- Research
- Open Access
Power-efficient distributed resource allocation under goodput QoS constraints for heterogeneous networks
- Riccardo Andreotti^{1, 2}Email author,
- Paolo Del Fiorentino^{1},
- Filippo Giannetti^{1} and
- Vincenzo Lottici^{1}View ORCID ID profile
https://doi.org/10.1186/s13634-016-0428-x
© The Author(s) 2016
- Received: 1 April 2016
- Accepted: 23 November 2016
- Published: 1 December 2016
Abstract
This work proposes a distributed resource allocation (RA) algorithm for packet bit-interleaved coded OFDM transmissions in the uplink of heterogeneous networks (HetNets), characterized by small cells deployed over a macrocell area and sharing the same band. Every user allocates its transmission resources, i.e., bits per active subcarrier, coding rate, and power per subcarrier, to minimize the power consumption while both guaranteeing a target quality of service (QoS) and accounting for the interference inflicted by other users transmitting over the same band. The QoS consists of the number of information bits delivered in error-free packets per unit of time, or goodput (GP), estimated at the transmitter by resorting to an efficient effective SNR mapping technique. First, the RA problem is solved in the point-to-point case, thus deriving an approximate yet accurate closed-form expression for the power allocation (PA). Then, the interference-limited HetNet case is examined, where the RA problem is described as a non-cooperative game, providing a solution in terms of generalized Nash equilibrium. Thanks to the closed-form of the PA, the solution analysis is based on the best response concept. Hence, sufficient conditions for existence and uniqueness of the solution are analytically derived, along with a distributed algorithm capable of reaching the game equilibrium.
Keywords
- Power-efficiency
- Game theory
- Goodput
1 Introduction
Future wireless networks are required to offer extremely enhanced capabilities including very high achievable data rates, very low latency, ultra-high reliability, and the possibility to handle very high density of devices [1]. On the other hand, since this trend dramatically contributes to the pollution related to energy consumption [2], energy-efficient wireless communications emerged as a viable design concept to reduce the CO_{2} emission in the next years [3]. In addition, to manage the envisioned huge demand of traffic, a very promising solution is offered by the concept of heterogeneous network (HetNet), where small-cell (SC) networks, characterized by low-cost, low-power, and low-coverage base stations (BSs), i.e., small BSs (SBSs), are massively deployed over the macrocell (MC) areas. Their adoption allows offloading the traffic of the macro network, thus increasing the offered data rate and spectrum re-usage and providing, at the same time, a more power-efficient architecture thanks to their reduced coverage [4]. Since in HetNets SCs share the same frequency bands of the MC [5], one of the main technical effort to be made is the management of the interference, either between SC and MC users (cross-tier interference) or between users in different SCs (co-tier interference). Hence, the random deployment of SCs together with the heterogeneity of these networks naturally calls for decentralized resource allocation (RA) strategies [6], where every user, independently from the other ones, maximizes its performance taking into account interference experienced on its transmission band. Indeed, distributed RA strategies only require the knowledge of local channel state information (CSI), exchanged between each user and its associated BS, thereby avoiding the waste of energy associated with centralized algorithms based instead on a huge information exchange between the users and/or the network administrator [6, 7].
1.1 Related works
Most of the works on RA in HetNets focus on the improvement of the energy efficiency (EE) as in [8], where the authors study the RA problem optimizing the EE over the downlink of an orthogonal frequency division multiplexing (OFDM) cognitive radio (CR) HetNet. A joint subcarrier allocation (SA) and power allocation (PA) solution is provided to guarantee quality of service (QoS) requirements for both secondary and primary users in the cognitive HetNet. Moreover, the authors propose a low-complexity RA algorithm in presence of imperfect CSI. In [9], authors address a joint EE RA and interference alignment problem for downlink multiple-input multiple-output (MIMO) transmissions in HetNets. The EE maximization problem is solved in order to provide time-slot allocation, PA, and beamforming. Unlike [8, 9], paper [10] analyzes RA techniques with different levels of CSI for uplink orthogonal frequency-division multiple access (OFDMA) transmissions in HetNets with one macrocell and cognitive small cells. Here, the purpose of the RA is to maximize the weighted sum of instantaneous rates of all the users, providing a joint PA and SA. Article [11] discusses possible future developments for HetNet in fifth-generation (5G) communications, where massive MIMO and mmWave technologies may be included, emphasizing that RA solutions will once again be one of the most critical issues. Besides, in this work a distributed RA problem over interference channels is tackled.
