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A crosslayer optimization framework for congestion and power control in cognitive radio ad hoc networks under predictable contact
EURASIP Journal on Wireless Communications and Networking volumeÂ 2018, ArticleÂ number:Â 57 (2018)
Abstract
In this paper, we investigate the crosslayer optimization problem of congestion and power control in cognitive radio ad hoc networks (CRANETs) under predictable contact constraint. To measure the uncertainty of contact between any pair of secondary users (SUs), we construct the predictable contact model by attaining the probability distribution of contact. In particular, we propose a distributed crosslayer optimization framework achieving the joint design of hopbyhop congestion control (HHCC) in the transport layer and perlink power control (PLPC) in the physical layer for upstream SUs. The PLPC and the HHCC problems are further formulated as two noncooperative differential game models by taking into account the utility function maximization problem and the linear differential equation constraint with regard to the aggregate power interference to primary users (PUs) and the congestion bid for a bottleneck SU. In addition, we obtain the optimal transmit power and the optimal data rate of upstream SUs by taking advantage of dynamic programming and maximum principle, respectively. The proposed framework can balance transmit power and data rate among upstream SUs while protecting active PUs from excessive interference. Finally, simulation results are presented to demonstrate the effectiveness of the proposed framework for congestion and power control by jointly optimizing the PLPCHHCC problem simultaneously.
1 Introduction
1.1 Background and motivation
Cognitive radio (CR) [1] has been widely recognized as a critical technique to mitigate the spectrum scarcity problem and enhance the overall efficiency of spectrum usage, aiming to accommodate for the evolution of wireless systems towards 5G [2]. In a CR network (CRN), unlicensed secondary users (SUs) are allowed to opportunistically access the spectrum allocated to licensed or primary users (PUs) without interfering with the coexisting PUs. That is, the SUs do not violate the quality of service (QoS) requirements of the PUs. Most of the existing research efforts in CRNs mainly focus on the issues of the physical and media access control (MAC) layers for an infrastructurebased singlehop scenario, including spectrum sensing, spectrum access, and sharing techniques [3,4,5]. In addition, SUs can also form a multihop decentralized ad hoc network without the support of infrastructure. In a multihop cognitive radio ad hoc network (CRANET) [6], SU can access the licensed spectrum by seeking to overlay, underlay, or interweave its signal with the existing PUsâ€™ signals [7]. For the underlay approach, SUs are permitted to concurrently share the licensed spectrum with PUs while guaranteeing the power of interference and noise at the PU not beyond the interference temperature limit. In this context, the interference caused by the SUs should be controlled and mitigated through effective power control strategies. Many studies on power control for CRNs have been reported from different perspectives, such as imperfect channel knowledge [8], arbitrary input distributions [9], and social utility maximization [10].
In comparison with the lower layer solutions as stated before, recent work indicates that there are many new challenges towards routing problem at the network layer in multihop CRANETs, aiming to give more insights into the impact of spectrum uncertainty on routing strategies [11, 12]. However, the constraints and challenges with regard to SUs including random mobility, low deployment density, and limited resource along with discontinuous spectrum availability will give rise to intermittent connectivity of links among SUs in a decentralized CRANET [13]. Clearly, stochastic link outage further has a bearing on the successful transmission of data packets between a pair of SUs. To describe effective continuous transmission of SUs, the paradigm contact has been presented from different types [14], e.g., persistent contact, ondemand contact, and scheduled contact. Conceptually, a contact can be defined as a communication opportunity during which two adjacent SUs can communicate with each other. In a scheduled contactbased CRANET, multiple contacts or the set of communication opportunities can be easily derived from the statistical data of a priori available contact. In this case, the scheduled contact can be predicted and calculated accurately.
Similar to a wireline Internet or most other traditional wireless networks, network congestion in CRNs will also occur when the offered data load exceeds the available capacity of SU due to buffer overflow caused by accumulated data packets injected from upstream SUs. This therefore leads to aggressive retransmission, queuing delay, and blocking of new flows from upstream SUs. Indubitably, congestion control policy in the transport layer is essential to balance resource load and to avoid excessive congestion. However, the conventional Transmission Control Protocol (TCP) as a congestion control mechanism via windowbased or acknowledgementtriggered methods is initially designed and optimized to perform in reliable wired links with constrained bit error rates (BERs) and round trip times (RTTs) [15]. Recent study by [16] has reported that the performance of HTTP download deteriorates as much as 40% under the TCP window control in an IEEE P1900.4based cognitive wireless system using User Datagram Protocol (UDP) and TCP transport protocols. Alternatively, to accommodate for the challenging multihop wireless environments, some other research efforts about congestion control techniques have been conducted from the perspective of finding methods to modify TCP protocol [17]. Unfortunately, it has been also shown that these schemes of TCP modifications and extensions cannot be applied into CRANETs because of sudden largescale bandwidth fluctuation and periodic interruption caused by spectrum sensing and channel switching [18, 19].
It is also noted that the TCP congestion control is targeted to regulate the data rate of upstream SUs so that the total accumulated data load does not exceed the available capacity of SU. In principle, the link capacity between a pair of SUs depend strongly on transmit power of SU coupled with wireless channel conditions [20]. By leveraging the congestion control technique, on the one hand, the attainable data rate on a wireless link between a pair of SUs depends on the interference level, which in turn rests on power control policy. On the other hand, each SU is expected to increase its transmit power in order to obtain as much link capacity that each flow requires [21]. However, increasing the link capacity on one link may reduce the link capacities on other links owing to mutual interference of SUs. From the above discussions, we can see that jointly optimizing transmit power in the physical layer and data rate in the transport layer for attaining the optimal link capacity becomes highly valuable. With a joint crosslayer design, the physical layer is able to share its information and configuration about optimal transmit power with the transport layer without breaking the hierarchical structure of the traditional layered architecture [22]. This motivates us to reinvestigate the crosslayer coupling between capacity supply by power control and rate demand by rate control.
1.2 Related works
Congestion control in wireless multihop networks has been widely discussed via the NUM optimization problem maximizing the total utility, subject to some different constraints including the efficiency and fairness of resource allocation [22], heterogeneous traffic [22], lossy link [23], and multipath transmission [24]. Under the condition of outage probability caused by lossy links, another work [25] investigates the rateeffective network utility maximization problem to meet with delayconstrained data traffic requirement. However, although all of the aforementioned studies consider some realistic constraints, they apply to the traditional wireless multihop networks only and do not consider the spectrum uncertainty in CRNs. To the best of our knowledge, some studies on congestion control for CRNs have been reported recently, although the mainstream research effort is aimed at the problems of the physical and MAC layers. In [26], Xiao et al. developed a robust active queue management scheme to stabilize the TCP queue length at base station in an infrastructurebased CRN. By using the multiple model predictive control, the proposed scheme absorbed the disturbances caused by busty background traffic and capacity variation. It is found that [26] is not suitable for decentralized CRANET scenario due to a lack of centralized control and global information. Unlike the condition of infrastructurebased CRN, other studies in [18, 27, 28] have been undertaken in multihop CRN scenarios. In [27], Song et al. proposed an endtoend congestion control framework without the aid of common control channel by taking into account the nonuniform channel availability. The explicit feedback mechanism without timeouts and the timeout mechanism were also investigated. In [28], Zhong et al. presented a TCP network coding dynamic generation size adjust scheme by jointly considering network coding gain and delay. The proposed scheme can significantly reduce the retransmissions and guarantee the QoS and enhance the TCP performance. In [18], AlAli et al. proposed an endtoend equationbased TCP friendly rate control mechanism, which achieves rate adjustment by identifying network congestion. However, the endtoend control policy in [18, 27] is ill suited for operation over wireless transmission links characterized by higher RTTs, particularly if the links present the feature of intermittent connectivity in CRANET under predictable contact. On the contrary, the hopbyhop control reacts to congestion faster where the rates are adjusted at upstream nodes by feedback information about the congestion state of intermediate nodes.
Other recent schemes that exploit the crosslayer interaction information try to deal with congestion control problem in decentralized CRNs from a crosslayer design perspective. The objective of these schemes is to improve the overall network utility while protecting active PUsâ€™ communications from excessive interference introduced by SUs. In [29], Cammarano et al. presented a distributed crosslayer framework for joint optimization of MAC, scheduling, routing, and congestion control in CRAHNs, by maximizing the throughput of a set of multihop endtoend packet flows. However, similar to [18, 27], it is not clear how good the performance of the endtoend rate control is compared under a wireless transmission environment with higher RTTs. In [30], Nguyen et al. proposed a crosslayer framework to jointly attain both congestion and power control in OFDMbased CRNs through nonconvex optimization method. By means of the adaptation of dual decomposition technique also used by [20], the distributed algorithm was developed to obtain the global optimization. In [31], Nguyen et al. further devised an optimization framework achieving tradeoff between energy efficiency and network utility maximization for CRAHNs. By adjusting transmit power, persistence probability, and data rate simultaneously via the interaction between MAC and other layers, the proposed framework can jointly balance interference, collision, and congestion among SUs However, both of the frameworks in [30, 31] fail to take into account the impact of predictable contact or priori available contact between any pair of SUs on overall crosslayer performance.
1.3 Our approach and contributions
Our work in this paper mainly focuses on a decentralized CRANET under predictable contact in that it is easy to obtain the set of communication opportunities derived from the statistical available contacts among SUs. Owing to a lack of global information to achieve the centralized schedule, the crosslayer coupling between capacity supply by power control and rate demand by rate control needs to be carried out distributively by each SU via local information. Distributed implementation for power control and rate control depends on interactive processes among competitive SUs to figure out the crosslayer coupling relationships. Moreover, the objectives of SUs to maximize their utility functions are conflicting and their decisions are interactive. Apparently, it will be far more realistic to dynamically adjust transmit power and data rate according to the current instant time in the practical dynamic environment. The reason for adopting a differential game model rather than other decentralized optimization approaches is that the differential game is a continuous time dynamic game to investigate interactive decision making over time. In a differential game, the interactions among individual players are characterized by time dependency. This is in line with the nature of dynamic spectrum environment in practical CRANET scenario. Therefore, motivated by crosslayer coupling between capacity supply by power control and rate demand by rate control, we present a crosslayer optimization framework for CRANET under predictable contact by achieving the joint congestion and power control using a differential game theoretic approach. The main contributions of this paper are summarized as follows:

