency-channel scheduler, and no CSI exchange between the network nodes. Our approach to opportunistic spectrum allocation is related to noncooperative game theory and to the concept of pricing.

The game-theoretic scheduling for OFDMA has been considered in the literature as centralized and distributed SCs allocation. The centralized schemes allow for more efficient and fair spectrum utilization; however they require centralized management and a considerable amount of control traffic related to the CSI of all possible links in all considered frequency channels and to the information on the allocated channels. This information has to be exchanged or to be available at a central unit (e.g., at a base station of a cellular network) every time the channels qualities change for the nodes in the network area. Newest results for such centralized solutions based on cooperative complete information game models have been presented in [24]. Distributed decision making, on the contrary, deploys noncooperative games and seeks for Nash Equilibrium (NE) as a game solution. However, for the spectrum allocation, only the complete-information games have been considered in the literature so far. We believe that such models cannot be considered for practical applications in dynamically changing wireless networks, since the complete knowledge of the CSI related to all links to be available at every other node would require a lot of control traffic between the nodes in dedicated control channels. This information would have to be sent every time the channels change, so in mobile environment, the control traffic would be comparable to the information-data traffic. Thus, noncooperative complete-information game models are only suitable for multicell environment, where the players are the base stations, which have the CSI of all links in their cell areas [5, 6], or in static wireless scenarios.

The concept of resource pricing (or coercive taxation) has been considered in the literature extensively for power allocation, for example, for OFDM and OFDMA in [79]. There, the resource that is taxed is the power used by the network nodes, and the goal is to maximize the sum throughput given the total allowable transmission power in the network. In these papers however, it has been assumed that the SCs distribution among the users has already been done somehow centrally.

Pricing has been also applied for distributed power and interference management in the network, for example, in [10] for the code-division multiple access. For this purpose the complete-information noncooperative game models have been formulated. Such game-theoretic problem formulation has practical application for interference management, due to the fact that the complete information on the interference level can be available for each player, since the nodes can measure it locally. However, for the spectrum management the problem is different.

Some papers, for example, [1113] consider distributed allocation of resources based on pricing in a multicell scenario, where the base stations act as players. The pricing concepts developed for the multicell scenario, cannot be considered for the distributed resource allocation in decentralized networks, because contrary to a base station that may have the CSI of all links, a network node may only have the CSI of its own link. Another approach to price based spectrum management is based on iterative water filling, which allows all players to use the same frequency channels and adjust their power levels in these channels based on pricing function [14]. Definition of the pricing function for each player requires the CSI knowledge and exchange between the neighboring players, as well as the number of iterations. Similarly, in [15] the information exchange between the secondary and primary users is assumed for the spectrum leasing. In [16] the spectrum sharing is modeled as the oligopoly competition between the primary users and the Bertrand game model, which again requires the knowledge of the secondary users’ CSI by the players (primary users). Although the above-mentioned works have contributed to significant advance in the game-theoretic price-based models for spectrum sharing, they all make an assumption on the complete information available for all players that relate to their links CSI or they narrow to the power and interference management.

In our earlier work [17], we have presented distributed SCs allocation interference-free method in a network of the OFDMA-based opportunistic radios. It aimed at rational and efficient spectrum utilization in both the uplink or downlink transmission. Rationality in our case means that apart from maximizing the spectral efficiency, the network and each individual node aim at lowering the cost of this efficiency (resulting from taxation of resources) and at increasing the QoE (resulting from the number of served nodes). These rationality measures have been reflected in the definition of the noncooperative game model with complete information, and in the utility function defined for each player (the network node). The definition of this game involved aggregation of the players, in such a way that each player (the network node) can view all other players as one, named the network-nodes community (NNC). The complete information required in this game does not include the individual CSI of the other network nodes, but only the local (single-link) CSI and the taxation rate. This way noncooperative game with full information is reasonable and practically applicable in the dynamically changing network scenarios.

Here, in this paper, we present a generalized framework for the taxation-based allocation of orthogonal frequency channels in the opportunistic radio. First, we show inappropriateness of the complete-information game models in the considered framework. Then, we consider selfish and social behavior of the players by appropriate definitions of the utility functions reflecting such behaviors. These utility functions include the linear-taxation summand dependent on the amount of the acquired spectrum resources. We aim at finding the optimal taxation rate to come up with a high overall network efficiency defined in two ways. We provide the mathematical description of the problem of finding the optimal tax rate, show that the problem complexity is NP hard, and present and examine the reduced-complexity algorithm for solving it.

In Section 2, we present the main idea of the proposed game-theoretic approach to distributed spectrum allocation and provide formal definitions of the considered games. In Section 3, we mathematically derive the amount of bandwidth each player is inclined to acquire. In Section 4, we present the reduced-complexity algorithm to obtain the optimal taxation, where the optimality is defined in a number of ways. The simulation results are presented in Section 5, and the work is concluded in Section 6.

2. Taxation-Based Models of Distributed Spectrum Allocation

We consider the scenery of multiple cognitive-radio nodes (or users) appearing in the opportunistic network area, that make use of the orthogonal frequency channels, for example, OFDMA subcarriers. It implies that the nodes do not have to apply any guard frequency bands to limit the out-of-band interference. The frequency correction at the receiver is also assumed to be perfect. This scenery of the opportunistic and cognitive radio network is presented in Figure 1. The nodes are able to sense the radio environment, detect the parameter called tax rate available in a given area, detect available spectrum resources, and acquire a subset of these resources usable for their intended transmission, for example, peer-to-peer communication, an access to a cellular network or to any wireless network in general. The goal of each node is to make the best use of these resources, that is, obtain high data rate at the lowest cost. As a proof of our concept, we consider the freedom in the spectrum allocation, that is, theoretically even the smallest part of the spectrum can be used by a player. This theoretical assumption can be refined for practical applications, if we assume that the nodes demand the spectrum only if they can make use of it, that is, if there is a minimal contiguous part of the spectrum available for their intended communication (such as one OFDM subcarrier band) that may include the protection band to mitigate the interference to other transmissions.

Figure 1: The scenery of the cognitive radio network with the taxation-based opportunistic spectrum access.