The most suited framework to study this kind of problem is identified in the theory of non-cooperative games [12], as outlined in [13, 14] about the uplink power control (PC) problem for flat and frequency-selective channels, respectively. Since then, the literature on this topic has increased more and more. Worth to be cited are [15, 16], where the energy-efficient PC problem for wireless data and code division multiple access (CDMA) networks is addressed, respectively, [17, 18], wherein the NCG framework is exploited to tackle the distributed PA problem for EE maximization in OFDM channels and MIMO HetNets, respectively, whereas in [19] the issue of dynamic RA is investigated in the context of multi-user cognitive networks, by exploiting a NCG for signal waveform design combined with sparsity constraints. In [20–22], the equilibrium analysis of the NCG describing the rate maximization problem in multicarrier and MIMO channels is discussed. A minimum PA game over OFDM channels with rate constraints is proposed in [23], whereas the PA problem based on the EE maximization under rate constraints is studied for HetNets in [6]. A final interesting work is presented in [24], where the authors investigate the RA problem for device-to-device (D2D) uplink communications in MIMO cellular networks. The RA is evaluated by means of a non-cooperative game that provides a joint optimization of channel allocation, power control, and precoding of the D2D users, by maximizing the sum-rate of the cellular uplink network. The distributed RA solution in [24] is exploited in [25] for D2D communications in MIMO HetNets.
A common feature to all the above works is that they rely on the assumption of infinite-length Gaussian codebooks and thus employ the channel capacity as figure of merit. However, when dealing with real systems characterized by practical and finite-sized modulation and coding schemes and automatic repeat request (ARQ) mechanisms, such information theoretical performance metric may offer an unreliable picture of the actual link performance [26]. In this cases, a suitable performance measure is the number of information bits delivered in error-free packets per unit of time, named goodput (GP) for short [26, 27]. To our best knowledge, GP-based distributed RA strategies have been addressed so far in few works only, as in [28, 29]. In [28], a PC strategy for CDMA ad hoc networks is proposed, aiming at maximizing the GP to power ratio under transmission rate and power constraints, whereas in [29], a network utility maximization problem with GP flow variables under queues stability constraints is proposed for flat fading mobile ad-hoc networks.
1.2 Contributions
- 1)
As first step, the RA problem is analyzed for the point-to-point case (P2P), i.e., for a single transmitter receiver pair without interference. The optimal solution as well as an approximate yet accurate closed-form one, which is reminiscent of the water-filling solution for Gaussian signaling, is derived for the PA.
- 2)
The RA problem is then extended to the the interference-limited HetNet scenario. The PA problem is solved in distributed manner as a function of the BL vector and the coding rate, by modeling it as a NCG, where players, strategy, and the utility function correspond to users, PA vector, and power consumption, respectively. Due to the QoS constraints imposed on the GP level, the set of strategies of each user depends on the strategies of the other users. This is the basic reason why the solution is the generalized Nash equilibrium (GNE) [32]. To be more specific, the GNE is described by a set of fixed-point equations based on the best response (BR) strategy of each player, wherein the BR (depending on the target EGP level) is obtained by capitalizing on the water-filling-like solution to the PA. Sufficient conditions for the feasibility of the problem, as well as for the existence and uniqueness of the GNE, are analytically derived. Additionally, some insights on the relationship between the GNE and the pareto optimal (PO) solution are provided.
- 3)
A distributed RA algorithm, that allows the network to reach the equilibrium, is proposed. To support our findings, the algorithm performance is corroborated by physical layer simulations of the HetNet over realistic wireless scenarios.
Notations. Matrices are in upper case bold while column vectors are in lower case bold; [ ·]^{T} is the transpose of a matrix or a vector; ⌈x⌉ denotes the nearest greater integer than x; × denotes the Cartesian product; calligraphic mathematical symbols, e.g., \({\mathcal {A}}\), represent sets; \(|{\mathcal {A}}|\) denotes the cardinality of the set \({\mathcal {A}}\); \({\mathcal {A}}(i)\) is the ith element of the set \({\mathcal {A}}\); \(y=[\!x]_{a}^{b}\) means y=x if a<x<b, x=a (x=b) if x≤a (x≥b); y=[ x]^{+} means y=x (y=0) if x>0 (x≤0); the square root of a vector x is intended as the vector including the square root entries of x; a⊙b denotes the element-wise multiplication between a and b; and a⊥b means a·b=0; inequalities between vectors are evaluated element-wise.