To measure the uncertainty of contact between a pair of SUs, a predictable contact model is presented by deriving the probability distribution of contact via a mathematical statistics theory. By using Shannon entropy theory, we further devise an entropy paradigm to characterize quantitatively the probability distribution of contact.

We propose a distributed crosslayer optimization framework for hopbyhop congestion control (HHCC) and perlink power control (PLPC) for upstream SUs. The HHCC and the PLPC problems are formulated as two noncooperative differential game models, by taking into account the utility function maximization and linear differential equation constraint with regard to the aggregate power interference to PUs and congestion bid for bottleneck SU.

We convert the noncooperative differential game models for the PLPC and the HHCC problems into two dynamic optimization problems. By adopting dynamic programming and maximum principle, we obtain the optimal transmit power and the optimal data rate of upstream SUs, respectively. The crosslayer optimization framework is implemented in a distributed manner through the crosslayer coordination mechanism between capacity supply by power controller and rate demand by rate controller.
1.4 Organization and notation
The rest of this paper is organized as follows. We firstly describe the system model in Section 2. Then, the problem formulation is presented in Section 3. In Section 4, we derive the optimal solutions to the proposed noncooperative differential game models and propose the distributed implementation approach to construct the crosslayer optimization framework. Simulation results are provided in Section 5, followed by the conclusions in Section 6.
Notation: \( \mathcal{A} \) denotes a set, and \( \left\mathcal{A}\right \) denotes the cardinality for any set \( \mathcal{A} \). We use a boldface capital to denote vector A to discriminate vectors from scalar quantities. â€‰â‹…â€‰ and â€–â‹…â€– represent the absolute value of a polynomial function and the Euclidean distance between the pair of variables, respectively. \( \mathbb{E}\left[\cdot \right] \) stands for the statistical expectation operator.
2 System model
2.1 Network model
Consider an underlay multihop CRANET coexisting with a cellular primary network as depicted in Fig. 1, wherein PUs can send their data traffic to a primary base station (PBS) via the licensed uplink channels. We denote the set of uplink channels by â„‹â€‰=â€‰{ch_{1},â€‰ch_{2},â€‰â‹¯,â€‰ch_{ Ï• }} where Ï• is the number of uplink channels. The uplink channel is either occupied by PUs or unoccupied. We employ the independent and identically distributed alternating ONOFF process to model the occupation time length of PUs in uplink channels. Specifically, the OFF state indicates the idle state where the uplink channels can be freely occupied by SUs. By performing spectrum sensing on all the uplink channels periodically, S SUs leverage the OFF state to access the unoccupied uplink channels by PUs. Let \( \mathcal{V}=\left\{{v}_1,{v}_2,\cdots, {v}_S\right\} \) refer to the set of S SUs. Each SU is equipped with two radio transceivers. One with a cognitive radio is used to opportunistically access the uplink channels for transmissions of data packets. The other is used for exchange of control signaling. Due to the randomness of data traffic and the dynamic behavior of PUs, we assume that the licensed uplink channels are available for usage by SU v_{ i } with a probability of Î´_{ i }, for \( {v}_i\in \mathcal{V} \). Based on the aforementioned ONOFF process, the occupancy probability of uplink channel ch_{ Î¾ } by PUs is defined as Î±_{ Î¾ }/(Î±_{ Î¾ }â€‰+â€‰Î²_{ Î¾ }) [32], where Î±_{ Î¾ } is a probability that uplink channel Î¾ transits from OFF to ON state, and Î²_{ Î¾ } is probability that uplink channel Î¾ transits from ON to OFF state, for Î¾â€‰=â€‰1, 2, â‹¯, Ï•. It is assumed that SUs can determine the occupancy probability of uplink channels by PUs through a priori knowledge of PUsâ€™ activities and local spectrum sensing. Owing to the mutually independent occupancy probability of uplink channel ch_{ Î¾ }, the probability Î´_{ i } of uplink channels used by SU v_{ i } can be expressed as:
Different from the assumption that time is divided into fixed time slots in a discrete way, we exploit the continuous time model to represent the operation duration of the CRANET. The continuoustime operation is confined to a predefined time interval [t_{0},â€‰T]. Use Ï‘_{ i }(t) and z(t) to denote the positions of SU v_{ i }, PU z at time tâ€‰âˆˆâ€‰[t_{0},â€‰T], respectively. Let R_{ T } and R_{ I } stand for the maximum transmission range of SU and the interference range of PU, respectively. Without interference with PUs, a pair of SU v_{ i } and SU v_{ j } can successfully communicate with each other on channel ch_{ Î¾ } at time t only if the Euclidean distance between SU v_{ i } and SU v_{ j } satisfies â€–Ï‘_{ i }(t)â€‰âˆ’â€‰Ï‘_{ j }(t)â€–â€‰â‰¤â€‰R_{ T } and when there is no any PU z on channel ch_{ Î¾ }, i.e., â€–Ï‘_{ i }(t)â€‰âˆ’â€‰z(t)â€–â€‰>â€‰R_{ I } and â€–Ï‘_{ j }(t)â€‰âˆ’â€‰z(t)â€–â€‰>â€‰R_{ I }, for \( {v}_i,{v}_j\in \mathcal{V} \) and Î¾â€‰=â€‰1, 2, â‹¯, Ï•. In this context, there exists a successful transmission link denoted by l_{(i,â€‰j)} from SU v_{ i } to SU v_{ j } on channel ch_{ Î¾ } at time t. For the sake of conciseness, instant time t will be restricted to the time interval [t_{0},â€‰T] henceforth.
Under the constraint of successful transmission links, we assume that there are multiple different sessions from source SUs to destination SUs. Each session is associated with a route from a source SU to a destination SU. Figure 2 illustrates an example of logical topology of the underlay multihop CRANET, where a series of red solid line denote a session along a route from source SU1 to destination SU5, which is one of the different routes. It is assumed that a session consists of several perlink flows with elastic traffic. We use the term perlink flow to describe a sequence of data packets with elastic traffic transmitted along a successful transmission link. With regard to the route from source SU1 to destination SU5, data packets of a flow enter upstream SU2, travel via single hop, then converge at bottleneck SU3 and, finally, move to downstream SU4. We focus on a scenario that multiple different sessions converge at bottleneck SU, aiming to reinvestigate the crosslayer coupling between capacity supply by power control and rate demand by rate control at upstream SUs. From Fig. 2, the convergence of multiple flows from upstream SU2, SU6, and SU8 via single hop may result in a possible congestion at bottleneck SU3 when the offered data load exceeds the available capacity of SU3 due to a buffer overflow, although the amount of data packets has been delivered to downstream SU4, SU7, and SU9. We assume that there are N flows of elastic traffic along the successful transmission links from N upstream SUs to bottleneck SU v_{ b } via single hop, for \( {v}_b\in \mathcal{V} \) and Nâ€‰<â€‰S. Let \( {\mathcal{V}}_{UP} \) and \( \mathcal{N}=\left\{1,2,\cdots, N\right\} \) represent the set of N upstream SUs and the set of flows of elastic traffic from N upstream SUs to bottleneck SU v_{ b }, for \( {\mathcal{V}}_{UP}\subset \mathcal{V} \). For notational simplicity, the flow of elastic traffic along link l_{(i,â€‰b)} from upstream SU v_{ i } to bottleneck SU v_{ b } is described by flow i, for \( i\in \mathcal{N} \) and \( {v}_i\in {\mathcal{V}}_{UP} \). Assuming that flow i of elastic traffic along link l_{(i,â€‰b)} arrives as a Poisson process of flow arrival intensity Î»_{ i } with a size drawn independently from a common distribution of mean \( \mathbb{E}\left[{\lambda}_i\right] \) [33]. When Î¨_{(i,â€‰b)}â€‰<â€‰1, the transmission link load, denoted by Î¨_{(i,â€‰b)}, induced by elastic traffic along link l_{(i,â€‰b)} is equal to [33]:
where C_{(i,â€‰b)}(P) denotes the capacity of link l_{(i,â€‰b)}, and Pâ€‰=â€‰{p_{1}(t),â€‰p_{2}(t),â€‰â‹¯,â€‰p_{ N }(t)} corresponds to the transmit power vector of N upstream SUs at time t. Here, we use p_{ i }(t) to represent the instant transmit power of upstream SU v_{ i } at time t. Noticing that the transmit power p_{ i }(t) of upstream SU v_{ i } can be adjusted in a continuous way but is also limited by a maximum transmit power threshold denoted by \( {\overline{p}}_i \), i.e., \( {p}_i(t)\in \left[0,{\overline{p}}_i\right) \). Based on elastic traffic model for each flow, the expected duration D_{ i } of flow i with size \( \mathbb{E}\left[{\lambda}_i\right] \) is given by [33]:
Under spectrum underlay scenario, SUs can simultaneously transmit with PUs but have to strictly control their transmit power to avoid interfering with coexisting PUs. Note that the simultaneous transmissions among SUs along a successful transmission link must be undertaken on the same channel, which will further incur the cochannel multiple access interference. We assume that the simultaneous transmissions among N SUs along a successful transmission link on channel ch_{ Î¾ } can be undertaken under the CDMAbased medium access in the physical layer [34]. The reason for adopting the CDMAbased medium access model is that transmit power of upstream SU can be controlled to induce a different signaltointerferenceplusnoise ratio (SINR) of successful transmission link due to a cochannel multipleaccess interference [20, 35, 36]. In principle, link capacity under this scenario cannot remain fixed but depends on SINR of successful transmission link between a pair of SUs. Let SINR_{(i,â€‰b)}(P) be the received SINR of bottleneck SU v_{ b } along link l_{(i,â€‰b)} on channel ch_{ Î¾ }. Therefore, the capacity of link l_{(i,â€‰b)} from upstream SU v_{ i } to bottleneck SU v_{ b } can be characterized by a global and nonlinear nonconvex function of the transmit power vector and channel conditions as follows [20]:
where Î¤_{ s } is a symbol period and Ï‡â€‰=â€‰â€‰âˆ’â€‰Ï†_{1}/log_{2}(Ï†_{2}â€‰â‹…â€‰BER) is a constant processing gain factor with Ï†_{1} and Ï†_{2} depending upon an acceptable BER along with the specific modulation and coding scheme. We assume that a largescale slowfading channel model is adopted to describe the lineofsight wireless transmission environment. In this case, channel gain is subject to distancedependent power attenuation or lognormal shadowing. As for the practical nonlineofsight scenario, we use a Rayleigh fading model, in which the channel gain is assumed to be independent exponentially distributed random variables with unit mean [37]. Let G_{(i,â€‰b)} and F_{(i,â€‰b)} denote the largescale slowfading and the Rayleigh fading channel gain of link l_{(i,â€‰b)} from upstream SU v_{ i } to bottleneck SU v_{ b }, respectively. Thus, we have the normalized Rayleigh fading channel gain \( \mathbb{E}\left[{F}_{\left(i,b\right)}\right]=1 \). By using the certaintyequivalent transmit power and interference power [34, 37], the received SINR of link l_{(i,â€‰b)} at bottleneck SU v_{ b } can be expressed as:
where n_{0} is the thermal noise power at bottleneck SU v_{ b }, I_{ p } is the interference caused by the PBS, and I_{ i } is the aggregate power interference introduced by other upstream SUs except upstream SU v_{ i }. The aggregate power interference is given by \( {I}_i={\sum}_{j\in \mathcal{N}\backslash i}{p}_j(t){G}_{\left(j,b\right)} \). In what follows, we are targeted at the lineofsight wireless transmission environment with the largescale slowfading channel gain. How to apply the dynamic fastfading or Rayleigh fading model under the nonlineofsight scenario into the crosslayer optimization framework for CRANET will be our further work in the future. Under spectrum underlay scenario, the interference power constraint shall be imposed to protect active PUsâ€™ communications from harmful interference caused by all the upstream SUs. We assume that the interference measurement point is located at bottleneck SU v_{ b } for convenience. Hence, the total interference caused by all the upstream SUs should be kept below the interference temperature limit Ï–_{PBS} at the interference measurement point of PBS:
2.2 Predictable contact model
Considering that a contact is viewed as a communication opportunity during which two adjacent SUs can communicate with each other, we move on to model the predictable contact between a pair of SUs from a priori available contact perspective. Based on the insight into successful transmission link l_{(i,â€‰j)} as noticed earlier, an encounter e^{(i,â€‰j)} is defined as an effective continuous transmission between SU v_{ i } and SU v_{ j } with a certain duration, for \( {v}_i,{v}_j\in \mathcal{V} \). It is worth pointing out that an encounter rests on the time of incidence and the duration of an effective continuous transmission between a pair of SUs [38]. Let t^{0, (i,â€‰j)} and Î”t^{(i,â€‰j)} represent the time of incidence and the duration of an encounter e^{(i,â€‰j)}, respectively, for 0â€‰<â€‰Î”t^{(i,â€‰j)}â€‰<â€‰Tâ€‰âˆ’â€‰t_{0}. Therefore, an encounter e^{(i,â€‰j)} between SU v_{ i } and SU v_{ j } can be formulated as:
Suppose that there exist K encounters between SU v_{ i } and SU v_{ j } within a predefined time interval [t_{0},â€‰T]. In particular, the â„“th encounter \( {e}_{\mathrm{\ell}}^{\left(i,j\right)} \) between SU v_{ i } and SU v_{ j } with a duration \( \Delta {t}_{\mathrm{\ell}}^{\left(i,j\right)} \), for â„“â€‰=â€‰1, 2, â‹¯, K, is given as:
where \( {t}_{\mathrm{\ell}}^{0,\left(i,j\right)} \) refers to the time of incidence of the â„“th encounter \( {e}_{\mathrm{\ell}}^{\left(i,j\right)} \). For mathematical tractability, we use the duration \( \Delta {t}_{\mathrm{\ell}}^{\left(i,j\right)} \) in (8) to characterize the â„“th encounter, i.e., \( {e}_{\mathrm{\ell}}^{\left(i,j\right)}\triangleq \Delta {t}_{\mathrm{\ell}}^{\left(i,j\right)} \). Thus, within time interval [t_{0},â€‰T], contact \( {\mathcal{C}}^{\left(i,j\right)} \) between SU v_{ i } and SU v_{ j } can be rigorously regarded as the set of all encounters, i.e., \( {\mathcal{C}}^{\left(i,j\right)}=\left\{{e}_1^{\left(i,j\right)},{e}_2^{\left(i,j\right)},\cdots, {e}_K^{\left(i,j\right)}\right\} \) and \( \left{\mathcal{C}}^{\left(i,j\right)}\right=K \). Note that the â„“th encounter \( {e}_{\mathrm{\ell}}^{\left(i,j\right)} \) in contact \( {\mathcal{C}}^{\left(i,j\right)} \) can be referred to a random variable due to the uncertainty of communication opportunity between SU v_{ i } and SU v_{ j }. In this way, we turn to employ a mathematical statistics theory to attain the probability distribution \( {\varUpsilon}^{\left(i,j\right)}=\left\{{\rho}_1^{\left(i,j\right)},{\rho}_2^{\left(i,j\right)},\cdots, {\rho}_M^{\left(i,j\right)}\right\} \) of contact \( {\mathcal{C}}^{\left(i,j\right)} \), which has been derived from Algorithm 1. In Algorithm 1, we introduce a coefficient M to denote the number of the subintervals, which is obtained by dividing interval [a,â€‰b] equally. According to the approximate derivation of the sample distribution in mathematical statistics, coefficient M should be reasonably assigned, depending on the number of encounters of K. That is, when Kâ€‰â‰¤â€‰100, coefficient M can range from 5 to 12. Obviously, it will be possible to measure the uncertainty of contact \( {\mathcal{C}}^{\left(i,j\right)} \) between SU v_{ i } and SU v_{ j } by the aid of the probability distribution Ï’^{(i,â€‰j)}. Thus, by analyzing the statistical data of a priori available contact or all encounters between a pair of SUs, it is implicitly understood that the contact can be in a sense predicted very accurately.
Table 1 summarizes the constrained relationship between the number of encounters within subinterval (d_{lâ€‰âˆ’â€‰1},â€‰d_{ l }] and contact probability \( {\rho}_l^{\left(i,j\right)} \) in Algorithm 1, for lâ€‰=â€‰1, 2, â‹¯, M. It is important to emphasize that the probability distribution Ï’^{(i,â€‰j)} belongs to a complete probability distribution, i.e., \( {\sum}_{l=1}^M{\rho}_l^{\left(i,j\right)}=1 \). Technically, the entropy paradigm is widely used for a measure of the uncertainty or randomness associated with a random variable in information theory [39]. In order to characterize quantitatively the probability distribution Ï’^{(i,â€‰j)}, we put forward an entropy paradigm by using Shannon entropy theory to measure the uncertainty of contact \( {\mathcal{C}}^{\left(i,j\right)} \). Specifically, the entropy H(Ï’^{(i,â€‰j)}) of the probability distribution Ï’^{(i,â€‰j)} can be given as:
Based on Algorithm 1, it is obvious to find that coefficient M impacts the structure of the probability distribution Ï’^{(i,â€‰j)}. Thereby, the entropy H(Ï’^{(i,â€‰j)}) will depend on the selection of coefficient M. Recall that the probability Î´_{ i } of uplink channels used by SU v_{ i } determines the stability of successful transmission link l_{(i,â€‰j)} or even the contact \( {\mathcal{C}}^{\left(i,j\right)} \) between SU v_{ i } and SU v_{ j } due to the impact of PUsâ€™ activities on the licensed uplink channels. As such, we formally devise a contact affinity metric to describe the stability of the contact between a pair of SUs. Without losing generality, the contact affinity metric A_{(i,â€‰j)} between SU v_{ i } and SU v_{ j } within time interval [t_{0},â€‰T] is formally expressed as:
3 Problem formulation
In this section, we intend to employ the differential game theoretic approach to formulate the PLPC problem in the physical layer and the HHCC problem in the transport layer. Clearly, the distributed strategy needs to be used to design the crosslayer optimization framework for congestion and power control due to the lack of centralized control and global information under an underlay CRANET scenario. As depicted in Fig. 3, each upstream SU will serve as power and rate controller in charge of joint optimized allocation of transmit power in the physical layer and data rate in the transport layer. Note that the change of power and rate will be continuous in time due to the fact that dynamic congestion and power control will be more realistic in a practical environment.
3.1 Perlink power control in the physical layer
Given the channel conditions, the capacity of successful transmission link between a pair of SUs is a nonconvex function of transmit power vector P. In fact, increasing the link capacity on one link may reduce the link capacities on other links because of the mutual interference caused by SUs [20]. Instead, each SU is expected to increase its transmit power to provide as much link capacity that perlink flow requires [21]. However, this adjustment of power will generate extra interference to other SUs. It is necessary to achieve an optimal perlink power allocation in the physical layer for upstream SUs to meet link capacity supply for all the flows. By letting the transmit power of upstream SU v_{ i } equals the maximum transmit power threshold, we can easily obtain the maximum transmit power vector \( \overline{\mathbf{P}}=\left\{{\overline{p}}_1,{\overline{p}}_2,\cdots, {\overline{p}}_N\right\} \). The transmission loss along link l_{(i,â€‰b)} from upstream SU v_{ i } to bottleneck SU v_{ b } on channel ch_{ Î¾ } is denoted by Î·_{(i,â€‰b)}. Due to the lineofsight wireless transmission environment with slowfading channel model, the transmission loss along link l_{(i,â€‰b)} is represented by Î·_{(i,â€‰b)}â€‰=â€‰[c/(4Ï€f_{ Î¾ }â€‰â‹…â€‰â€–Ï‘_{ i }(t)â€‰âˆ’â€‰Ï‘_{ b }(t)â€–)]^{2}, where f_{ Î¾ } is the carrier frequency operating on channel ch_{ Î¾ } and c is the speed of light. Therefore, the maximum transmit power threshold \( {\overline{p}}_i \) of upstream SU v_{ i } along link l_{(i,â€‰b)} from upstream SU v_{ i } to bottleneck SU v_{ b } can be formulated as:
where \( {\mathrm{p}}_{\left(i,b\right)}^{\mathrm{ref}} \) is the received reference power at bottleneck SU v_{ b } along link l_{(i,â€‰b)}. Given the maximum transmit power threshold \( {\overline{p}}_i \), the capacity of link l_{(i,â€‰b)} from upstream SU v_{ i } to bottleneck SU v_{ b } can be denoted by \( {C}_{\left(i,b\right)}^{\ast}\left(\overline{\mathbf{P}}\right) \). Owing to maximum transmit power threshold, the value of power reduction for upstream SU v_{ i } is equal to \( {\overline{p}}_i{p}_i(t) \). Recall that bitsperJoule capacity usually serves as a metric to measure the energy efficiency of a communication system [40]. Considering the impact of power reduction on energy efficiency with a link capacity constraint, energy efficiency for power reduction is formally defined as the power reduction achieved per capacity obtained under the maximum transmit power threshold. Thus, we plan to use the energy efficiency for power reduction to characterize a pricing factor of energypercapacity, aiming to design the revenue function of power reduction for upstream SU. Given the maximum transmit power vector \( \overline{\mathbf{P}} \), the pricing factor Î¦_{ i }(t) of energypercapacity of upstream SU v_{ i } at time t can be formally defined as:
Revisiting the pricing factor of energypercapacity of upstream SU v_{ i }, we define the revenue function of power reduction for upstream SU v_{ i } at time t by attaining the product of the pricing factor together with power reduction value, i.e., \( {\varPhi}_i(t)\left({\overline{p}}_i{p}_i(t)\right) \). Let Ï‰ denote the pricing factor announced by upstream SU v_{ i } to measure the cost of the amount of aggregate power interference to PUs. The amount of aggregate power interference to PUs is denoted by I(t). The cost function of aggregate power interference to PUs for upstream SU v_{ i } at time t is given by Ï‰I(t). Note that I(t) is a dynamic variable influenced by transmit power p_{ i }(t) of upstream SU v_{ i } and instant level I(t) within time interval [t_{0},â€‰T]. Thus, the aggregate power interference I(t) can be characterized as a linear differential equation given as:
where Î³â€‰>â€‰0 is a penalty factor of the amount of aggregate power interference and \( {I}_{t_0} \) is an initial aggregate power interference to PUs at time t_{0}. Therefore, based on both revenue and cost functions as mentioned, the utility function \( {U}_1^i(t) \) of upstream SU v_{ i } at time t is given as:
Note that utility function \( {U}_1^i(t) \) is a continuously differentiable function of p_{ i }(t) and I(t). We can find that utility function in (14) mainly relies on pricing factor of energypercapacity in that the marginal effect on utility function stems from aggregate power interference. As shown in Fig. 3, we figuratively define pricing factor Î¦_{ i }(t) as a shadow price â„µ_{ i } which is a function of transmit power of upstream SU v_{ i }. Our optimization objective is to maximize utility function \( {U}_1^i(t) \) by choosing optimal transmit power \( {p}_i^{\#}(t) \) of upstream SU v_{ i } according to Î¦_{ i }(t) and Ï‰:
where a is a discount factor, for 0â€‰<â€‰aâ€‰<â€‰1. Note that discount factor a is an exponential factor between 0 and 1 by which the future utility must be multiplied in order to obtain the present value under the underlying structure of differential game. Hence, utility function \( {U}_1^i(t) \) has to be discounted by the factor \( {e}^{a\left(t{t}_0\right)} \). Formally, the PLPC problem in the physical layer is formulated as a differential game model Î“_{ PLPC }:
where