Let us consider the resource acquisition procedure as a game, which each network node plays against the other nodes (the players). Let us assume that there are 𝐾 players, and the available bandwidth is 𝐵. (For the simplicity of our considerations, in the remainder of this paper, we assume that 𝐵 and 𝐾 are fixed, although in dynamically changing network environment, the number of players, their demands, and the available bandwidth vary.) A single player decides what portion of the available bandwidth she is going to use. (Note that such a personification and female pronouns are established in the game-theoretic convention.) Selfish player aiming at her throughput maximization would occupy the whole available spectrum; however, such a behavior decreases the spectral efficiency and the capacity of the network, as well as the QoE of other players who cannot access the network. The problem is known as the Tragedy of Commons described in [18].

As a countermeasure for the problem of commons sharing and utilization, taxation of the resources is introduced. In our network scenery presented in Figure 1, the taxation rate is the same for all network nodes or users (the players). This tax rate is known in the considered area. It can be stored in an area database (among other parameters, required for the efficient operation of the cognitive or opportunistic users, e.g., the spectrum masks for available bands in a given location and time, primary-users detection thresholds, etc.) or transmitted by the elected master node in case, when the considered network operates independently from the area database in an ad hoc manner. It is being updated periodically and broadcasted in this area as one parameter among many other ones in a typical Broadcast Control CHannel (BCCH), or specifically defined Cognitive Pilot Channel (CPC) [19]. Let us stress that this broadcast transmission of a single parameter occupies really minor resources, contrary to the situation of transmitting full CSI of all links in the considered frequency band using dedicated channels.

2.1. Inappropriateness of the All Links Complete-Information Game Model

Let us first show that the complete-information game model which makes use of the CSI of all involved links is not suitable for our scenario. The utility function for player 𝑘 in such a game, in which the concept of resource taxation is applied, reflects her throughput (revenue) and its related cost, and is defined as (Note that from this point, the mathematical analysis in this paper is performed in a continuous space for the sake of generality, however it can be easily translated into a discrete orthogonal channel scenario.) 𝜍𝑤𝑘=(𝑓)𝐵2𝐵1log21+𝛿𝑘𝑃𝑘(𝑓)𝛾𝑘(𝑓)𝑤𝑘(𝑓)𝑑𝑓𝑟0𝑏𝑘,(1) where 𝑤𝑘(𝑓) is the function indicating the occupancy of the frequencies by player 𝑘 (𝑤𝑘(𝑓)=1 if frequency 𝑓 is assigned to user 𝑘 and 𝑤𝑘(𝑓)=0 otherwise), 𝐵1 and 𝐵2 are the lower and the upper bounds of the available spectrum (𝐵2𝐵1=𝐵), 𝑏𝑘 is the amount of bandwidth the player acquires, 𝛾𝑘(𝑓)=|𝐻𝑘(𝑓)|2/𝒩0 is the Carrier-to-Noise Ratio (CNR) measured at the frequency 𝑓, 𝐻𝑘(𝑓) and 𝑃𝑘(𝑓) are the 𝑘th user’s channel characteristic and the power spectral density allocated to this frequency, respectively, and 𝒩0 is the noise power spectral density. Let us note that for the case of orthogonal channels, interference that is usually added to noise equals zero. Moreover, in (1), 𝛿𝑘 is the factor (often called the Signal-to-Noise Ratio (SNR)-gap) depending on the assumed player’s Bit Error Probability (BEP) 𝑃𝑒𝑘. (In case of M-QAM, 𝛿𝑘=1,5/ln(0,5𝑃𝑒𝑘), while for 𝑀4 and the SNR in the range of 0–30 dB it can be set more precisely as 𝛿𝑘=1,5/ln(5𝑃𝑒𝑘) [20].) Finally, in the above equation, 𝑟0 is a linear tax rate.

Since every channel can be used by a sole player: 𝑓!𝑘𝑤𝑘𝑏(𝑓)=1,𝑘=𝐵2𝐵1𝑤𝑘(𝑓)𝑑𝑓.(2)

To find the NE in such a game, we shall be solving this problem numerically. This numerical representation of finding the NE is the binary linear programming problem, that is, for a given set of 𝛾𝑘(𝑓) values for all 𝐾 nodes and for a considered number of available channels we shall find binary values 𝑤𝑘(𝑓) for a given 𝑟0. Then, looking for optimum 𝑟0 to maximize some goal function, for example, the network sum throughput, would be even more complex. Note, that solution of such a defined problem would require the knowledge of all 𝛾𝑘(𝑓) by every node, and thus, as mentioned before such a model for resource allocation is not practical.

2.2. Game Models Using Only the Local CSI

To narrow the space of this analysis and to eliminate the necessity for complete information concerning other players’ strategies and payoffs, we propose to treat the rest of the players as a whole (the NNC). Note that NNC is not formally organized in any way. It is only viewed as such by a single player. Moreover, we let the players take decisions independently and subsequently one after another, as they appear in the network. It is a usual case in all wireless networks that some collision avoidance mechanism is implemented, or an access to decision-making entity (e.g., base-station) makes use of the random access channel to avoid taking decisions at the same time. We also limit the players in the maximum amount of bandwidth they can take at a time as a countermeasure for their greedy behavior, and let this maximum allowed bandwidth be 𝐵𝐼. The rationale behind such a limitation is proved in [21].

Let us first look at the utility defined for the 𝑘th player so, as to reflect the player’s throughput: 𝜁𝑘𝑏𝑘=𝑓𝐒𝑘log21+𝛿𝑘𝑃𝑘(𝑓)𝛾𝑘(𝑓)𝑑𝑓𝑟0𝑏𝑘,(3) where 𝐒𝑘 is the (noncountable) set of frequencies player 𝑘 occupies. Let us note that before a decision is taken by player 𝑘, she senses the available spectrum resources and knows which frequency bands are already occupied (the amount of available bandwidth for the 𝑘th player is 𝐵𝑘, and there is no possibility that different players acquire the same frequencies: 𝑘j𝑆𝑘𝑆𝑗= Ø). Thus, the game model is dynamic. Moreover, the game for each player is two-dimensional only, what noticeably simplifies the problem.