2 HetNet model
In this section, the HetNet scenario is first introduced and then the BIC-OFDM uplink channel with multiple access interference (MAI) is described.
2.1 HetNet scenario description
2.2 BIC-OFDM system description
Subsequently, IFFT operation is performed, cyclic prefix (CP) is inserted and after the digital-to-analog conversion, the OFDM signals of all the K users active in both the MC and the SCs are each transmitted in uplink over (different) frequency-selective block-fading channels, which are assumed stationary for the whole packet transmission duration.
3 Performance metrics background and problem formulation
In this section, a brief description of the adopted LPP metric is first given, then the optimization problem (OP) for minimum power consumption under EGP constraint satisfaction is introduced.
3.1 LPP background
As discussed in Section 1, practical modulation and coding schemes are employed; therefore, the GP metric is more suitable in giving a reliable picture of the actual link performance. Here, in order to apply the RA algorithm at the transmitter, the key issue is to get a prediction of the GP, i.e., the EGP, which in turn depends on the estimate of the link packet error rate (PER) [28, 31].
where \(\phantom {\dot {i}\!}\text {PER}_{{\boldsymbol {\varphi }}_{k}}\) and \(\phantom {\dot {i}\!}\Phi _{r_{k}}\) denote the PER of the coded BIC-OFDM system over frequency-selective channel employing TM φ _{ k } and that of the equivalent coded BPSK system over AWGN channel experiencing the ESNR \(\overline {\gamma }_{k}\), respectively. Worth of being observed, \(\Phi _{r_{k}}\), according to [33, 34], is an analytic, monotonically decreasing, and convex function in the region of interest.
Values of κESM coefficients for 4-, 16-, and 64-QAM
μ | |||||
---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||
2^{ m−1} α _{ μ } | 4-QAM | 4 | 0 | 0 | 0 |
16-QAM | 24 | 8 | 0 | 0 | |
64-QAM | 112 | 48 | 16 | 16 | |
ψ _{ μ } | 4-QAM | 2 | 1/2 | 2/9 | 1/8 |
16-QAM | 10 | 5/2 | 10/9 | 5/8 | |
64-QAM | 42 | 21/2 | 14/3 | 21/8 |
3.2 Problem formulation
- 1)
The OP (9), when applied to a P2P BIC-OFDM link, has not yet been taken into consideration, and so, it lacks analysis. Therefore, the above issue is first addressed in Section 4, where we derive a closed-form solution for the PA, being it useful for the subsequent case of multiple users addressed in Section 5.
- 2)
In view of the QoS contraints, there exists a competition among the multiple active users allowed to transmit over the same band. Thus, the problem falls in the NCG framework, which is efficiently employed to study the strategic inter-user interactions [35]. Hence, the OP (9) can be formulated as a NCG, whose conditions of existence and uniqueness of the solution are analytically derived in Section 5, together with a distributed implementation of the RA algorithm.
- 3)
The total power constraint (1) for each user is skipped in the OP (9), since its presence makes the game analysis impractical. Hence, we assume that the optimal PA, satisfying the QoS constraint, satisfies (1) as well. Nevertheless, when introducing the distributed RA algorithm in Section 5.3, such a power constraint will be restored, giving additional comments and insights on it.
4 PEGE OP in point-to-point links
Let us focus on the P2P communication link, that is \(|{\mathcal {Q}}|=1\) and \(|{\mathcal {K}}|=1\) (for this reason in the reminder of this section, the index k will be neglected in the quantities of interest). Specifically, Section 4.1 formulates the strategy to select the optimal TM and PA vector solving OP (9), whereas Section 4.2 solves the PA problem in closed-form.
4.1 Problem formulation
As for the solution of OP (9), it can be pointed out that (i) even if uniform BL is chosen for each user, the choice of the active subcarriers entails again a BL procedure, since the nth component of the BL m is m _{ n }=m if subcarrier n is active, m _{ n }=0 otherwise, as described in Section 3.1; (ii) the BL and PA problems cannot be jointly solved, in that the problem is NP-hard. Given the QoS constraint and the SINR vector γ, however, the OP can be efficiently yet suboptimally tackled relying on the following strategy: first, the optimal PA vector \(\mathbf {p}^{*}\triangleq [p_{1}^{*},\cdots,p_{N}^{*}]^{\mathrm {T}}\) is derived as a function of the generic TM φ, i.e., p ^{∗}≡p ^{∗}(φ), and, then, the pair TM and PA that minimizes the power consumption while satisfying the QoS constraint is selected as solution to the OP.
being constant values, both depending only on φ. From Eqs. (11)–(12), it is seen that, given the BL vector and the coding rate, the QoS constraint \(\bar \zeta \) can be equivalently expressed as a function of the target ESNR γ ^{∗}(φ).