Player set \( \mathcal{N} \): \( \mathcal{N}=\left\{1,2,\cdots, N\right\} \) is the set of all the upstream SUs in the PLPC problem as power controllers playing the game. Note that upstream SU v_{ i } stands for the ith player which is a rational policy maker and acts throughout time interval [t_{0},â€‰T].

Set of strategies \( {\left\{{p}_i(t)\right\}}_{i\in \mathcal{N}} \): The strategy of the ith player refers to its instant transmit power limited by the maximum transmit power threshold, i.e., \( {p}_i(t)\in \left[0,{\overline{p}}_i\right) \).

State variable I(t): The state variable of the ith player corresponds to the amount of aggregate power interference to PUs.

Set of utility functions \( {\left\{{U}_1^i(t)\right\}}_{i\in \mathcal{N}} \): \( {U}_1^i(t) \) is the utility function of the ith player. The objective of the ith player is to maximize its utility function by rationally selecting optimal strategy \( {p}_i^{\#}(t) \) and optimal state I^{#}(t).
3.2 Hopbyhop congestion control in the transport layer
Under the scenario that multiple flows from upstream SU2, SU6, and SU8 via single hop converge at bottleneck SU3 in Fig. 2, bottleneck SU3 is a little more inclined to be a congested SU when offered data load exceeds available capacity of SU3 due to buffer overflow. The amount of data packets with elastic traffic in the buffer of bottleneck SU v_{ b } at time t is denoted by Ï†_{ b }(t). Given time t, t^{â€²}â€‰âˆˆâ€‰[t_{0},â€‰T] and tâ€‰<â€‰t^{â€²}, the amount of data packets Ï†_{ b }(t) of bottleneck SU v_{ b } within the time interval [t,â€‰t^{â€²}] satisfies the following iterative equation:
where Îº_{ b } is the buffer size of bottleneck SU v_{ b }, and \( {\varphi}_b^{t\to {t}^{\prime }}\left({t}^{\prime}\right) \) and \( {\varLambda}_b^{t\to {t}^{\prime }}\left({t}^{\prime}\right) \) are the amount of data packets accumulated in the buffer of bottleneck SU v_{ b } and the amount of data packets that could be delivered successfully to downstream SUs, respectively. Considering the constraint of the saturation value \( {\widehat{L}}_b \) of the buffer of bottleneck SU v_{ b }, we have the buffer constraint \( {\varphi}_b(t)\le {\widehat{L}}_b \) to guarantee that bottleneck SU v_{ b } will not become the real congested SU.
In the endtoend congestion control, the congestion detection information is piggybacked over data packets to the destination and then sent to the source through the acknowledgement packet from the destination. However, to feed the congestion detection information back to the upstream SU2, SU6, and SU8, bottleneck SU3 will generate a backpressure signal to notify that the congestion occurs as shown in Fig. 2. Compared with the endtoend mechanism, the backpressure signal is directly sent back to the corresponding upstream SUs from the bottleneck SU3 via single hop. Instead of passing the congestion detection information sent to the source in an endtoend approach, the idea of hopbyhop congestion control policy in this paper is that upstream SU v_{ i } directly adjusts the data rate r_{ i }(t) according to the backpressure signal of bottleneck SU v_{ b } when \( {\varphi}_b(t)>{\widehat{L}}_b \). Based on the radio transceiver equipped by each SU, the backpressure signal is assumed to be transmitted through a common control channel.
Given the received SINR of link l_{(i,â€‰b)}, we proceed to derive the required bandwidth of upstream SU v_{ i } for transmissions of data packets with elastic traffic. According to Shannonâ€™s capacity formula, the required bandwidth of upstream SU v_{ i } can be expressed by r_{ i }(t)/log_{2}(1â€‰+â€‰Ï‡â€‰â‹…â€‰SINR_{(i,â€‰b)}(P)). We further assume that bottleneck SU v_{ b } acts as a bidder and pays for upstream SU v_{ i } to accommodate the consumption of its network resources while regulating the data rate r_{ i }(t). We use x(t) to represent the congestion bid that bottleneck SU v_{ b } is willing to pay. We then characterize a bandwidthtobid ratio aiming to describe the efficiency of bid with regard to the required bandwidth, i.e., [r_{ i }(t)/log_{2}(1â€‰+â€‰Ï‡â€‰â‹…â€‰SINR_{(i,â€‰b)}(P))]/x(t). Hence, the cost function C_{ i }(t) of upstream SU v_{ i } at time t can be defined as:
We also use the bandwidthtobid ratio to figuratively express the shadow price â„µ_{ i } which is a function of the data rate of upstream SU v_{ i }. Note that the shadow price â„µ_{ i } depends on an optimal perlink power allocation in the PLPC problem due to the constraint of the received SINR of link l_{(i,â€‰b)} in bandwidthtobid ratio. In this way, we turn our attention to the crosslayer coordination mechanism between capacity supply by power controller and rate demand by rate controller based on shadow prices â„µ_{ i } and â„µ_{ i } as shown in Fig. 3. By taking into account the stability of the contact between a pair of SUs, we conclude that A_{(i,â€‰b)}x(t) is the accumulated revenue obtained by upstream SU v_{ i } that bottleneck SU v_{ b } needs to pay. We also remark that the congestion bid x(t) is a dynamic variable influenced by rate r_{ i }(t) as well as by instant level of x(t) within time interval [t_{0},â€‰T]. Accordingly, the congestion bid x(t) can be formulated as a linear differential equation:
where \( {x}_{t_0} \) is an initial congestion bid that bottleneck SU v_{ b } needs to pay at time t_{0} and Ï… is an average bid per rate, which is assumed to be a unit value for all the upstream SUs, i.e., Ï…â€‰=â€‰1. Formally, based on both revenue and cost functions as stated before, the utility function \( {U}_2^i(t) \) of upstream SU v_{ i } at time t can be expressed as:
Noticing that utility function \( {U}_2^i(t) \) is also a continuously differentiable function of r_{ i }(t) and x(t). Our optimization objective is to maximize utility function \( {U}_2^i(t) \) by choosing optimal data rate \( {r}_i^{\#}(t) \) of upstream SU v_{ i } while satisfying the buffer constraint at the same time:
where Ï„ is a discount factor, for 0â€‰<â€‰Ï„â€‰<â€‰1. Similarly, \( {U}_2^i(t) \) will also be discounted by the factor \( {e}^{\tau \left(t{t}_0\right)} \). Correspondingly, the HHCC problem in the transport layer can be also defined as a differential game model Î“_{ HHCC }:
where

Player set \( \mathcal{N} \): \( \mathcal{N}=\left\{1,2,\cdots, N\right\} \) is the set of all the upstream SUs in the HHCC problem as rate controllers playing the game. Upstream SU v_{ i } is also known as the ith player which is a rational policy maker and act throughout time interval [t_{0},â€‰T].

Set of strategies \( {\left\{{r}_i(t)\right\}}_{i\in \mathcal{N}} \): The strategy of the ith player corresponds to its instant data rate r_{ i }(t).

State variable x(t): The state variable of the ith player refers to the congestion bid that bottleneck SU v_{ b } is willing to pay.

