Abstract
This paper investigates the performance of coordinated multipoint (CoMP) networks in the presence of hybrid automatic repeat request (HARQ) feedback. With an information theoretic point of view, the throughput and the outage probability of different HARQ protocols are studied for slowfading channels. The results are compared with the ones obtained in the presence of repetition codes and basic HARQ, or when there is no channel state information available at the base stations. The analytical and numerical results demonstrate the efficiency of the CoMPHARQ techniques in different conditions.
1 Introduction
Coordinated multipoint (CoMP), also known as network multipleinput multipleoutput (MIMO), is one of the most promising techniques for improving the data transmission efficiency of wireless cellular networks [14]. The main idea of a CoMP network is to allow geographically separated base stations (BSs) to cooperate in serving the users. The cooperation is achieved through highspeed backhaul links such that the users’ data and channel state information (CSI) can be shared between the BSs. Theoretically, coordinated systems should outperform noncooperative schemes. In practice, however, the performance gain of the CoMP networks largely depends on (1) the amount of CSI available at the transmitters (CSIT) and receivers (CSIR) and (2) the limited capacity of the backhaul links. This is the main motivation for studying the CoMP systems under limited CSI and backhaul capacity conditions, which has become a hot topic recently.
Performance of CoMP networks has been studied, e.g., by [510] under a perfect CSI assumption and by [1114] in finite backhaul capacity conditions. The papers dealing with imperfect CSI in CoMP networks can be divided into two categories. The first group is the ones in which both the CSIT and the CSIR are assumed to be partially known [1417]. In these works, the focus is on investigating the effect of training signals, providing imperfect CSIR, on the network performance. The second group, on the other hand, is the papers in which, while the receivers are assumed to perfectly estimate the channels (perfect CSIR), the BSs have access to imperfect CSIT obtained by quantized CSI feedback [1830]. Here, it is concentrated on the effects of the CSIT quantization and the quantized CSIT error is normally modeled as Gaussian noise. However, some other CSI quantization schemes have been summarized in [2128] as well. Finally, to study the effect of BS synchronization, [2730] have considered the cases where the BSs are provided with quantized CSI of the channel phase, while the amplitudes are perfectly fed back. Some of the most important conclusions drawn from these works are as follows:
• Compared to noncooperative systems, the cooperationbased schemes suffer more from inaccurate CSIR. That is, the CoMP systems need larger number of pilot symbols than singlecell transmission to make use of its potential [1417]. Therefore, the CoMP networks are more appropriate for lowmobility communication setups, where the channels change slowly and can be estimated accurately.
• The presence of phase ambiguity and asynchronous data transmission can degrade the system performance severely and even make it worse than a nonCoMP model [2830]. Particularly, an important phase ambiguity source, not considered by [2830], is phasenoise and frequency offset [31,32] which makes the synchronization very difficult.
• For realistic results, simple cooperation schemes should be considered, where not only the amount of backhaul capacity and feedback resources is minimized but also the implementation complexity is as low as possible.
These points are the main motivations for our paper. Here, the goal is to develop a simple cooperation scheme for lowmobility CoMP networks which leads to minimum feedback requirement, acceptable backhauling resources, and affordable implementation complexity. Therefore, we propose that hybrid automatic repeat request (HARQ) feedback is selected and, at every moment, each user is served by a single BS, while the serving BSs are switched in the successive time slots. The reasons for our selection are
• HARQ is a technique in the data link layer already provided in many wireless protocols, e.g., IEEE 802.11n [33], IEEE 802.16e [34], and 3GPP LTE [35]. Hence, it needs no new additional design which introduces it as a cost and complexityefficient approach.
• From an information theory point of view, HARQ is a sequential feedback approach; in each (re)transmission round, only 1bit feedback is sent by the users which, compared to the quantized CSI schemes reporting all corresponding fading coefficients of the channel, reduces the feedback load substantially. Moreover, there is no quantized CSI feedback to be shared between the BSs, decreasing the backhauling requirements (Meanwhile, the HARQ feedback bits are shared between the BSs).
• In the proposed scheme, each user is served by only one BS in each time slot. Therefore, there is no need to synchronize the BSs, which decreases the implementation complexity substantially.
In summary, the proposed scheme achieves some advantages related to coordinated data transmission but can still avoid many problems that may limit the practical implementation of CoMP networks.
The results are of particular interest when we remember that, although HARQ schemes have been widely studied in singleuser [3641] and MIMO systems with asymptotically high signaltonoise ratios (SNRs) [4247], there are very few results for CoMPHARQ [48]. Specifically, reviewing the third generation partnership project (3GPP) reports, e.g. [49], there are many aspects of the CoMPHARQ that have not been studied yet. Therefore, the final conclusions of the paper should be useful for the people involved in limited feedback issues in CoMP network standardization.