Nevertheless OP (10), having a linear objective function and convex constraints, be a convex OP, due to the presence of the QoS constraint it does not present a closed-form solution. Therefore, solving (10) can be approached via conventional numerical methods [36], though at the price of a high computational load. Afterward, we propose an alternative approximate yet efficient method to get a closed-form solution of OP (10). Hence, the numerical computationally heavy solution will be only used as benchmark to test the accuracy of the proposed approximated closed-form solution.
TM selection algorithm
1. | Input: \({\mathcal {D}}_{\mathbf {\overline m}}\), γ |
2. | Initialize u _{opt}=P |
3. | For \(i=1,\cdots,|{\mathcal {D}}_{m}|\) |
4. | For \(j=1,\cdots,|{\mathcal {D}}_{r}|\) |
5. | For n=1,⋯,N |
6. | Set φ={r,m}, with \(r={\mathcal {D}}_{r}(j)\), \(m={\mathcal {D}}_{m}(i)\) and \(\mathbf {m}=\mathbf {\overline m}^{(n,m)}\) |
7. | Evaluate p ^{∗}(φ) |
8. | If \(\zeta ({\boldsymbol {\varphi }},\mathbf {p}^{*}({\boldsymbol {\varphi }}))\ge \bar \zeta \) and u(p ^{∗}(φ))<u _{opt} |
9. | Set u _{opt}=u(p ^{∗}(φ)) and φ ^{∗}=φ |
10. | Go To Step 4. |
11. | End If |
12. | End For |
13. | End For |
14. | End For |
15. | Output: φ ^{∗}, p ^{∗}(φ ^{∗}) |
4.2 Closed-form solution for the PA problem
Let us now point out that the summation over μ in the QoS constraint prevents to easily obtain a closed-form solution to the PA problem. Thus, the idea we pursue is to approximate the QoS expression in (10) by introducing a scalar β such that a possible closed-form PA solution matches the optimal one earned by solving (10) via a numerical (computationally heavy) method.
Proposition 1
where Θ is such that p ^{⋆} satisfies the QoS constraint in (14) with equality and represents the “water-level.”
Proof
The proof is given in Appendix A. □
where \(\delta ^{(\mathrm {l})} \triangleq \psi _{1}\), \(\beta ^{(\mathrm {l})} \triangleq \alpha _{1}\), \(\delta ^{(\mathrm {u})} \triangleq \psi _{1}\) and \(\beta ^{(\mathrm {u})} \triangleq \sum \limits _{\mu =1}^{\sqrt {2^{m}}/2} \alpha _{\mu } \). The strict equality holds for m=2, since in this case the sum over μ reduces exactly to only one term.
Denoting with p ^{(l)} and p ^{(u)} the optimal solutions (15) when the pair {β,δ} is set in Eqs. (14)–(15) to {β ^{(l)},δ ^{(l)}} and {β ^{(u)},δ ^{(u)}}, respectively, the drawback is that the solution associated to the lower bound (LB) tends to underestimate the QoS constraint and, thus, to allocate less power compared to the optimal solution. On the other hand, the solution associated to the upper bound (UB) overestimates the QoS constraint and, thus, the performance in term of EGP are always satisfied, although at the price of spending more power than the strictly required one. Hence, the idea is to find the optimal pair {β ^{(o)},δ ^{(o)}} that minimizes the mean square error (MSE) between the exact expression of the QoS constraint in (10) and the proposed expression of the QoS constraint in (14), for every modulation order.
BIC-OFDM link setup
Parameter | Value/description |
---|---|
Payload length \(N_{p}^{(p)}\) | 1024 bits |
RLC-PDU length N ^{(i)} | 1056 bits |
N. of active subcarriers N | 1320 |
FFT size | 2048 |
RF bandwidth/subcarrier spacing | 20 MHz/15.152 kHz |
Coding scheme | Parallel concatenated convolutional coding (PCCC) turbo codes \({\mathcal {D}}_{r}=\left \{\frac {1}{3},\frac {2}{5},\frac {1}{2},\frac {4}{7}\right \}\) |
Modulation | 4-,16-,64-QAM |
Available power P | 33 dBm |
Channel model | 6-tap power profile of ITU-Pedestrian B |
Optimal values of {β ^{(o)},δ ^{(o)}} for every modulation order
4-QAM | 16-QAM | 64-QAM | |
---|---|---|---|
β ^{(o)} | 2 | 3.716 | 8.244 |
δ ^{(o)} | 2 | 5.407 | 25.9134 |
5 PEGE OP over multi-access interference links
We assume now that there not exist any centralized unit, and so, users coordinate among themselves in a distributed manner in order to reach a stable RA configuration. The goal is to design a distributed algorithm so that each user minimizes its power consumption, while satisfying its QoS constraint and accounting for the interference caused by the other users exploiting the same frequencies.