Set of utility functions \( {\left\{{U}_2^i(t)\right\}}_{i\in \mathcal{N}} \): \( {U}_2^i(t) \) is the utility function of the ith player. The objective of the ith player is to maximize its utility function by rationally choosing optimal strategy \( {r}_i^{\#}(t) \) and optimal state x^{#}(t).
4 Optimal solution and distributed implementation
Conventionally, upstream SUs as players of the game are expected to act cooperatively and maximize their joint utility functions with fairness for players by constituting the collaborative coalition. As a result, the global optimization of transmit power and data rate will be attained through cooperation among players with group rationality, which has been recently reported in a cooperative bargaining game [41]. However, each upstream SU is unwilling to jointly adjust the power and rate because of the selfish behavior in forwarding data packets. This is a natural idea due to the fact that the transmissions lead to the consumption of network resources of upstream SUs, such as energy and spectrum. Therefore, the crosslayer optimization framework for congestion and power control will be restricted to noncooperation scenario. In the noncooperative differential game models Î“_{ PLPC } and Î“_{ HHCC }, the ith player competes to maximize the present value of its utility function derived over time interval [t_{0},â€‰T]. For mathematical tractability, we define the starting time of the differential game models Î“_{ PLPC } and Î“_{ HHCC } as t_{0}â€‰=â€‰0 hereinafter, but the results can be easily extended to more general cases.
4.1 Optimal solution to Î“ _{ PLPC }
For the noncooperation scenario, we formulate a dynamic optimization problem â„™1 to derive the optimal solution to the noncooperative differential game model Î“_{ PLPC } by taking into account the utility function maximization problem coupled with the linear differential equation constraint in (13):
We aim at deriving an optimal solution to â„™1 by employing the theory of dynamic programming developed by Bellman [42]. Remark that the optimal solution is also viewed as a Nash equilibrium solution to â„™1 if all the players play noncooperatively. Here, we relax the terminal time of Î“_{ PLPC } to explore when T approaches âˆž (i.e., Tâ€‰â†’â€‰âˆž) as an infinite time horizon. It is more realistic to obtain the longterm optimal power allocation for upstream SUs due to spectrum underlay strategy with cellular primary network. We use \( {p}_i^{\#}(t) \) to represent the optimal solution to â„™1 and assume that there exists a continuously differentiable function V^{i}(p_{ i },â€‰I) satisfying the following partial differential equation:
Theorem 1
A vector of optimal transmit power \( {\mathbf{P}}^{\#}=\left\{{p}_1^{\#}(t),{p}_2^{\#}(t),\cdots, {p}_N^{\#}(t)\right\} \) of upstream SUs constitutes a Nash equilibrium solution to â„™1 if and only if the optimal transmit power \( {p}_i^{\#}(t) \) of the ith player and the continuously differentiable function V^{i}(p_{ i },â€‰I) can be formulated as follows:
Proof: See Appendix 1. â– .
From Theorem 1, we can observe that the existence and uniqueness of the Nash equilibrium point to â„™1 are guaranteed under the constraint of analytical solution in (25) and (26). It is also revealed that the optimal transmit power \( {p}_i^{\#}(t) \) has been characterized by a fixed and unique value in (25). Evidently, Theorem 1 mathematically ensures the convergence of \( {p}_i^{\#}(t) \) to a Nash equilibrium point. The key point to derive the optimal solution to the differential game model Î“_{ PLPC } is illustrated with a block diagram shown in Fig. 4a.
Proposition 1
For the given largescale slowfading channel model, by letting G_{1}â€‰=â€‰Ï–_{PBS}/(10^{6}g_{0})^{N}, the optimal transmit power \( {p}_i^{\#}(t) \) of the ith player should follow the interference power constraint:
Proof: See Appendix 2. â– .
Note that the optimal transmit power \( {p}_i^{\#}(t) \) of the ith player is fully constrained by the Euclidean distance between upstream SU v_{ i } and bottleneck SU v_{ b } under the given channel model. Substituting for \( {p}_i^{\#}(t) \) with its expression from Theorem 1 and taking into account the previous expression of shadow price â„µ_{ i }, we can easily rewrite â„µ_{ i } as:
Apparently, shadow price â„µ_{ i } tends to be a constant value for all the upstream SUs. Although (25) and (26) offer an analytical solution to â„™1, it still remains to design an algorithm to ensure fast convergence of the update of optimal transmit power. Therefore, we devise a distributed optimal transmit power update (OTPU) strategy given in Algorithm 2 to update the optimal transmit power vector P^{#} for upstream SUs. Similar to [43], shadow price â„µ_{ i } in Algorithm 2 needs to carefully be chosen to ensure fast convergence of the update of instant transmit power \( {p}_i^{\odot }(t) \). It is also noted that the update of instant transmit power \( {p}_i^{\odot }(t) \) for upstream SU v_{ i } can be made locally according to its optimal transmit power \( {p}_i^{\#}(t) \) along with interference power constraint.
4.2 Optimal solution to Î“ _{ HHCC }
For notational simplicity, we begin by defining a notation B_{(i,â€‰b)}â€‰â‰œâ€‰â€‰âˆ’â€‰D_{ i }/log_{2}(1â€‰+â€‰Ï‡â€‰â‹…â€‰SINR_{(i,â€‰b)}(P)). For the noncooperation scenario, we formulate a dynamic optimization problem â„™2 to derive the optimal solution to the noncooperative differential game model Î“_{ HHCC } by taking into account both the utility function maximization problem and the linear differential equation constraint in (19):
We turn to take advantage of the theory of maximum principle developed by Pontryagin [42] to derive an optimal solution or a Nash equilibrium solution to â„™2. We further use \( {r}_i^{\#}(t) \) to represent the optimal solution to â„™2 and assume that there exists a continuously differentiable function W^{i}(r_{ i },â€‰x) satisfying the partial differential equation as follows:
For tractability, we introduce two extra introduced auxiliary variables Y_{ i }(t) and J_{ i }(t) to characterize W^{i}(r_{ i },â€‰x). Specifically, we define W^{i}(r_{ i },â€‰x)â€‰â‰œâ€‰(Y_{ i }(t)x(t)â€‰+â€‰J_{ i }(t))e^{âˆ’Ï„t}.
Theorem 2
A vector of optimal data rate \( {\mathbf{R}}^{\#}=\left\{{r}_1^{\#}(t),{r}_2^{\#}(t),\cdots, {r}_N^{\#}(t)\right\} \) of upstream SUs constitutes a Nash equilibrium solution to â„™2 if and only if the optimal data rate \( {r}_i^{\#}(t) \) of the ith player can be expressed as:
where Y_{ i }(t) and J_{ i }(t) satisfy the following differential equations:
Proof: See Appendix 3. â– .
For notational simplicity, we set \( {\varOmega}_{\left(i,b\right)}=1/\left(4{B}_{\left(i,b\right)}\right)+0.5{\sum}_{j\in \mathcal{N}\backslash i}1/{B}_{\left(j,b\right)} \) and \( \varepsilon =\sqrt{{\left(\tau 1\right)}^2+4{\varOmega}_{\left(i,b\right)}{A}_{\left(i,b\right)}} \), for 4Î©_{(i,â€‰b)}A_{(i,â€‰b)}â€‰+â€‰(Ï„â€‰âˆ’â€‰1)^{2}â€‰>â€‰0. We also denote G_{2} as a constant number. Substituting Î©_{(i,â€‰b)} into Îµ, we can rewrite Îµ as follows:
Proposition 2
The auxiliary variable Y_{ i }(t) in the Nash equilibrium solution \( {r}_i^{\#}(t) \) to â„™2 can be further given as:
Proof: See Appendix 4. â– .
Combining Y_{ i }(t) in (35) and Îµ in (34) yields the expression for Y_{ i }(t) as:
Substituting B_{(i,â€‰b)}â€‰â‰œâ€‰â€‰âˆ’â€‰D_{ i }/log_{2}(1â€‰+â€‰Ï‡â€‰â‹…â€‰SINR_{(i,â€‰b)}(P)) into (36), we can rewrite Y_{ i }(t) as follows:
Let G_{3} be a constant number. Based on (31) and (29), the optimal data rate \( {r}_i^{\#}(t) \) and the optimal state variable x^{#}(t) associated with \( {r}_i^{\#}(t) \) can be described as:
From (38), we can see that the optimal data rate \( {r}_i^{\#}(t) \) is determined by both B_{(i,â€‰b)} and auxiliary variable Y_{ i }(t) under the received SINR SINR_{(i,â€‰b)}(P) of link l_{(i,â€‰b)}. Unfortunately, it is difficult to directly obtain the relationship between SINR_{(i,â€‰b)}(P) and \( {r}_i^{\#}(t) \) through an analytical derivation because Y_{ i }(t) cannot be further simplified into a concise structure. Thereby, given the channel gain G_{(i,â€‰b)} of link l_{(i,â€‰b)} under the largescale slowfading channel model, we use the numerical simulations to validate the effectiveness of the optimal data rate \( {r}_i^{\#}(t) \) of upstream SU v_{ i }. We further remark that Theorem 2 and Proposition 2 characterize the existence of the Nash equilibrium point to â„™2. It should be also admitted that the optimal data rate \( {r}_i^{\#}(t) \) has been formulated as a fixed and unique value in (38) by using auxiliary variable Y_{ i }(t) in (37). Correspondingly, Theorem 2 mathematically ensures the convergence of \( {r}_i^{\#}(t) \) to a Nash equilibrium point. The key point to derive the optimal solution to the differential game model Î“_{ HHCC } is also illustrated with a block diagram depicted in Fig. 4b. With the help of vectors R^{#} and P^{#}, shadow price â„µ_{ i } can be calculated as:
Proposition 3
For the given vector R^{#} and vector P^{#}, a strict lower bound of shadow price â„µ_{ i } can be approximately calculated as follows:
Proof: See Appendix 5. â– .
Let \( {\mathrm{r}}_{\left(i,b\right)}^{\mathrm{base}} \) represent the baseline rate along link l_{(i,â€‰b)} from upstream SU v_{ i } to bottleneck SU v_{ b }. To guarantee the constraint of the saturation value of the buffer of bottleneck SU v_{ b }, we design a distributed algorithm to obtain the optimal data rate update (ODRU) for upstream SUs as summarized in Algorithm 3, where Râ€‰=â€‰{r_{1}(t),â€‰r_{2}(t),â€‰â‹¯,â€‰r_{ N }(t)} denotes the data rate vector of N upstream SUs at time t. A simple yet effective way to locally adjust the instant data rate \( {r}_i^{\odot }(t) \) of upstream SU v_{ i } is to employ the optimal data rate \( {r}_i^{\#}(t) \) under the condition of buffer constraint \( {\varphi}_b(t)\le {\widehat{L}}_b \). Also, baseline rate \( {\mathrm{r}}_{\left(i,b\right)}^{\mathrm{base}} \) in Algorithm 3 should be carefully chosen to ensure the effectiveness of instant data rate.
4.3 Distributed implementation
So far, we have devised Algorithms 2 and 3 to satisfy the interference power constraint along with buffer constraint by locally adjusting the optimal transmit power and the optimal data rate, respectively. In what follows, we would like to describe the distributed implementation strategy to realize the crosslayer optimization framework for congestion and power control by jointly optimizing PLPCHHCC simultaneously. In conclusion, the crosslayer optimization scheme for joint PLPCHHCC design is implemented in a distributed manner as follows:
4.3.1 Shadow price
Update shadow price â„µ_{ i } using (28) and shadow price â„µ_{ i } using (40), respectively.
4.3.2 Power controller in the physical layer
For each upstream SU, we initially assign optimal transmit power \( {p}_i^{\#}(t) \) using (25) to update instant transmit power \( {p}_i^{\odot }(t) \) at power controller. Due to the interference power constraint in (6) to protect PUs, \( {p}_i^{\odot }(t) \) should satisfy the following distributed powerupdate function when \( {\sum}_{i\in \mathcal{N}}{p}_i(t){G}_{\left(i,b\right)}>{\varpi}_{\mathrm{PBS}} \) via OTPU algorithm:
4.3.3 Rate controller in the transport layer
For each upstream SU, we also initially assign optimal data rate \( {r}_i^{\#}(t) \) using (38) to update instant data rate \( {r}_i^{\odot }(t) \) at rate controller. Owing to the buffer constraint \( {\varphi}_b(t)\le {\widehat{L}}_b \) to guarantee that bottleneck SU v_{ b } will not become congested, \( {r}_i^{\odot }(t) \) should be subject to the following distributed rate update function when \( {\varphi}_b(t)>{\widehat{L}}_b \) via ODRU algorithm:
4.3.4 Crosslayer coordination mechanism
With the aid of the updated power \( {p}_i^{\odot }(t) \), the link capacity supply C_{(i,â€‰b)}(P^{âŠ™}) with respect to each upstream SU is regulated by power controller by using (4) as shown in Fig. 3. The rate demand depends on instant data rate \( {r}_i^{\odot }(t) \) regulated by rate controller, which is nonlinear function of instant transmit power vector P^{âŠ™} according to (37) and (38).
5 Simulation results
5.1 Simulation settings
The simulation scenario is shown in Fig. 5, which consists of one bottleneck SU and Nâ€‰=â€‰6 randomly distributed upstream SUs transmitting data packets with elastic traffic towards bottleneck SU within a range of 100 mâ€‰Ã—â€‰100 m. The scenario is easily extendable to a general case which involves much more randomly distributed upstream SUs. Our simulations pay more attention to evaluate the effect of the crosslayer optimization framework for congestion and power control on both optimal data rate and optimal transmit power of six different perlink flows. Different from the channel with carrier frequency of 890.4 MHz used by SUs, we assume that active PUs in simulation scenario occupy other uplink channels from set â„‹.This can make possible the successful transmissions of data packets from upstream SUs to bottleneck SU. The probability Î´_{ i } of uplink channels used by each SU is assumed to be 0.65. The channel gain of link l_{(i,â€‰b)} is defined with largescale slowfading model, given by G_{(i,â€‰b)}â€‰=â€‰100g_{0}â€–Ï‘_{ i }(t)â€‰âˆ’â€‰Ï‘_{ b }(t)â€–^{âˆ’4} [34], where the reference channel gain g_{0} is set to 9.7â€‰Ã—â€‰10^{âˆ’4} [44]. We adopt a processing gain factor Ï‡â€‰=â€‰â€‰âˆ’â€‰1.5/log_{2}(5BER) where the target bit error rate is BERâ€‰=â€‰10^{âˆ’3} for multiple quadrature amplitude modulation with a symbol period of Î¤_{ s }â€‰=â€‰52.5â€‚Î¼s. The thermal noise power at bottleneck SU and interference caused by PBS are assumed to be n_{0}â€‰=â€‰â€‰âˆ’â€‰50â€‚dBm and I_{ P }â€‰=â€‰10â€‚dBm, respectively. In addition, the receiving reference power at bottleneck SU is chosen as \( {p}_{\left(i,b\right)}^{ref}=37\kern0.5em \mathrm{dBm} \) for each upstream SU. With regard to the elastic traffic modeled by Poisson process [32], the flow arrival intensity is set to a normalized value Î»_{ i }â€‰=â€‰125â€‚bps for each upstream SU, and the mean of flow size is given as \( \mathbb{E}\left[{\lambda}_i\right]=2\kern0.5em \mathrm{Mbits} \). Under our differential game models Î“_{ PLPC } and Î“_{ HHCC }, we choose the pricing factor Ï‰â€‰=â€‰22, the penalty factor Î³â€‰=â€‰0.7, two constant numbers G_{2}â€‰=â€‰5.5 and G_{3}â€‰=â€‰360.
Due to the lack of empirical data about available contact or all encounters between a pair of SUs, to evaluate the uncertainty of contact \( {\mathcal{C}}^{\left(i,b\right)} \) along link l_{(i,â€‰b)}, we assume that the minimum and maximum values of encounter duration for all encounters within contact \( {\mathcal{C}}^{\left(i,b\right)} \) is offered in Fig. 6. Note that Flow i in Fig. 6 corresponds to contact \( {\mathcal{C}}^{\left(i,b\right)} \) along link l_{(i,â€‰b)}, for iâ€‰=â€‰1, 2, â‹¯, 6. We set the number of subintervals in Algorithm 1 to Mâ€‰=â€‰8 for all upstream SUs in that the number of encounters K seems to be a lower value because of the short time interval in the simulations. In fact, game time or time interval is just set to [0,â€‰5]â€‚s in the following simulations. Under this setting, the probability distribution Ï’^{(i,â€‰b)} of contact \( {\mathcal{C}}^{\left(i,b\right)} \) derived from Algorithm 1 is assumed to comply with the contact distribution as provided by Fig. 7.
The proposed OTPU algorithm for the PLPC problem is compared with the existing classical distributed constrained power control (DCPC) algorithm in [45]. The DCPC algorithm is a SINRconstrained power control algorithm which distributively and iteratively searches for transmit power updated from the Ï‚â€‚th iteration to the (Ï‚â€‰+â€‰1)â€‚th iteration. Let \( {\mathrm{SINR}}_i^{tar} \) denote the target SINR for upstream SU v_{ i } to maintain a certain QoS requirement. In the simulations, the target SINR can be set as \( {\mathrm{SINR}}_i^{tar}=8\kern0.5em dB \). Therefore, the iterative function of transmit power update in the DCPC algorithm with number of iteration Ï‚â€‰=â€‰0, 1, 2, â‹¯ is specifically given as [45]:
6 Results
6.1 Performance of OTPU algorithm