The model defined above is not really a commonly understood game, since the decision of the NNC is not represented in the utility function (unless we call it a game against the radio environment, or a game against the nature as in [22]). This model reflects selfish behavior of the players, who care only for their own throughput. In the reminder of the paper we will call this model Selfish-Behavior Model.

In [17] we have proposed a noncooperative complete-information two-dimensional game, where the payoffs of both players (player 𝑘 and the NCC at the 𝑘th game stage) reflect their decisions. In such a case, being the alternative to the framework discussed above, the strategy space of the considered 𝑘th player is the same (consists of possible number of the orthogonal channels the player acquires). The strategies of the NNC are the numbers of channels this community may occupy all together apart from the considered player. The utility function of the considered 𝑘th player is defined as follows: 𝜉𝑘𝑏𝑘,𝑐𝑘=1𝐵𝑓𝐒𝑘log21+𝛿𝑘𝑃𝑘(𝑓)𝛾𝑘𝐵(𝑓)𝑑𝑓𝑘𝑏𝑘𝑐𝑘𝑟0𝑏𝑘,(4) where 𝑐𝑘 is the amount of bandwidth that will be occupied by NNC, and 𝐵𝑘 is the amount of bandwidth available at the 𝑘th game stage. One may interpret formula (4) as the total normalized throughput (throughput per frequency unit of the total available bandwidth 𝐵) which could be obtained by the new incoming players in case they occupied the remaining bandwidth and had the same average spectral efficiency as the considered player. This way, in the decision making on how much of the spectrum to occupy, the players factor the social aspect of the network (to serve multiple nodes) and not just their own benefit. In the reminder of the paper the game model with the 𝑘th player’s utility function defined by (4) will be called the Social-Behavior Model.

Finally, the payoff for the NNC at the 𝑘th game stage can be defined as 𝜀𝑘=𝑐𝑘, where 𝑐𝑘[0,max(0,𝐵𝑘𝐵𝐼)], that is, the number of resources that can be potentially occupied by NNC.

3. Optimal Choice of Spectrum Resources: The Players’ Perspective

Let us now consider how the players choose their strategic options, and how to coerce their desired behavior to obtain specifically defined network benefit.

3.1. Selfish-Behavior Model

As the practical approach to the Selfish-Behavior Model (SelBM) described by the utility function (3), we propose to eliminate dominated (disadvantageous) strategies of the players. Taking the 𝑘th player’s strongest frequencies (the ones that have the highest 𝛾𝑘(𝑓) values) into account is in fact the elimination of dominated strategies of the considered player. Note, that from both the individual node and the whole network perspective, making use of the strongest channels results in higher spectral efficiency. Thus, the strategies of a single node are all possible channels of the strongest frequencies (from 0 to 𝐵𝐼). In order to find the optimum taxation, let us consider rewriting formula (3): ̇𝜁𝑘𝑏𝑘=𝑏𝑘0log21+𝛿𝑘̇𝑃𝑘𝑓𝑘̇𝛾𝑘𝑓𝑘𝑑𝑓𝑘𝑟0𝑏𝑘,(5) where ̇𝛾𝑘(𝑓𝑘) is the function resulting from ordering (in the descending manner) of the continuous values of 𝛾𝑘(𝑓): (Ordering operation described by (6) is a hypothetical bijective mapping, which cannot be generically defined for any continuous space of values and depends on 𝛾𝑘(𝑓). This operation involves mapping of both the domain arguments and the codomain images of 𝛾𝑘(𝑓) to new arguments and images of ̇𝛾𝑘(𝑓𝑘) belonging to the same domain and codomain, respectively. For a discrete set of values ordering can be done by a standard sorting algorithm.) ̇𝛾𝑘𝑓𝑘𝛾𝑘(𝑓)𝑛,𝑚𝑓𝑘𝑛𝑓𝑘𝑚̇𝛾𝑘𝑓𝑘𝑛̇𝛾𝑘𝑓𝑘𝑚,(6) and ̇𝑃𝑘(𝑓𝑘) is the power spectral density resulting from the optimal power allocation for a total-power constraint, that is, from the water filling. Note that this ordering takes place at every stage, so in (5), the lower integration endpoint is always equal to zero. Moreover, as mentioned before 𝑏𝑘𝐵min𝐼,𝐵𝑘,(7) where 𝐵I is the maximum allowable bandwidth one player can take at a time, and 𝐵𝑘 is the bandwidth available at the 𝑘th game stage (for the 𝑘th player). We also limit our choices of 𝑏𝑘 to only useful frequencies, that is, 𝑏𝑘𝐵WF𝑘,(8) where 𝐵WF𝑘 is the useful bandwidth after the water-filling, that is, the bandwidth, in which: ̇𝑃𝑘(𝑓𝑘)>0 for all 𝑓𝑘[0,𝐵WF𝑘]. In such a case: 𝑓𝑘𝑏𝑘̇𝑃𝑘𝑓𝑘=1𝛾0𝑘𝑏𝑘1𝛿𝑘̇𝛾𝑘𝑓𝑘,(9) where 1/𝛾0𝑘(𝑏𝑘) is the water level obtained in the water-filling algorithm over the acquired bandwidth 𝑏𝑘 of player 𝑘. The utility function ̇𝜁𝑘(𝑏𝑘) can be thus expressed as: ̇𝜁𝑘𝑏𝑘=𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘/𝛾0𝑘𝑏𝑘𝑑𝑓𝑘𝑟0𝑏𝑘.(10) It can be easily shown that the above-defined function is concave, (For all 𝑏𝑘 defined by (7) and (8), the first summand is concave and monotonically increasing because 𝛿𝑘̇𝛾𝑘(𝑓𝑘)/𝛾0𝑘(𝑏𝑘)>1 for all 𝑓𝑘, and the second summand is linearly decreasing with 𝑏𝑘) so we may find its maximum (as each rational player would do) by solving the following equation: 𝜕𝜕𝑏𝑘̇𝜁𝑘𝑏𝑘=0.(11) As derived in Appendix A (formula (A.10)): 𝜕𝜕𝑏𝑘̇𝜁𝑘𝑏𝑘=𝑘𝑏𝑘𝑟0,(12) where 𝑘(𝑏𝑘) can be defined in a number of ways depending on the player’s CNR characteristic 𝛾𝑘(𝑓), and its resulting sorted values at the 𝑘th game stage ̇𝛾𝑘(𝑓𝑘). As derived in Appendix A in (A.11) for the two-path propagation model it can be approximated as 𝑏𝑘log2𝐴𝑘𝛿𝑘1+cos2𝜋𝑏𝑘𝜏𝑘𝜑𝑘+log2𝑏𝑘1Γ𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘2+𝑏𝑘×Γln2𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘212𝐴𝑘𝛿𝑘cos2𝜋𝑏𝑘𝜏𝑘𝜑𝑘21,(13) where the 𝐴𝑘, 𝜑𝑘, and 𝜏𝑘 are the parameters of the considered multipath propagation model depending on the signal attenuation, average phase difference between the arriving multipath signal components, and the multipath delay spread (see Appendix A). Thus, by solving (11) we obtain the amount of bandwidth 𝑏𝑘 the 𝑘th player is inclined to acquire 𝑏𝑘=𝑏𝑘𝑏𝑘=𝑟0.(14) Note, that for a given 𝑟0, user 𝑘 can find 𝑏𝑘, not knowing other players CSI, and this finding is independent from other players’ choices.