5.1 Game formulation
and β _{ k } and δ _{ k } assume values reported in Table 4. OP (17) can be solved by modeling it as a NCG [35]. Such a framework offers a powerful analytical tool that describes how rational entities interact and make appropriate choices so that they can find their own maximum utility.
- 1)
\(\mathcal {K}\triangleq \mathcal {K}_{1}\times \cdots \times \mathcal {K}_{Q}\) is the overall set of users, i.e., players.
- 2)\(\mathcal {P}\triangleq \mathcal {P}_{1}\times \cdots \times \mathcal {P}_{K}\) is the set strategies, where the strategy of user k is its feasible PA set, defined as$$ {}\begin{aligned} \mathcal{P}_{k}\triangleq\left\{p_{k,n}\;|\; g_{k}(\mathbf{p}_{k},\mathbf{p}_{-k})\right. &\le 0, \;\;\; p_{k,n} \\ & \left. \ge 0, \; \forall n \in \mathcal{N}_{k} \right\}, \;\; \forall k\in \mathcal{K}, \end{aligned} $$(19)
with \(g_{k}(\mathbf {p}_{k},\mathbf {p}_{-k})\triangleq \sum _{n\in \mathcal {N}_{k}} \beta _{k} \mathrm {e}^{-\frac {p_{k,n}}{\rho _{k,n}(\mathbf {p}_{-k,n})}} - \kappa _{k}\).
- 3)
\(\mathcal {U}\triangleq \{u_{1},\cdots,u_{K}\}\) is the set collecting the utility functions defined in (9).
In order to study the game equilibrium, let us point out the following remark.
Remark 1
and \(\Theta _{k}^{*}\) is such that the optimal PA \(\mathbf {p}_{k}^{*}\) satisfies the QoS constraints with equality.
Thus, according to the definition of GNE and in view of Remark 1, the GNE of the game must satisfy the following condition.
Proposition 2
with the operator BR defined as in (22).
We remark here that the fixed-point system of Eq. (23) may lead to more than one solution, or the solution of game (17) may not exist, since there may not exist a p ^{∗} that satisfies all the QoS constraints of all the users at the same time. Thus, before going into details of how to solve (17), in the next section we focus on the feasibility conditions of OP (17), as well as on the existence and uniqueness of the GNE of the game describing OP (17).
5.2 GNE analysis
where \({\bar h}_{k,\phi (k),n}\triangleq h_{k,\phi (k),n}/\delta _{k}\) and \(\tilde \gamma _{k}^{*}\triangleq \gamma ^{*}_{k} + \log \beta _{k}\), being δ _{ k } and β _{ k } constant values defined in Table 4.
Proposition 3
Given γ ^{∗}, a sufficient condition for the feasibility of (17) is that Z _{ n }(γ ^{∗}), defined in (25), is a P-matrix.
Proof
See Appendix B. □
Capitalizing on the feasibility condition derived above, the existence of at least one GNE is guaranteed, as stated in the following proposition.
Proposition 4
where \(\bar {\mathbf {q}}_{n} \triangleq [\bar p_{1,n},\cdots,\bar p_{K,n}]^{\mathrm {T}}\).
Proof
See Appendix C. □
In order to better understand the physical meaning of the existence condition, let us first express the channel coefficient as a function of the path loss (PL) between the relevant transmitter and receiver pair, i.e., \(|{\tilde h}_{j,\phi (k),n}|^{2}\triangleq |h_{j,\phi (k),n}|^{2}L_{j,\phi (k)}\), where L _{ j,ϕ(k)} is the path loss between between user j and BS ϕ(k) and \({\tilde h}_{j,\phi (k),n}\in {\mathcal {CN}(0,1)}\). Since a sufficient condition for the matrices {Z _{ n }} to be P-matrices is that they satisfy the diagonal dominance condition [38], then the following corollary holds.
Corollary 1
which can be derived by exploiting the definition of diagonal dominance of a matrix.