As described both theoretically and numerically, the proposed scheme leads to considerable performance improvement in terms of system throughput and the user outage probability. Particularly, the BS cooperation makes it possible to combine the advantages of fastfading channels, having a large diversity gain, with the potential for accurate channel estimation in slowfading channels. For sufficiently large number of users, the proposed CoMPHARQ scheme can be modeled as a collection of singleuser interferencefree networks experiencing a modified SNR. Moreover, to have a positive multiplexing gain, the number of retransmissions should be scaled with the transmission power. Finally, compared to codecombining HARQ protocols, the diversity combining schemes are preferable at low powers, because they lead to the same throughput and outage probability with less complexity at the encoders and decoders.
2 System model
Consider a CoMP communication setup consisting of N geographically distributed BSs and K users, with K≥N. The BSs are connected via delayfree backhaul links such that the users’ data can be shared between the BSs. The BSs are limited to a peak power constraint P. Let h_{j,i} be the channel coefficient between the jth BS and the ith user. Also, define the channel gains . We study the lowmobility, also called slowfading, scenario where the channel gains remain constant in a fading block, determined by the channel coherence time L_{c}, and then change independently to other values according to the probability density functions (pdfs) . The simulation results are given for both homogenous and heterogenous links. However, to be more tractable, the analytical results are presented for homogenous links where the channels experience the same fading pdfs. Note that considering the same pdfs does not mean that the channel realizations are the same in a time slot, as they have independent random values. However, for ergodic channels, the same pdf indicates that in the longrun, i.e., over infinitely many time slots, they experience the same behavior. Also, extension of the results for heterogenous networks is straightforward.
The length of the fading block, L_{c}, is assumed to be so long that many packets^{a} are transmitted in a single fading block. As a result, the channel gains can be assumed to be perfectly known by the receivers [1830,3641]. On the other hand, there is no CSI available at the BSs (CSIT), except the HARQ feedback bits. The complex white Gaussian noises added at the receivers are supposed to have distributions . Finally, the results are presented in natural logarithm basis, and the throughput is obtained in nats per channel use (npcu).
2.1 Data transmission model
A maximum of M HARQbased retransmission rounds are considered, i.e., each codeword is (re)transmitted a maximum of M+1 times. In each fading block, N users are selected randomly^{b}. From a specific user perspective, the user receives the data from one BS in each time slot. Then, in the next slot, the serving BSs are switched and the user is served by another BS. The BS switching is done independently of whether new codewords are going to be sent to the users or the previous messages should be retransmitted in an HARQbased fashion. By this technique, the HARQ protocol has a large diversity gain compared to the nonCoMP case.
Remark 1
Four different schemes have been considered for the CoMP networks in the 3GPP community during Release 11 [50,51]: (1) joint transmission, (2) dynamic point selection (DPS), (3) dynamic point blanking, and (4) coordinated scheduling/beamforming. The proposed scheme belongs to the DPS approach of the CoMP networks where the transmission points are varied according to the considered cooperation rules.
Here, although some discussions about the basic ARQ and repetition codes are presented, the main focus is on two stopandwait HARQ protocols:
1) Repetition time diversity (RTD). This scheme belongs to the diversity combining category of HARQ protocols [52] where the same data is repeated in the (re)transmission rounds and, in each round, the receiver performs maximum ratio combining (MRC) of all received signals.
2) Incremental redundancy (INR). The INR belongs to the category of code combining protocols [52]. Here, a codeword is sent with an aggressive rate in the first round. Then, if the user cannot decode the initial codeword, further parity bits are sent in the next retransmission rounds and in each round, the receiver decodes the data based on all received signals.
In harmony with, e.g., [3641], we study the system performance for the complex Gaussian codes which have been proved to be optimal for powerlimited data transmission in fading AWGN channels with long codewords ([53], Chapter 9). The same approach as in, e.g., [54], can be used to extend the results to the case with, e.g., lowdensity paritycheck codes.
2.2 Figures of merit
We consider two, namely, outage probability and longterm throughput, evaluation yardsticks. A user outage probability is the probability of the event that the data cannot be decoded by the user when all retransmission rounds are used. The longterm throughput, on the other hand, is defined as follows. Let the total number of channel uses and the number of successfully decoded information nats up to the end of the kth slot be τ^{(k)} and Q^{(k)}, respectively. Then, the longterm throughput [3641] is defined as
where E{.} is the expectation operator and and are the number of successfully decoded information nats and the number of channel uses in a fading block, respectively. Here, the continuous communication model is considered, where the users are always actively requesting new information. In this case, all of the channel uses during a fading block are used. Therefore, defining as the achievable rate random variable of the HARQ protocols in a fading block, the longterm throughput is rephrased as
3 Performance analysis for the RTD HARQ protocol
In this section, we derive the throughput and outage probability of the network in the presence of RTD HARQ protocol. The simulation results for the RTD are summarized in the figures presented in Section 6.