3.2. Social-Behavior Model

Let us now consider formula (4) reflecting the Social Behavior Model (SocBM) in the form with ordered values of 𝛾𝑘(𝑓) (similarly as in the previous section): ̇𝜉𝑘𝑏𝑘,𝑐𝑘=1𝐵𝑏𝑘0log21+𝛿𝑘̇𝑃𝑘𝑓𝑘̇𝛾𝑘𝑓𝑘𝑑𝑓𝑘𝐵𝑘𝑏𝑘𝑐𝑘𝑟0𝑏𝑘,(15) Based on (6)–(9), the above formula can be easily (again similarly as in the previous section) converted to ̇𝜉𝑘𝑏𝑘,𝑐𝑘=1𝐵𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘/𝛾0𝑘𝑏𝑘𝑑𝑓𝑘𝐵𝑘𝑏𝑘𝑐𝑘𝑟0𝑏𝑘.(16) It can be shown that function (16) is concave, (For all 𝑏𝑘 defined by (7) and (8), the first factor in the first summand is concave and monotonically increasing, the second factor in the first summand is linearly decreasing with 𝑏𝑘, and so is the second summand.) so we may find its maximum (as each rational player would do) by solving the following equation: 𝜕𝜕𝑏𝑘̇𝜉𝑘𝑏𝑘,𝑐𝑘=0.(17) The derivative (𝜕/𝜕𝑏𝑘)̇𝜉𝑘(𝑏𝑘,𝑐𝑘) is defined by formula (B.1) obtained in Appendix B, whose simplified form is the following: 𝜕𝜕𝑏𝑘̇𝜉𝑘𝑏𝑘,𝑐𝑘=𝒢𝑘𝑏𝑘,𝑐𝑘𝑟0,(18) where 𝒢𝑘(𝑏𝑘,𝑐𝑘) can be defined as in (B.3) for rural areas. (We do not repeat its long formula here. See Appendix B for its definition.) Thus, by solving (16) we obtain the amount of bandwidth 𝑏𝑘 which the 𝑘th player is inclined to acquire taking into account the considered amount of bandwidth to be occupied by the NNC at the 𝑘th game stage: 𝑏𝑘𝑐𝑘=𝑏𝑘𝑐𝑘𝑏𝒢𝑘=𝑟0.(19) In other words, 𝑏𝑘(𝑐𝑘) is the best-response function in the considered two-dimensional game.

Let us recall that the payoff of the NNC is defined as 𝜀𝑘=𝑐𝑘, where 𝑐𝑘[0,max(0,𝐵𝑘B𝐼)] and is not dependent on the strategy of the 𝑘th player 𝑏𝑘. Thus, the NNC would always choose the strategy 𝑐𝑘=max(0,𝐵𝑘𝐵𝐼) resulting in its highest possible payoff. For this NNC strategy (𝑐𝑘) and for 𝑏𝑘(𝑐𝑘) we obtain the NE. Thus, for the calculated equilibrium strategies the players acquire a portion of bandwidth for their transmission.

4. Optimal Tax Rates

To obtain the desired behavior of the players and high overall network efficiency, the tax rate for the considered games (presenting either the SelBM or SocMB) should be properly chosen to obtain the maximum benefit for the whole network in the considered framework, for example, the maximum sum throughput reflecting the efficiency of the spectrum distribution. We can define our objective function as 𝜂ST=1𝐵𝐾1𝑘=0𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘/𝛾0𝑘𝑏𝑘𝑑𝑓𝑘,(20) which is the sum throughput (ST) of all players averaged over the total available bandwidth 𝐵. Alternatively, we may look at maximizing the actual spectral efficiency (SE) of the transmission in the network (the sum throughput averaged over the actually used frequency bandwidth): 𝜂SE=1𝐾1𝑘=0𝑏𝑘𝐾1𝑘=0𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘/𝛾0𝑘𝑏𝑘𝑑𝑓𝑘.(21) The next step would be to find the optimum value of 𝑟0 to maximize either function (20) or (21). Note that many other definitions of the objective function are possible, that could reflect the fairness or proportional fairness in the distribution of resources, as well as other factors, for example, the percentage of used bandwidth or the percentage of served players. Below, we will examine the two objective functions defined above by (20) and (21) and show that some fairness in resource distribution is also achieved with a tax-rate optimal for (20).

