A cross-layer optimization framework for congestion and power control in cognitive radio ad hoc networks under predictable contact

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 infrastructure-based single-hop scenario, including spectrum sensing, spectrum access, and sharing techniques [3–5]. In addition, SUs can also form a multi-hop decentralized ad hoc network without the support of infrastructure. In a multi-hop 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 PUs’ 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 multi-hop 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, on-demand 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 contact-based 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 window-based or acknowledgement-triggered 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 P1900.4-based cognitive wireless system using User Datagram Protocol (UDP) and TCP transport protocols. Alternatively, to accommodate for the challenging multi-hop 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 large-scale 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 cross-layer 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 cross-layer coupling between capacity supply by power control and rate demand by rate control.

1.2 Related works

Congestion control in wireless multi-hop 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 rate-effective network utility maximization problem to meet with delay-constrained data traffic requirement. However, although all of the aforementioned studies consider some realistic constraints, they apply to the traditional wireless multi-hop 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 infrastructure-based 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 infrastructure-based CRN, other studies in [18, 27, 28] have been undertaken in multi-hop CRN scenarios. In [27], Song et al. proposed an end-to-end congestion control framework without the aid of common control channel by taking into account the non-uniform 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], Al-Ali et al. proposed an end-to-end equation-based TCP friendly rate control mechanism, which achieves rate adjustment by identifying network congestion. However, the end-to-end 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 hop-by-hop 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 cross-layer interaction information try to deal with congestion control problem in decentralized CRNs from a cross-layer design perspective. The objective of these schemes is to improve the overall network utility while protecting active PUs’ communications from excessive interference introduced by SUs. In [29], Cammarano et al. presented a distributed cross-layer framework for joint optimization of MAC, scheduling, routing, and congestion control in CRAHNs, by maximizing the throughput of a set of multi-hop end-to-end packet flows. However, similar to [18, 27], it is not clear how good the performance of the end-to-end rate control is compared under a wireless transmission environment with higher RTTs. In [30], Nguyen et al. proposed a cross-layer framework to jointly attain both congestion and power control in OFDM-based 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 trade-off 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 cross-layer performance.

1.3 Our approach and contributions

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 cross-layer 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. | ⋅ | and ‖⋅‖ 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.

2.1 Network model

Consider an underlay multi-hop 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 ℋ = {ch₁, ch₂, ⋯, ch _ϕ } where ϕ is the number of uplink channels. The uplink channel is either occupied by PUs or unoccupied. We employ the independent and identically distributed alternating ON-OFF 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 δ _i , for \( {v}_i\in \mathcal{V} \). Based on the aforementioned ON-OFF process, the occupancy probability of uplink channel ch _ξ by PUs is defined as α _ξ /(α _ξ  + β _ξ ) [32], where α _ξ is a probability that uplink channel ξ transits from OFF to ON state, and β _ξ is probability that uplink channel ξ transits from ON to OFF state, for ξ = 1, 2, ⋯, ϕ. It is assumed that SUs can determine the occupancy probability of uplink channels by PUs through a priori knowledge of PUs’ activities and local spectrum sensing. Owing to the mutually independent occupancy probability of uplink channel ch _ξ , the probability δ _i of uplink channels used by SU v _i can be expressed as:

$$ {\delta}_i=\prod \limits_{\xi =1}^{\phi}\left(1-\frac{\alpha_{\xi }}{\alpha_{\xi }+{\beta}_{\xi }}\right). $$

(1)

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 continuous-time operation is confined to a predefined time interval [t₀, T]. Use ϑ _i (t) and z(t) to denote the positions of SU v _i , PU z at time t ∈ [t₀, 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 _ξ at time t only if the Euclidean distance between SU v _i and SU v _j satisfies ‖ϑ _i (t) − ϑ _j (t)‖ ≤ R _T and when there is no any PU z on channel ch _ξ , i.e., ‖ϑ _i (t) − z(t)‖ > R _I and ‖ϑ _j (t) − z(t)‖ > R _I , for \( {v}_i,{v}_j\in \mathcal{V} \) and ξ = 1, 2, ⋯, ϕ. In this context, there exists a successful transmission link denoted by l_(i, j) from SU v _i to SU v _j on channel ch _ξ at time t. For the sake of conciseness, instant time t will be restricted to the time interval [t₀, 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 multi-hop 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 per-link flows with elastic traffic. We use the term per-link 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 cross-layer 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 N < 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, 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, b) arrives as a Poisson process of flow arrival intensity λ _i with a size drawn independently from a common distribution of mean \( \mathbb{E}\left[{\lambda}_i\right] \) [33]. When Ψ_(i, b) < 1, the transmission link load, denoted by Ψ_(i, b), induced by elastic traffic along link l_(i, b) is equal to [33]:

$$ {\varPsi}_{\left(i,b\right)}=\frac{\lambda_i\times \mathbb{E}\left[{\lambda}_i\right]}{C_{\left(i,b\right)}\left(\mathbf{P}\right)}, $$