With no loss of generality, assume that the new data transmission of the kth user is started from the kth BS. Also, let us focus on a specific, say the first, user while the final throughput is obtained by summation on all users throughput. In this way, the signaltointerferenceandnoise ratio (SINR) received at the first user in the mth (re)transmission round is
Here, (3) is based on the fact that starting from the first BS, the user is served by the mth BS in the mth (re)transmission round. Considering Rayleigh fading channels , on which we focus, the SINR cumulative distribution function (cdf) is obtained as
where (a) is found by integration on the gains pdfs and λ represents the fading parameter determined by the path loss and shadowing between the BSs and the terminals^{c}.
Using MRC at the receiver, the equivalent received SINR at the end of the mth (re)transmission round of the RTD is . Also, as the same data is repeated in the retransmission rounds of the RTD, the equivalent data rate at the end of the mth round is where Q is the number of information nats considered for the initial codeword, L denotes the codewords length, and represents the initial codeword rate.
The data is decoded at the end of the mth round if (1) it has not been decoded before and (2) retransmitting the data at the mth round, the channel SINR, accumulated after MRC, supports the data rate. In this way, the achievable data rate random variable for the RTD protocol is obtained as
where . Here, (5) follows from the fact that with accumulated SINR x at the end of the mth round the maximum decodable data rate is if the codeword is repeated m times. In this way, the longterm throughput for the first user is
where F_{γ}^{(m)} is the cdf of auxiliary variable γ^{(m)}. Hence, as the packet starts from each BS with probability and the network is homogenous, the system longterm throughput is found as
Here, (7) gives the users expected achievable rates for different message decoding conditions. Moreover, (7) indicates that the performance of the HARQ protocol improves as the number of retransmissions increases (as seen in the following, the throughput is scaled with the number of retransmission rounds at most logarithmically). Moreover, the user outage probability is
The optimal parameter R, in terms of, e.g., throughput, can be found numerically. Finally, bounds of the throughput and comparisons between the RTD and other schemes are presented in Section 5.
4 Performance analysis for the INR HARQ protocol
As mentioned before, the INR protocol is based on an aggressive codeword transmission in the first round, sending further parity bits in the retransmissions and combining all received representations of the signal at the receiver. Therefore, the results of, e.g., [3639,41] can be used to find the achievable rate random variable of the first user with packet transmission starting from the first BS as
Here, (9) is based on the fact that with SINR γ^{n} in the nth, n=1,…,m, time slot the maximum decodable data rate is if different codewords are sent in each round. In this way, the first user longterm throughput is found as
and the system longterm throughput is
According to (7) and (11), the only difference between the RTD and INR protocols is in the achievable rate terms where of the RTD is replaced by in the INR (Further comparisons between the RTD and INR protocols are presented in Section 5). Again, R is optimized to maximize the throughput for each transmission power. Also, the user outage probability is obtained by
In Section 5, we use (10) to (12) to analyze the throughput at asymptotically low and high powers, and evaluate the effect of BS cooperation on the performance of the network.
5 On the performance of the proposed schemes
In this section, we derive performance bounds for various cases and compare the performance of different schemes operating in the CoMP scenario.
5.1 Performance bounds
This subsection presents performance bounds/approximations for the throughput and outage probability of the proposed CoMPARQ approach. Theorem 1 shows that the performance of the considered protocols becomes bounded if the number of retransmission rounds is finite.
Theorem 1
With a fixed number of retransmissions, the throughput achieved by the considered CoMPHARQ schemes is upper bounded even if the transmission power goes to infinity.