The values of 𝜂ST or 𝜂SE depend on 𝑟0, and on the values of ̇𝛾𝑘(𝑓𝑘). (This is because 𝑟0 and ̇𝛾𝑘(𝑓𝑘) have the implication on 𝑏𝑘 and on the throughput obtained by player 𝑘.) Moreover, the order of appearance of the players in the game matters, since ̇𝛾𝑘(𝑓𝑘) depends on ̇𝛾𝑗(𝑓𝑗) for all 𝑗<𝑘 (frequencies allocated to players taking decisions before player 𝑘 must be excluded from this player considerations). Unfortunately, both 𝜂ST and 𝜂SE are neither strictly concave or convex functions of 𝑟0. In general, for low 𝑟0 it pays off for all players to acquire the highest possible amount of bandwidth, irrespective to their channel qualities. As 𝑟0 increases, it becomes affordable to acquire some bandwidth only for the players with good channel conditions (high 𝛾𝑘(𝑓) and ̇𝛾𝑘(𝑓𝑘)). Thus, in such a case, the spectral efficiency 𝜂SE is increased, and the average sum-throughput 𝜂ST may be decreased. However, when 𝑟0 is too high, that is, close to the barrage tax rate, only very few players can afford some small portion of the frequency band, so a lot of available bandwidth is not used, and thus both the average sum-throughput 𝜂ST and the spectral efficiency of the network 𝜂SE are low. Thus, there exist some optimum values for 𝑟0 to maximize either 𝜂ST or 𝜂SE: 𝑟0=𝑟ST0𝜂ST𝑟ST0=max𝜂ST𝑟0,𝑟0=𝑟SE0𝜂SE𝑟SE0=max𝜂SE𝑟0.(22) However, it is not straightforward to determine these optimum values. (As mentioned before, 𝜂ST and 𝜂ST depend on ̇𝛾𝑘(𝑓𝑘) functions, which have different arguments 𝑓𝑘 for different players, and on the order of players’ appearance.) To find this optimum, even numerically, is a complex NP-hard problem, and the optimization procedure has to be performed every time the users’ channels change. Some simplifications can be obtained in finding this optimum in a proper time span (not necessarily shorter than the coherence time of the tracked processes: 𝛾𝑘(𝑓)), because the value of the optimum in the next time instant should be found close to the optimum value in the previous time instant. For this purpose we may apply a method that actually traces the variations of the objective function (𝜂ST or 𝜂SE) rather than the variations of the players’ channel conditions.

The considerations presented in this section for continuous orthogonal channels can be easily translated to discrete orthogonal channels, that is, to the case of having a set of 𝑁 available channels (e.g., OFDM subcarriers) to be acquired by the players. In such a case, the integrations in (5), (10), (15), (16), (20), and (21) should be replaced by summations, and the value of 𝑏𝑘 and 𝑏𝑘 should be approximated by the discrete number of channels 𝑖𝑘 of a particular bandwidth Δ𝑓 (with a particular resolution). Moreover, as shown in [17], there exists the NE for the discrete orthogonal channels (like in OFDMA). Below, for such a case of the available bandwidth discretization, we present the optimal tax-rate-searching algorithm with reduced complexity tracing the instantaneous variations of the network-objective function 𝜂ST around its maximal value.

Step 1. Initialize algorithm:
Determine 𝑁 available channels of Δ𝑓 bandwidth,
Determine the range of 𝑟0[𝑟min,𝑟max],
Determine the increment of 𝑟0: Δ𝑟0,
Determine acceptable value of 𝜂ST: 𝜂STmin,
Determine 𝐾, 𝐵𝐼, measure the players’ 𝛾𝑘(𝑓),
Determine the order of players’ appearance.

Step 2. Find the 𝑏𝑘 values for all considered tax-rates 𝑟0.

Step 3. Calculate 𝜂ST for all r0.

Step 4. Find optimal tax-rate 𝑟ST0 that maximizes 𝜂ST.

Loop 1:
Step L1.1. Monitor the network-objective function 𝜂ST.Step L1.2. If 𝜂ST<𝜂STmin, go to Loop 2,else go to Loop 1.

Step 5. Update (increase) 𝑟ST0: 𝑟ST+=𝑟ST0+Δ𝑟0,

Step 6. Calculate the resulting 𝜂ST+=𝜂ST(𝑟ST+),

Step 7. If 𝜂ST+>𝜂ST, go to Step 9
else go to Loop 3.

Loop 2:
RepeatStep L2.1. 𝑟ST0=𝑟ST+,Step L2.2. 𝜂ST=𝜂ST+,Step L2.3. 𝑟ST+=𝑟ST0+Δ𝑟0,Step L2.4. Calculate 𝜂ST+=𝜂ST(𝑟ST+),Until 𝜂ST+>𝜂ST.

Step 8. Go to Step 12.

Step 9. Update (decrease) 𝑟ST0: 𝑟ST=𝑟ST0Δ𝑟0,

Step 10. Calculate the resulting 𝜂ST=𝜂ST(𝑟ST),

Step 11. If 𝜂ST>𝜂ST, go to Loop 3
else go to Step 12.

Loop 3:
RepeatStep L3.1. 𝑟ST0=𝑟ST,Step L3.2. 𝜂ST=𝜂ST, Step L3.3. 𝑟ST=𝑟ST0Δ𝑟0,Step L3.4. Calculate 𝜂ST=𝜂ST(𝑟ST),Until 𝜂ST>𝜂ST.

Step 12. If 𝜂ST𝜂STmin
communicate new tax-rate and go to Loop 1,
else Warn: “No tax-rate meeting the objectives”

As it will be shown in the next section, 𝜂ST is more appropriate as the network-objective function for the fairness of resource distribution among the players in the case of both SelBM and SocBM, and always has a maximum when 𝐵I is properly chosen (not too small) for a given 𝐾 and 𝐵. Analogous algorithm to the one presented below can be performed for searching the maximum of 𝜂SE. The presented algorithm has reduced complexity due to the application of the following methods: optimum tax-rate searching around the previous optimum and optimization procedure running only when 𝜂ST drops below the required value: 𝜂STmin.

Alternatively, to reduce the rate of necessary calculations to solve the optimization problem, we may maximize the expected values of (20) or (21) over the set of random variables ̇𝛾𝑘(𝑓𝑘): 𝜂ST=𝔼̇𝛾𝑘{𝜂ST}, 𝜂SE=𝔼̇𝛾𝑘{𝜂SE}. Such definitions of the objective functions could be useful if we were able to approximate the expectation values with the average values and use them in a static (or slowly changing) environment. The resulting tax rate would approximate the optimum one (either 𝑟ST0 or 𝑟SE0) with unknown accuracy, while the optimization procedure can be performed off-line. This option is to be investigated in the future.