Thus, the above condition states that as long as the ratio between the interfering channel and the direct one, i.e, the direct channel between the transmitter and the receiver, is lower than a certain threshold, or, in other words, the interference is small enough, then a solution for game (17) exists. The threshold depends on the EGP constraints, expressed in terms of optimal ESNR, along with the modulation order adopted, via the coefficients δ _{ k } and β _{ k } (the latter contained in \(\tilde \gamma ^{*}_{k})\). Thus, the more distant the competing transmitters, the lower the interference and, thus, the higher the probability of having a non-empty solution set [21,23]. Further, it is worth noting that the condition formulated in (27) is in agreement with that derived in [23] with Gaussian signaling. In this latter case, indeed, δ _{ k }=1 and β _{ k }=1 as well, thus implying \(\tilde \gamma ^{*}_{k}=\gamma ^{*}_{k}\), and \(\gamma _{k}^{*}= \mathrm {e}^{R_{k}^{*}}-1\), where \(R_{k}^{*}\) is the desired rate.
Finally, we focus on the sufficient conditions for the uniqueness of the GNE. This analysis, however, is made difficult by the presence of the QoS constraints, that introduces an interdependency among the strategy sets of the users. Thus, partly inspired by [23], the derivation of the uniqueness conditions is carried out as follows: first, a change of variable is done in order to obtain an equivalent formulation of the problem as a variational inequality (VI) problem; then, the conditions of uniqueness of the solution for the original problem are derived.
Proposition 5
Given γ ^{∗} and under the assumption that the problem is feasible, a sufficient condition for the uniqueness of the GNE for game (17) is that the matrix V(γ ^{∗}), defined in (66), is a P-matrix.
Proof
See Appendix D. □
As expected, also the uniqueness condition depends on the target ESNR γ ^{∗} and on the ratio among the direct and interference channels, which appears on the off-diagonal elements of V(γ ^{∗}) through the parameter \(\chi _{j,k}^{(\max)}\) defined in (64). Thus, although condition of Proposition 5 is less easy to check than the one corresponding to the existence of the GNE, from (66) it can be inferred that whenever the ratio between each direct and interference channel is small, matrix V(γ ^{∗}) is likely to satisfy the diagonal dominance property and thus to be a P-matrix, ensuring the uniqueness of the GNE.
5.3 Distributed algorithm
Distributed RA algorithm
1. | Input: \(\mathbf {p}^{(0)}_{k}\), \({{\boldsymbol {\varphi }}}^{(0)}_{k}\forall k \in \mathcal {K}\). |
2. | Initialize: j=0, ε _{ p }. |
3. | Do |
4. | For k=1,⋯,K |
5. | Evaluate \({\mathcal {D}}_{\overline {\mathbf {m}}}^{(k,j)}\) |
6. | Evaluate \(\mathbf {p}^{(j+1)}_{k}\) and \({{\boldsymbol {\varphi }}}^{(j+1)}_{k}\) applying Algorithm in Table 2 with |
Input \({\mathcal {D}}_{\overline {\mathbf {m}}}^{(k,j)}\) and \({\boldsymbol {\gamma }}_{k}(\mathbf {p}_{-k}^{(j)})\) and evaluating the PA vector | |
according to (22) | |
7. | End For |
8. | Set j←j+1; |
9. | Until ||p ^{(j)}−p ^{(j−1)}||≤ε _{ p } or j=N _{it} |
10. | Output: \({{\boldsymbol {\varphi }}}_{k}^{*}={{\boldsymbol {\varphi }}}_{k}^{(j)}\), \(\mathbf {p}_{k}^{*}=\mathbf {p}_{k}^{(j)}\), \(\forall k \in {\mathcal {K}}\). |
Remark 2
The proposed best response-based algorithm converges under the same conditions for which the GNE is unique (stated in Proposition 5), which in turn requires the condition stated in Proposition 3 in order to have a non-empty solution set. In fact the condition of convergence can be demonstrated with the same approach used to derive the uniqueness condition, simply replacing the two solutions of the GNE with two PA vectors produced by the algorithm at two consecutive iterations.