(2)

where C_(i, b)(P) denotes the capacity of link l_(i, b), and P = {p₁(t), p₂(t), ⋯, 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]:

$$ {D}_i=\frac{\mathbb{E}\left[{\lambda}_i\right]}{C_{\left(i,b\right)}\left(\mathbf{P}\right)\times \left(1-{\varPsi}_{\left(i,b\right)}\right)}. $$

(3)

2.2 Predictable contact model

Based on Algorithm 1, it is obvious to find that coefficient M impacts the structure of the probability distribution ϒ^(i, j). Thereby, the entropy H(ϒ^(i, j)) will depend on the selection of coefficient M. Recall that the probability δ _i of uplink channels used by SU v _i determines the stability of successful transmission link l_(i, j) or even the contact \( {\mathcal{C}}^{\left(i,j\right)} \) between SU v _i and SU v _j due to the impact of PUs’ 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, j) between SU v _i and SU v _j within time interval [t₀, T] is formally expressed as:

$$ {A}_{\left(i,j\right)}=H\left({\varUpsilon}^{\left(i,j\right)}\right)\times {\delta}_i\times {\delta}_j. $$

(10)

3.1 Per-link power control in the physical layer

3.2 Hop-by-hop congestion control in the transport layer

In the end-to-end 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 end-to-end 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 end-to-end approach, the idea of hop-by-hop 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.

4.1 Optimal solution to Γ _PLPC

Proof: See Appendix 1. ■.

From Theorem 1, we can observe that the existence and uniqueness of the Nash equilibrium point to ℙ1 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 Γ _PLPC is illustrated with a block diagram shown in Fig. 4a.

Proof: See Appendix 2. ■.

Apparently, shadow price ℵ _i tends to be a constant value for all the upstream SUs. Although (25) and (26) offer an analytical solution to ℙ1, 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 ℵ _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 Γ _HHCC

For tractability, we introduce two extra introduced auxiliary variables Y _i (t) and J _i (t) to characterize W ⁱ (r _i , x). Specifically, we define W ⁱ (r _i , x) ≜ (Y _i (t)x(t) + J _i (t))e^−τt.

Proof: See Appendix 3. ■.

Proof: See Appendix 4. ■.

Proof: See Appendix 5. ■.

Let \( {\mathrm{r}}_{\left(i,b\right)}^{\mathrm{base}} \) represent the baseline rate along link l_(i, 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 R = {r₁(t), r₂(t), ⋯, 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 cross-layer optimization framework for congestion and power control by jointly optimizing PLPC-HHCC simultaneously. In conclusion, the cross-layer optimization scheme for joint PLPC-HHCC design is implemented in a distributed manner as follows:

5.1 Simulation settings

The simulation scenario is shown in Fig. 5, which consists of one bottleneck SU and N = 6 randomly distributed upstream SUs transmitting data packets with elastic traffic towards bottleneck SU within a range of 100 m × 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 cross-layer optimization framework for congestion and power control on both optimal data rate and optimal transmit power of six different per-link 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 ℋ.This can make possible the successful transmissions of data packets from upstream SUs to bottleneck SU. The probability δ _i of uplink channels used by each SU is assumed to be 0.65. The channel gain of link l_(i, b) is defined with large-scale slow-fading model, given by G_(i, b) = 100g₀‖ϑ _i (t) − ϑ _b (t)‖⁻⁴ [34], where the reference channel gain g₀ is set to 9.7 × 10⁻⁴ [44]. We adopt a processing gain factor χ =  − 1.5/log₂(5BER) where the target bit error rate is BER = 10⁻³ for multiple quadrature amplitude modulation with a symbol period of Τ _s  = 52.5 μs. The thermal noise power at bottleneck SU and interference caused by PBS are assumed to be n₀ =  − 50 dBm and I _P  = 10 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 λ _i  = 125 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 Γ _PLPC and Γ _HHCC , we choose the pricing factor ω = 22, the penalty factor γ = 0.7, two constant numbers G₂ = 5.5 and G₃ = 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, 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, b), for i = 1, 2, ⋯, 6. We set the number of subintervals in Algorithm 1 to M = 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, 5] s in the following simulations. Under this setting, the probability distribution ϒ^(i, 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.

6.1 Performance of OTPU algorithm

Figure 8 shows the optimal transmit power comparison for six per-link flows between the OTPU algorithm with the evolution of discount factor a and the DCPC algorithm with ς = 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 per-link 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 ϖ_PBS =  − 10 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 a = 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 per-link flows obtained by the ODRU algorithm versus game time t ∈ [0, 5]s under the condition of discount factor τ = 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 τ = 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 per-link flows with the aid of the ODRU algorithm on the condition of discount factor τ = 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 t = 4.3 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 per-link flows via the ODRU algorithm versus discount factor τ under the condition of two fixed instant game time point t (i.e., t = 3 and t = 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^−τt at time t under the differential game structure Γ _HHCC . As we expected, the optimal data rate of flow 6 can obtain the minimum rate level within approximately interval [6, 26] kbps, and the optimal data rate of flow 2 can gain the maximum value of data rate with nearly interval [30, 140] 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 cross-layer optimization framework. Essentially, this signifies the importance of cross-layer coordination mechanism on the coupling between rate demand regulated by rate controller and capacity supply regulated by power controller.