5. Numerical Results

Our simulation setup is the following. We have considered an available bandwidth 𝐵 with the resolution Δ𝑓=𝐵/256, where Δ𝑓 can be considered as the smallest spectrum unit, that can be occupied by orthogonal signals, for example, OFDM subcarriers. In our considered scenario, the total transmission power has been fixed. The power constraint for each link results from the distance between the transmitter and the receiver and from the power-control mechanism. (Usually this mechanism is applied to combat the near-far effect and the interference between the users; however, here, we assume orthogonal frequency channels, so this mechanism is only used to assure the appropriate quality of the link, i.e., the required average SNR, which in our case has been set to 30 dB). For our simulation purposes, it has been assumed that the order of appearance of the players in a game is random. Furthermore, we assume that the power control mechanism has a tolerance of 3 dB, so that random deviation from the average SNR is possible for any node (average SNR =(30±3) dB). This average SNR deviation (which also reflects the accuracy of power-control in modern radio systems) has been chosen to differentiate possible link qualities. Moreover, two example channel models have been compared. The first one is the two-path Rayleigh-fading channel with the delay spread ranging from 0 to 1/64 of 1/𝐵, and the average power of the second path being −3 dB relative to the first path (such a model can be considered as suitable for rural environment). The second considered model is the six-path channel, with paths having the same power, and delays uniformly spread between 0 to 1/𝐵. (This is a test-channel model often used for the test of equalizers that reflects particularly hostile environment with very small coherence bandwidth and very deep fading.) We have observed 1000 channel realizations and assumed the target BEP 𝑃𝑒𝑘=103 for all links (for all 𝑘).

For the comparison purposes, we present results of our proposed framework and the reduced-complexity algorithm of finding the optimal tax rate together with the results of the greedy algorithm (that assigns the frequencies to the players with the highest CNR values at these frequencies) and Round-Robin algorithm of resource distribution. Although both of these algorithms can be only implemented in a centralized manner, they give the two opposite extremes: either maximum spectral efficiency or maximum fairness for the case of the whole used bandwidth.

Let us first analyze the network performance in the case of the users’ SelBM and the influence of the tax rate 𝑟0, the restricted amount of bandwidth 𝐵I, and the number of players 𝐾 on the network behavior. In Figures 2 and 3 we observe the averaged sum throughput 𝜂ST defined by (20) and the network spectral efficiency 𝜂SE defined by (21), respectively, for the two-path channel model. As we can see, there is some optimum tax rate 𝑟ST0 that maximizes 𝜂ST when 𝐵I is not too small for a given 𝐾. Otherwise, 𝜂ST is constant for low tax rates, and then, for higher tax rates decreases to zero. The optimal tax rate 𝑟SE0 that maximizes 𝜂SE is close to the barrage tax.

Figure 2: Average sum throughput 𝜂ST versus the tax rate for SelBM; two-path channel model.
Figure 3: Spectral efficiency 𝜂SE versus the tax rate for SelBM; two-path channel model.

To better understand the mechanism of increasing tax rates in the network, already discussed in the previous section, let us analyze Figures 4 and 5. There, the percentage of served nodes and the percentage of used bandwidth are shown versus the tax rate 𝑟0. (We assume that the node is served if it is able to acquire a portion of bandwidth satisfying her target BEP.) As we can see, low taxes allow to utilize most of the bandwidth and serve most of the users, again when 𝐵I is properly chosen. In general, restricting the players in the amount of bandwidth they can take at their turn has negative influence on 𝜂ST (the maximum is always achieved for 𝐵𝐼=𝐵) and on the percentage of used bandwidth, and positive influence on 𝜂SE and the percentage of served nodes, but only for relatively low tax rates. For higher tax rates, for which 𝜂ST and the percentage of used bandwidth dramatically drop, both 𝜂SE and the percentage of served nodes are not dependent on 𝐵𝐼. Thus, our first conclusion is that for the fairness of the resource distribution, it is better to apply 𝐵𝐼=𝐵, and calibrate just the tax rate to optimize 𝜂ST rather than 𝜂SE. Similar (analogous) conclusions can be derived for the SocBM and for the other channel model.

Figure 4: The percentage of served nodes versus the tax rate for SelBM; two-path channel model.
Figure 5: The percentage of used bandwidth versus the tax rate for SelBM; two-path channel model.

In Figures 6 and 7 we can observe the tax rates 𝑟ST0 optimizing 𝜂ST for both the SelBM and the SocBM and in the case of two-path and six-path channel models. These tax rates have been found using the algorithm defined in the previous section. As we can see, for the SelBM the optimum tax rates 𝑟ST0 for different 𝐵𝐼, converge to the same value as 𝐾 increases. It is not the case for the SocBM. Moreover, for the two-path channel model, the values of the optimum tax rate 𝑟ST0 are higher than for the six-path channel model.

Figure 6: Optimal tax rate for sum throughput 𝑟ST0 versus the number of competing players for the SelBM and the SocBM; two-path channel model.
Figure 7: Optimal tax rate for sum throughput 𝑟ST0 versus the number of competing players for the SelBM and the SocBM; six-path channel model.

The tax-rate that optimizes 𝜂SE does not depend on 𝐾. For the SelBM, it also does not depend on 𝐵𝐼, but only varies for different channel models. For the SocBM, 𝑟SE0 depends on both, 𝐵𝐼 and the channel model. This can be observed in Table 1.

Table 1: Optimal tax rates 𝑟SE0 maximizing the network spectral efficiency 𝜂SE for all values of 𝐾.

In Figures 8 and 9 one can observe the average sum throughput 𝜂ST resulting from the optimal taxation versus the number of competing players 𝐾, for both the SelBM and the SocBM and in the case of two-path (Figure 8) and six-path (Figure 9) channel models. In Figures 10 and 11 the network spectral efficiency 𝜂SE is presented versus 𝐾 for the same cases. Note, that for a given channel model the achievable average sum throughput 𝜂ST is exactly the same for both behavior models: either SelBM or SocBM (although the respective optimal tax rates are different). The same holds for the achievable spectral efficiency 𝜂SE. The difference between the plots occurs for different channel models. The achievable 𝜂ST as well as 𝜂SE are higher for the six-path channel model than for the two-path channel model due to the fact that this six-path model presents higher diversity in the subbands qualities for each player, so the players can make better choices. Finally, we have observed that when the optimum tax rate 𝑟ST0 is applied in either scenario, 99–100% of nodes are served.