Remark 3
In the formulation of game (17), we did not consider the total power constraint per user (1). Even if this constraint is intrinsically present in wireless devices, since their power cannot grow infinitely, it makes the equilibrium analysis more involved. Then, we did not take into account this constraint in the theoretical analysis, leaving it for future research. Anyway, the following observations can be done. First, algorithm in Table 5 (as well as that for the P2P case) can easily account for constraint (1) as follows: for all \(k\in {\mathcal {K}}\), in the optimal PA expression (22), based on the water-filling like operator, the “water-level” must now be computed as \(\Lambda _{k} = \min \left \{ {\log \Theta _{k}^{*},\log \tilde \Theta _{k}^{*}} \right \}\), where \(\tilde \Theta _{k}^{*}\) is the “water-level” that, put into the water-filling expression, returns the PA vector \(\tilde {\mathbf {p}}_{k}\) that maximizes the EGP ζ _{ k }(p _{ k }) meeting with equality the total PA constraint (1) [31]. Poorly speaking, for a given strategy p _{−k }, the best response for user k is the PA vector that meets with equality the QoS constraint if the required total PA is not greater than the maximum one, otherwise, all the power is allocated returning the highest possible value of EGP. Finally, as shown in the simulation results, practical values of the PA never reach the maximum power limit, validating the theoretical results on the equilibrium analysis done.
5.4 Relationship between GNE and PO solution
In this section, we give some insights on the relation between the GNE and the PO solution, which represents the achievable performance upper bound. Specifically, the PO solution consists in solving a multi-objective OP, where the objective function is the sum of the utility functions of every user, i.e., \(u_{\text {PO}}(\mathbf {p})=\sum _{k=1}^{K}u_{k}(\mathbf {p}_{k})\), with \(\mathbf {p} \triangleq \left [\mathbf {p}_{1}^{\mathrm {T}},\cdots,\mathbf {p}_{K}^{\mathrm {T}}\right ]^{\mathrm {T}}\), and where the set of constraints is the same than that in (17). Let us refer to this problem as PO-OP. Besides being a non-convex OP, whose solution may be very hard to find, the PO-OP requires that every user, or a centralized entity, knows, among the others, every channel coefficient h _{ j,s,n } between any user and base station in the network. Obviously, this is unlikely to be feasible in the considered HetNet. However, in order to shed light on the relationship between the PO solution and the GNE, corresponding to the PA solution (23) based on the best response, we consider here a two-user two-subcarrier case (K=2, N=2), where user 1 is in cell 1 and user 2 is in cell 2, i.e., ϕ(1)=1 and ϕ(2)=2. The simulation setup is as follows: h _{1,1,1}=1.821, h _{1,1,2}=0.329, h _{1,2,1}=0.104, h _{1,2,2}=0.221, h _{2,2,1}=0.821, h _{2,2,2}=2.629, h _{2,1,1}=0.319, h _{2,1,2}=0.097, σ _{ w } _{ k,1}/P _{ k }=0.01, \(\forall n \in {\mathcal {N}}_{k}\), \(\forall k \in {\mathcal {K}}\), (m _{ k },r _{ k })=(2,1/3), and \(\forall k \in {\mathcal {K}}\). The PO solution is found solving the PO-OP with the Matlab Global Optimization Toolbox. Finally, we consider the target EGP of user 2 to be \(\bar \zeta _{2}=0.2\) (b/s/Hz), whereas we let \(\bar \zeta _{1}\) vary in the set [0.2,1] (b/s/Hz). When \(\bar \zeta _{1}=\bar \zeta _{2}\), only one GNE solution exists and is close to the PO one. Indeed, the ratio ς ^{(GNE-PO)} between the total power obtained with the GNE and PO solution, respectively, is about 0.77. When \(\bar \zeta _{1}=1\) (b/s/Hz), more power needs to be allocated by user 1, yielding a higher co-channel interference. In this case indeed 2 GNE solutions arise and ς ^{(GNE-PO)} drops down to about 0.56 in the worst case. From these observations, we can conclude that, as could be expected, the GNE is more efficient, i.e., closer to the PO, when the interference in the network is lower, whereas the more severe the interference, the less efficient the GNE solution. Nevertheless, we remark that the latter allows for a distributed solution suitable for the scenario under investigation.
6 Simulation results
where 𝜗(ℓ) equals 1 if the ℓth packet is correctly decoded and 0 when it is discarded, T _{ k }(ℓ) is the transmission interval of the ℓth packet, N _{a} is the number of trasmitted packets, set to 1000, each experiencing independent channel realizations and with the position of users randomly placed within their cells. In view of the effectiveness of the GP prediction based on the κESM model recalled in Section 3.1, it is apparent that the AGP performance meets for all the considered users, taken as example, the minimum target EGP. It can be noted that the same result holds for all the other users as well.