Figure 8 shows the optimal transmit power comparison for six perlink flows between the OTPU algorithm with the evolution of discount factor a and the DCPC algorithm with Ï‚â€‰=â€‰300 iterations. This figure clearly depicts that an increased discount factor from 0.1 to 0.9 will increase the optimal transmit power of each flow under the OTPU algorithm. Apparently, this is a direct consequence of discount factor a on the optimal transmit power according to (25). However, it is observed that the optimal transmit power of each flow via the DCPC algorithm presents a fixed constant value. This is due to the fact that the optimal transmit power of each flow via the DCPC algorithm converges to an expected equilibrium point after 300 iterations. It is worth mentioning that Theorem 1 mathematically makes the optimal transmit power of each upstream SU converge to a Nash equilibrium point distributively. From the results, we can also see that the optimal transmit power of each flow by the OTPU algorithm is obviously lower than that of the DCPC algorithm. This can be explained by the fact that DCPC algorithm gives rise to more power consumption for maintaining a certain SINR for each upstream SU. However, the optimal transmit power of each flow based on the OTPU algorithm mainly depends upon the maximum transmit power threshold of upstream SU. On the other hand, the instant power level can be further reduced via the change of discount factor a.
6.2 The impact of discount factor on optimal transmit power
Figure 9 illustrates the optimal transmit power comparison for six perlink flows via the OTPU algorithm under different discount factors. It is noted that the total interference caused by six upstream SUs satisfies the interference temperature limit Ï–_{PBS}â€‰=â€‰â€‰âˆ’â€‰10â€‚dBm according to the constraint of (6). As the discount factor increases, the optimal transmit power of six flows obtained by the OTPU algorithm will raise as well. As expected, the optimal transmit power of flow 6 can achieve the minimum transmit power level with approximately 50 mW, and the optimal transmit power of flow 2 can obtain higher transmit power level with the maximum value nearly 570 mW. However, the increasing rate of the optimal transmit power in regard to flows 6, 3, and 1 flattens out after discount factor aâ€‰=â€‰0.6. The reason is as follows. Firstly, based on (25), the discount factor effect is in direct proportion to the optimal transmit power. One the other hand, with even higher Euclidean distance between upstream SU and bottleneck SU, the maximum transmit power threshold will increase as well according to (11). Under the simulation scenario, the higher Euclidean distance of the successful transmission link of flow 2 leads to a higher transmit power level accordingly.
6.3 Optimal data rate performance of ODRU algorithm
Figure 10 exhibits the evolution of the optimal data rate for six perlink flows obtained by the ODRU algorithm versus game time tâ€‰âˆˆâ€‰[0,â€‰5]s under the condition of discount factor Ï„â€‰=â€‰0.2 and saturation value \( {\widehat{L}}_b=1\kern0.5em Mbps \). From the results, we can see the optimal data rate for six flows gradually increase with the growth of game time t. Meanwhile, the optimal data rate levels of six flows are very close from 0 to 4 s. When game time t is more than 4 s, the gaps among the optimal data rate levels will be enlarged. This demonstrates that the optimal data rate has large values during the game time of the end interval of the game. Under discount factor Ï„â€‰=â€‰0.2, the optimal data rate value of flow 2 is much larger than those of other flows with maximum value of 300 kbps, and the optimal rate of flow 6 has the lowest level within 50 kbps. It can also be observed that the optimal data rate of flow 2 yields significant performance gains than other flows under the condition of the fixed discount factor. According to saturation value \( {\widehat{L}}_b=1\kern0.5em Mbps \), we can observe that the total data rate generated by six upstream SUs is subject to the buffer constraint \( {\varphi}_b(t)\le {\widehat{L}}_b \) such that the instant data rate levels should not be adjusted through the ODRU algorithm.
Figure 11 depicts the optimal data rate update comparison for six perlink flows with the aid of the ODRU algorithm on the condition of discount factor Ï„â€‰=â€‰0.2 and saturation value \( {\widehat{L}}_b=490\kern0.5em kbps \). It is implicitly revealed that the total data rate caused by six upstream SUs fail to guarantee the buffer constraint \( {\varphi}_b(t)\le {\widehat{L}}_b \) such that the instant data rate levels must be updated according to the ODRU algorithm. Hence, the evolution of the optimal data rate levels of six upstream SUs will enter the rate update zone (i.e., shadow area in Fig. 11) when the constraint \( {\varphi}_b(t)>{\widehat{L}}_b \). From the results, we can see that the large values of the optimal data rate have been considerably dwindled according to the distributed rate update function in (41) when instant game time tâ€‰=â€‰4.3â€‚s in order to meet the buffer constraint of bottleneck SU.
6.4 The impact of discount factor on optimal data rate
Figure 12 displays the evolution of the optimal data rate for six perlink flows via the ODRU algorithm versus discount factor Ï„ under the condition of two fixed instant game time point t (i.e., tâ€‰=â€‰3 and tâ€‰=â€‰4) and saturation value \( {\widehat{L}}_b=950\kern0.5em \mathrm{kbps} \). From the results, we can see the total data rate generated by six upstream SUs accommodates for the buffer constraint \( {\varphi}_b(t)\le {\widehat{L}}_b \). It can be also observed that as the discount factor increases from 0.1 to 0.9, the optimal data rate of six flows obtained by the ODRU algorithm will decrease accordingly. The reason for this is that the utility function of each upstream SU must be discounted by the factor e^{âˆ’Ï„t} at time t under the differential game structure Î“_{ HHCC }. As we expected, the optimal data rate of flow 6 can obtain the minimum rate level within approximately interval [6,â€‰26]â€‚kbps, and the optimal data rate of flow 2 can gain the maximum value of data rate with nearly interval [30,â€‰140]â€‚kbps. This can be explained by the fact that the higher Euclidean distance of the successful transmission link of flow 2 will result in a higher transmit power level accordingly. This result of higher transmit power level of flow 2 will lead to the more link capacity supply. It implies that the upstream SU has the enough link capacity supply to achieve higher data rate in the proposed crosslayer optimization framework. Essentially, this signifies the importance of crosslayer coordination mechanism on the coupling between rate demand regulated by rate controller and capacity supply regulated by power controller.
7 Conclusions
In this paper, a distributed crosslayer optimization framework for congestion and power control for CRANETs under predictable contact has been proposed. Particularly, we introduced a predictable contact model by achieving the probability distribution of contact between any pair of SUs, aiming to measure the uncertainty of contact. Also, an entropy paradigm was presented to characterize quantitatively the probability distribution of contact. We employed a differential game theoretic approach to formulate the PLPC problem and the HHCC problem, and obtained the optimal transmit power and the optimal data rate of upstream SUs via dynamic programming and maximum principle. To guarantee the interference power constraint for active PUs and the buffer constraint of bottleneck SU, we developed two distributed update algorithms to locally adjust optimal transmit power and optimal data rate of upstream SUs. Finally, we presented a distributed implementation strategy to construct the crosslayer optimization framework for congestion and power control by jointly optimizing PLPCHHCC simultaneously and validated its performance with simulations. What we have discussed in this paper is the portion of foundation for the crosslayer optimization framework in CRANETs. In the future work, a joint objective function to achieve congestion and power control will be considered. Moreover, it will be interesting and important to investigate a tradeoff parameter as a whole to reflect the benefits of the proposed framework.
Abbreviations
 BER:

Bit error rate
 CR:

Cognitive radio
 CRANET:

Cognitive radio ad hoc network
 CRN:

CR network
 DCPC:

Distributed constrained power control
 HHCC:

Hopbyhop congestion control
 ODRU:

Optimal data rate update
 OTPU:

Optimal transmit power update
 PBS:

Primary base station
 PLPC:

Perlink power control
 PU:

Primary user
 QoS:

Quality of service
 RTT:

Round trip time
 SINR:

Signaltointerferenceplusnoise ratio
 SU:

Secondary user
 TCP:

Transmission control protocol
 UDP:

User datagram protocol
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Acknowledgements
We are grateful to the anonymous reviewers for their valuable comments and suggestions that have improved the paper.
Funding
This work was supported in part by the National Natural Science Foundation of China under Grants 61402147, 61402001, and 61501406; the Research Program for Topnotch Young Talents in Higher Education Institutions of Hebei Province, China, under Grant BJ2017037; the Research and Development Program for Science and Technology of Handan, China, under Grant 1621203037; and the Natural Science Foundation of Hebei Province of China under Grant F2017402068.
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LZ conceived the idea of this work and wrote the paper. HX provided valuable insights for the scheme and reviewed the manuscript. LZ and FZ performed the simulations and revised the paper. All authors read and approved the final manuscript.
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Appendices
Appendix 1
1.1 Proof of Theorem 1
According to the dynamic optimization problem â„™1, performing the maximization operation of the right hand side of (24) with respect to p_{ i }(t) yields the following optimal solution:
Substituting \( {p}_i^{\#}(t) \) in (45) into (24), we obtain:
Upon solving the differential equation in (46), \( {V}^i\left({p}_i^{\#},I\right) \) can be easily shown to be equivalent to the following equation:
Thus, an optimal transmit power \( {p}_i^{\#}(t) \) which constitutes a Nash equilibrium solution to â„™1 is given by:
Then, we can obtain the expression of \( {p}_i^{\#}(t) \) and V^{i}(p_{ i },â€‰I) as given by Theorem 1.
Appendix 2
1.1 Proof of Proposition 1
By substituting the vector of optimal transmit power \( {\mathbf{P}}^{\#}=\left\{{p}_1^{\#}(t),{p}_2^{\#}(t),\cdots, {p}_N^{\#}(t)\right\} \) into the interference power constraint inequality in (6), we can obtain:
After taking the logarithm of both sides of (49), we have:
Through rearranging terms, we have:
By taking into account the largescale slowfading channel model to describe the wireless transmission environment, the channel gain of link from upstream SU v_{ i } to bottleneck SU v_{ b } can be formulated as [30]:
where g_{0} is a reference channel gain at a distance of 100 m [34]. We substitute G_{(i,â€‰b)} in (52) into (51) and then we have:
Thus, (53) can be rewritten as follows:
Through defining G_{1}â€‰=â€‰Ï–_{PBS}/(10^{6}g_{0})^{N}, we can easily have the solution of (27).
Appendix 3
1.1 Proof of Theorem 2
According to the dynamic optimization problem â„™2, by performing the maximization operation of the right hand side of (30) with respect to r_{ i }(t), we can obtain:
Substituting W^{i}(r_{ i },â€‰x)â€‰â‰œâ€‰(Y_{ i }(t)x(t)â€‰+â€‰J_{ i }(t))e^{âˆ’Ï„t} and \( {r}_i^{\#}(t) \) in (55) into (30), we have:
Hence, this completes the proof.
Appendix 4
1.1 Proof of Proposition 2
Owing to the symmetric form of Y_{ i }(t) and Y_{ j }(t) in (57), we can immediately denote (57) by Riccati equation:
As such, the form of (59) can be rearranged by differential equation as:
Recall that we define \( {\varOmega}_{\left(i,b\right)}=1/\left(4{B}_{\left(i,b\right)}\right)+0.5{\sum}_{j\in \mathcal{N}\backslash i}1/{B}_{\left(j,b\right)} \), for 4Î©_{(i,â€‰b)}A_{(i,â€‰b)}â€‰+â€‰(Ï„â€‰âˆ’â€‰1)^{2}â€‰>â€‰0. Next, we turn to present a nonzero integrating factor â„‘(Y_{ i }(t),â€‰t) that can make equation in (61) an exact form by multiplying it on both sides of (61). Here, we can easily obtain:
So, (63) multiplied by â„‘(Y_{ i }(t),â€‰t) is exact, and then we obtain:
When \( {\varOmega}_{\left(i,b\right)}{Y}_i^2(t)+\left(\tau 1\right){Y}_i(t)>{A}_{\left(i,b\right)} \), we can easily define \( \varepsilon =\sqrt{{\left(\tau 1\right)}^2+4{\varOmega}_{\left(i,b\right)}{A}_{\left(i,b\right)}} \). By integrating (64) and (65) with respect to Y_{ i }(t), we have:
We can easily obtain h(t)â€‰=â€‰t. Let â„‘(Y_{ i }(t),â€‰t)â€‰=â€‰G_{2}, where G_{2} is a constant number. Upon solving (66) as follows:
Solving the above equation in (68) with respect to Y_{ i }(t), yields the desired result in (35).
Appendix 5
1.1 Proof of Proposition 3
For vector P^{#}, it is clear that the received SINR of link l_{(i,â€‰b)} at bottleneck SU v_{ b } satisfies the following inequality when all the upstream SUs with the equal Euclidean distance to bottleneck SU v_{ b }:
We can also approximate logarithmic function log_{2}(â‹…) by:
Thus, shadow price â„µ_{ i } should be subject to a strict lower bound:
which coincides with (41).
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Zhang, L., Zhuo, F. & Xu, H. A crosslayer optimization framework for congestion and power control in cognitive radio ad hoc networks under predictable contact. J Wireless Com Network 2018, 57 (2018). https://doi.org/10.1186/s136380181065x
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DOI: https://doi.org/10.1186/s136380181065x