Figure 8: Average sum throughput for the optimal tax rate for both SelBM and SocBM; two-path channel model.
Figure 9: Average sum throughput for the optimal tax rate for both SelBM and SocBM; six-path channel model.
Figure 10: Spectral efficiency for the optimal tax rate for both SelBM and SocBM; two-path channel model.
Figure 11: Spectral efficiency for the optimal tax rate for both SelBM and SocBM; six-path channel model.

Note, that our framework cannot result in the maximal achievable sum throughput, which can be only obtained when the problem described by (1) is solved, which assumes complete-information of all links CSI and 𝐾-dimensional game that can be solved in either cooperative or noncooperative manner.

6. Conclusions

Here above, we have presented a game-theory-related framework for distributed allocation of spectrum resources in the opportunistic radio access networks. Contrary to the methods presented in the literature so far, in our game models, we do not assume the complete knowledge of the players CSI. Each player has the information on her own CSI only. Additionally, the taxation-rate parameter available in a data base and mandatory in the considered area is made known to the players through the broadcast channel (BCCH or the CPC). This significantly reduces the amount of control traffic in the network when compared with the frequent exchange of the all links CSI in the dedicated channels. Above, we have proposed a reduced-complexity algorithm of finding and tracing the optimum tax-rate value maximizing the network objective function. Our presented framework and the algorithm of finding the optimal taxation-rate result in high network benefit reflected in the sum throughput, but also in fairness of resource distribution (understood as the number of served nodes). The simulation results show that it is more beneficial for the network and for the individual players to use taxation with the tax rate maximizing the network sum throughput rather than to additionally limit the users in the maximum bandwidth they can acquire at the time. It is also more beneficial than maximization of the network spectral efficiency due to better utilization of the spectrum resources and higher percentage of served nodes. Simulation results also show that in the considered scenarios, when the optimal tax rate is applied the achievable sum throughput per frequency unit is as high as 5.5–6 [bits/s/Hz] (depending on the considered propagation model) for sufficiently high number of players. Moreover, in such a case, 99-100% of nodes are served in the network, that is, are able to acquire some resources satisfying their target BEP.



Below, we calculate the derivative of ̇𝜁𝑘(𝑏𝑘): 𝜕𝜕𝑏𝑘̇𝜁𝑘𝑏𝑘=𝜕𝜕𝑏𝑘𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘/𝛾0𝑘𝑏𝑘𝑑𝑓𝑘𝜕𝜕𝑏𝑘𝑟0𝑏𝑘=𝜕𝜕𝑏𝑘𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘𝑑𝑓𝑘𝜕𝜕𝑏𝑘𝑏𝑘0log2𝛾0𝑘𝑏𝑘𝑑𝑓𝑘𝑟0.(A.1)

Because the one integrand in the first summand does not depend on 𝑏𝑘, and the one integrand in the second summand does not depend on 𝑓𝑘, further derivation of the above formula is the following: 𝜕𝜕𝑏𝑘̇𝜁𝑘𝑏𝑘=log2𝛿𝑘̇𝛾𝑘𝑏𝑘𝜕𝜕𝑏𝑘𝑏𝑘log2𝛾0𝑘𝑏𝑘𝑟0=log2𝛿𝑘̇𝛾𝑘𝑏𝑘𝑏𝑘𝜕𝜕𝑏𝑘log2𝛾0𝑘𝑏𝑘log2𝛾0𝑘𝑏𝑘𝑟0=log2𝛿𝑘̇𝛾𝑘𝑏𝑘1𝑏ln2𝑘𝛾0𝑘𝑏𝑘𝜕𝜕𝑏𝑘𝛾0𝑘𝑏𝑘log2𝛾0𝑘𝑏𝑘𝑟0.(A.2) Now, this formula does not have a closed form, that is, (𝜕/𝜕𝑏𝑘)̇𝜁𝑘(𝑏𝑘) depends on the particular shapes of ̇𝛾𝑘(𝑏𝑘) and of 𝛾0𝑘(𝑏𝑘) (where the later function in turn depends on the shape of the former). As an example, let us consider the rural area with the two-path propagation. In such a case, the channel power characteristic can be described as 𝛾𝑘||𝛼(𝑓)=𝑘||2||1𝛽𝑘exp𝑗2𝜋𝑓𝜏𝑘𝜑𝑘||2,(A.3) where 𝛼𝑘 is the complex amplitude attenuation of the first path, 𝛽𝑘 is the attenuation of the second path relative to the first path, 𝜏𝑘 is the multipath delay spread, and 𝜑𝑘 is average phase difference between the arriving multipath signal components. Let us denote 𝜙𝑘=2𝜋𝑓𝜏𝑘𝜑𝑘. For the sufficient distance between the transmitting and receiving antennas 𝛽𝑘1. Consequently, 𝛾𝑘(𝑓)𝛼2𝑘1cos𝜙𝑘+𝑗sin𝜙𝑘1cos𝜙𝑘𝑗sin𝜙𝑘=2𝛼2𝑘1cos𝜙𝑘=2𝛼2𝑘1cos2𝜋𝑓𝜏𝑘𝜑𝑘.(A.4) The above function is periodic and monotonically increasing for 𝜙[0,𝜋]. Therefore, its sorted (in a descending order) version ̇𝛾𝑘(𝑓𝑘) can be approximated as ̇𝛾𝑘𝑓𝑘𝐴𝑘1+cos2𝜋𝑓𝑘𝜏𝑘𝜑𝑘,(A.5) where 𝐴𝑘 is the proportionality constant.