7 Conclusions
This paper tackled the distributed RA problem, in the uplink of HetNets, aimed at minimizing the power consumption of each user under the satisfaction of a minimum GP constraint per user. The problem, tagged as PEGE OP, was faced by suitably decoupling the BL and coding rate allocation problem from the PA problem. First, the PEGE OP in P2P links was investigated and solved, obtaining in particular an approximate yet accurate closed-form expression of the PA, in a water-filling-like form, which depends on the QoS constraint. Then, we moved to the interference channel case, where the PA OP was described as a NCG. The relevant solution concept was identified in the GNE, due to QoS constraint that couples the strategies (i.e., the set of feasible PA vectors) of the players (i.e., the users in the HetNet transmitting over the same frequencies). Capitalizing on the closed-form expression for the PA, the analysis of the GNE was carried out through the BR concept, providing sufficient conditions for the existence and uniqueness of the solution. Finally, a distributed RA algorithm, which converges to the equilibrium of the game, was derived and its performance tested through extensive numerical simulations, certifying its good convergence properties, the reduction of power consumption in the HetNet, and the close match between EGP and AGP. Further line of research on this topic will be the analysis of the impact of the total power constraint on the equilibrium.
8 \thelikesection Endnote
^{1} The BL solution found applying the algorithm in [37] clearly depends on the initial PA taken over the subcarriers. Assuming an uniform PA, however, allows to have initially all the subcarriers being active, thus, to perform BL over all the subcarriers. This approach is suboptimal yet simple and efficient, as shown by the result obtained in [31].
9 Appendix A: Proof of Proposition 1
where p ^{∗} and {Θ ^{∗},ν ^{∗}} denote the optimal primal and dual variables, respectively. After some algebra, from (30) it easily follows that \(p^{*}_{n} = {\rho _{n}} \left [\log \Theta ^{*} - \log \frac {\rho _{n}}{\beta _{n}}\right ]^{+}\), where Θ ^{∗} is such that the QoS constraint in (14) holds with equality.
10 Appendix B: Proof of Proposition 3
where Z _{ n } and t _{ n } are defined in (25) and (24), respectively, and \(\mathbf {q}_{n} \triangleq [ p_{1,n},\cdots, p_{K,n}]^{\mathrm {T}}\).
Looking at the expression of Z _{ n } in (25), it is a Z-matrix, since all its off-diagonal entries are negative. Then, if Z _{ n } is also a P-matrix, its inverse is nonnegative [40] and thus, since t _{ n }≥0, condition (35) is satisfied, i.e., the problem is feasible since there exist at least one solution.
11 Appendix C: Proof of Proposition 4
According to the Nash existence theorem [41], given a game in strategic form with K players, each characterized by an action space \({\mathcal {P}}_{k}\) and an utility function u _{ k }, if, \(\forall k \in {\mathcal {K}}\), (i) \({\mathcal {P}}_{k}\) is non-empty, convex and compact, (ii) \(u_{k}: {\mathcal {P}} \rightarrow \mathbb {R}\) is continuous with \({\mathcal {P}} \triangleq {\mathcal {P}}_{1}\times \cdots \times {\mathcal {P}}_{K}\), and (iii) \(\forall \mathbf {p}_{-k} \in {\mathcal {P}} \backslash {\mathcal {P}}_{k}\), u _{ k } is concave on \({\mathcal {P}}_{k}\), then a Nash equilibrium exists. Conditions (ii) and (iii) and the convexity and compactness of each \({\mathcal {P}}_{k}\) easily follow by looking at their analytical expression. A sufficient condition for the non-emptiness of the sets \({\mathcal {P}}_{k}\), \(\forall k \in {\mathcal {K}}\), is given by the feasibility condition stated in Proposition 3. This proves the first part of Proposition 4.Under the feasibility condition, i.e., if Z _{ n } is a P-matrix, then there exist at least NK-sized vectors \(\mathbf {q}_{n}^{*} \triangleq [ p_{1,n}^{*},\cdots, p_{K,n}^{*}]^{\mathrm {T}}\), 1≤n≤N, which satisfy (26) in Appendix B and are thus the solution to OP (17). This proves the second part of Proposition 4.
12 Appendix D: Proof of Proposition 5
where \(\{{\bar {p}}_{k,n}\}\) coefficients are given by (26).
It can be noted that V is a Z-matrix. Then, if V is a P-matrix, it must have a nonnegative inverse, implying b=0. If b=0, then the equilibrium is unique. This proves Proposition 4.
Declarations
Acknowledgements
This work has been partially supported by the PRA 2016 research project 5GIOTTO funded by the University of Pisa.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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