Now, for our propagation model, let us find the formula for the water-level 𝛾0𝑘(𝑏𝑘) dependent on the shape of function ̇𝛾𝑘(𝑓𝑘) and on the considered bandwidth 𝑏𝑘. To this end we will integrate both sides of (9) resulting in 𝑏𝑘𝛾0𝑘𝑏𝑘=Γ𝑘+𝑏𝑘01𝛿𝑘̇𝛾𝑘𝑓𝑘𝑑𝑓𝑘,(A.6) where Γ𝑘 is the power limit for player 𝑘. Consequently 𝛾0𝑘𝑏𝑘𝑏𝑘Γ𝑘+𝑏𝑘01𝐴𝑘𝛿𝑘1+cos2𝜋𝑓𝑘𝜏𝑘𝜑𝑘𝑑𝑓𝑘1,(A.7) and because (1/cos𝜙𝑘)𝑑𝜙𝑘=tg𝜙𝑘/2, we obtain 𝛾0𝑘𝑏𝑘𝑏𝑘Γ𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘21.(A.8) Using the above expression we will derive the second term in (A.2): 𝑏𝑘𝛾0𝑘𝑏𝑘𝜕𝜕𝑏𝑘𝛾0𝑘𝑏𝑘Γ𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘2𝑏𝑘Γ𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘×tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘222𝐴𝑘𝛿𝑘cos2𝜋𝑏𝑘𝜏𝑘𝜑𝑘21=𝑏𝑘×Γ𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘212𝐴𝑘𝛿𝑘cos2𝜋𝑏𝑘𝜏𝑘𝜑𝑘21.(A.9) We can now substitute (A.5)–(A.9) to (A.2), which results in 𝜕𝜕𝑏𝑘̇𝜁𝑘𝑏𝑘=𝑘𝑏𝑘𝑟0,(A.10) where the first term on the right-hand side of the above equation 𝑘(𝑏𝑘) does not depend on 𝑟0, and for the considered channel model can be approximated as 𝑘𝑏𝑘log2𝐴𝑘𝛿𝑘1+cos2𝜋𝑏𝑘𝜏𝑘𝜑𝑘+log2𝑏𝑘1Γ𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘2+𝑏𝑘Γln2𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘21×2𝐴𝑘𝛿𝑘cos2𝜋𝑏𝑘𝜏𝑘𝜑𝑘21.(A.11) The above formula can be further simplified, when we assume that the phase difference between the arriving two-path waveform components is negligible due to similar distance that both waves travel, that is, when 𝜑𝑘0. Note, that formula (A.10) is very general, and function (𝑏𝑘) can be defined in a number of ways depending on the assumed propagation environment models.


Below, we calculate the derivative of ̇𝜉(𝑏𝑘,𝑐𝑘): 𝜕𝜕𝑏𝑘̇𝜉𝑘𝑏𝑘,𝑐𝑘=𝜕𝜕𝑏𝑘𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘/𝛾0𝑘𝑏𝑘𝑑𝑓𝑘×𝐵𝑘𝑏𝑘𝑐𝑘𝜕𝜕𝑏𝑘𝑟0𝑏𝑘=𝜕𝜕𝑏𝑘𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘/𝛾0𝑘𝑏𝑘𝑑𝑓𝑘×𝐵𝑘𝑏𝑘𝑐𝑘𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘/𝛾0𝑘𝑏𝑘𝑑𝑓𝑘𝑟0=𝜕𝜕𝑏𝑘𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘𝑑𝑓𝑘𝐵𝑘𝑏𝑘𝑐𝑘𝜕𝜕𝑏𝑘𝑏𝑘0log2𝛾0𝑘𝑏𝑘𝑑𝑓𝑘𝐵𝑘𝑏𝑘𝑐𝑘𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘/𝛾0𝑘𝑏𝑘𝑑𝑓𝑘𝑟0=log2𝛿𝑘̇𝛾𝑘𝑏𝑘𝜕𝜕𝑏𝑘𝑏𝑘log2𝛾0𝑘𝑏𝑘×𝐵𝑘𝑏𝑘𝑐𝑘𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘𝑑𝑓𝑘+𝑏𝑘0log2𝛾0𝑘𝑏𝑘𝑑𝑓𝑘𝑟0=log2𝛿𝑘̇𝛾𝑘𝑏𝑘log2𝛾0𝑘𝑏𝑘×𝐵𝑘𝑏𝑘𝑐𝑘1𝑏ln2𝑘𝛾0𝑘𝑏𝑘𝜕𝜕𝑏𝑘𝛾0𝑘𝑏𝑘×𝐵𝑘𝑏𝑘𝑐𝑘𝑏𝑘0log2𝛿𝑘̇𝛾𝑘𝑓𝑘𝑑𝑓𝑘+𝑏𝑘log2𝛾0𝑘𝑏𝑘𝑟0.(B.1) We can write the above expression in a simpler form: 𝜕𝜕𝑏𝑘̇𝜉𝑘𝑏𝑘,𝑐𝑘𝑏=𝒢𝑘,𝑐𝑘𝑟0,(B.2) and we can substitute expressions (A.5)–(A.9) to formula (B.1) to obtain the derivative of function ̇𝜉𝑘(𝑏𝑘,𝑐𝑘) for the rural channel model and the expression for 𝒢𝑘(𝑏𝑘): 𝒢𝑘𝑏𝑘log2𝐴𝑘𝛿𝑘1+cos2𝜋𝑏𝑘𝜏𝑘𝜑𝑘𝐵𝑘𝑏𝑘𝑐𝑘+log2𝑏𝑘1Γ𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘2×𝐵𝑘𝑏𝑘𝑐𝑘+𝑏𝑘×Γln2𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘21×2𝐴𝑘𝛿𝑘cos2𝜋𝑏𝑘𝜏𝑘𝜑𝑘21𝐵𝑘𝑏𝑘𝑐𝑘𝑏𝑘0log2𝐴𝑘𝛿𝑘1+cos2𝜋𝑓𝑘𝜏𝑘𝜑𝑘𝑑𝑓𝑘𝑏𝑘log21𝑏𝑘Γ𝑘+12𝜋𝜏𝑘𝐴𝑘𝛿𝑘tg𝜋𝑏𝑘𝜏𝑘𝜑𝑘2𝜑tg𝑘2.(B.3)


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