Proof
The theorem is proved for the INR protocol which, as stated in the following, results in higher throughput compared to the RTD. The throughput upper bound is found based on the following (in)equalities:
Here, (b) is obtained by some manipulation on (11) and (c) comes from the Markov’s inequality, [55]. Then, (d) follows from
where (f) is obtained by variable transform log(1+x)=z and partial integration and (g) is based on (4) and . Finally, (e) in (13) follows from which is obtained by Riemann integral and the fact that is a decreasing function^{d}. □
Intuitively, both the signal and the interference powers grow linearly with P. Hence, with high transmission power, the system becomes interferencelimited and, according to (4), the users received SINR is bounded. In other words, the theorem states that: (1) the multiplexingz gain of the proposed CoMP scheme, defined as [56], is zero if a fixed number of retransmission rounds are considered for the HARQ protocols. (2) To have a positive multiplexing gain, a necessary condition is to scale the number of retransmission rounds with the transmission power. (3) Also, according to (13), the throughput is scaled with the number of retransmission rounds at most logarithmically. Finally, denoting the channel instantaneous capacity by C, we can use (b) in (13) to write
which indicates that for sufficiently high initial transmission rate, where , the throughput scales with the initial rate R at most linearly. Note that (h) in (15) is obtained with the same procedure as in (b) of (13) and then implementation of the exponential Chebyshev’s inequality, Pr(X≥x)≤e^{−tx}E(e^{tX}),∀t>0 [55]. Also, (i) follows from the fact that , i.e., the maximum decodable rates is less than the channel instantaneous capacity C, and (j) comes from and where ξ is the Euler’s number. Finally, the same qualitative conclusions are valid for the RTD protocol.
Theorem 2
For Rayleigh fading channels, the proposed CoMPHARQ network can be modeled/underestimated by N interferencefree singleuser networks having a modified transmission SNR.
Proof
With the same procedure as in (13), the RTD and INRbased throughput, i.e., (7) and (11), can be rewritten as
Moreover, from (4), we can show that the auxiliary random variable γ^{n} is dominated^{e} by the random variable ω^{n} which follows the cdf^{f}
Therefore, using (16), (17), and the fact that F_{γ}^{n}(x)≤F_{ω}^{n}(x),∀x, i.e., Pr{γ^{n}≥x}≥ Pr{ω^{n}≥x}, ∀x, we have
Here, (k) in (19) and (20) is obtained by defining and the equivalent SNR , i.e., by appropriate scaling of the fading pdf and the transmission SNR. Also, η^{RTD,SU} and η^{INR,SU} denote the throughput achieved by the RTD and INR protocols, respectively, in the equivalent channel model of (18). In words, (19) and (20) imply that the CoMPHARQ network performance can be underestimated by N interferencefree singleinput singleoutput (SISO) Rayleigh fading channels with fading coefficient and transmission SNR . Interestingly, the approximation F_{γ}^{n}≃F_{ω}^{n} becomes very tight for moderate/high values of N. Thus, the collection of SISO channels becomes an accurate model for the proposed CoMPHARQ network, as stated in the theorem. □
One of the benefits of Theorem 2 is that there are closedform/approximate expressions for the probability terms of (19) and (20) such as
where V(x,y) is the normalized incomplete Gamma function (please see [57] for more details about (21)).
5.2 Comparisons
This subsection compares the performance of different schemes in terms of the throughput and the outage probability.
Remark 2
Let M<N. The spatial diversity gained in the proposed CoMPHARQ scheme with slowfading channel is the same as the time diversity achieved in the nonCoMPHARQ schemes when the channel is fastfading, i.e., when the channel gains change in each retransmission round independently.
In order to show this, consider a nonCoMP model where each user is always served by a specific BS and there is no cooperation between the BSs. In this way, the link between the ith user and its corresponding BS is an interferenceaffected SISO channel. Assuming a fastfading model, i.e., the channels take new values in each retransmission round, the received SINR of the ith user in round n is where g_{i}(n) represents the gain realization of the channel between the ith user and its corresponding BS at round n. Hence, a user throughput in the presence of the RTD and INR protocols is
respectively, which are the corresponding throughputs obtained for the ith user in the proposed CoMP model when the channel is slowfading (see (6) and (10)).
In other words, although each channel remains constant in all retransmission rounds of the slowfading model, by switching between the BSs, a different SINR realization is observed in each round, the same as in the fastfading channels. Thus, the CoMP model works well as (1) the slowfading behavior of the channel gives the opportunity to accurately estimate the channels at the receivers and (2) the same diversity as in the fastfading channels is gained by a simple cooperation approach. Finally, the spatial diversity exploited by the proposed CoMP scheme will be less than the time diversity achieved in the nonCoMP fastfading channel if M≥N. This is because it may occur that we return back to the same BSs when the number of retransmissions exceeds the number of BSs. However, this case is of less interest because the maximum number of possible HARQbased retransmissions is normally less than the number of cooperative BSs.
Remark 3
It has been previously shown that the INR outperforms the RTD in terms of outage probability and throughput ([38], lemma 1). However, as log(1+x)→x for small x’s, (7) and (11) can both be rewritten as
when P→0. That is, the same performance is achieved by the INR and RTD protocols at low transmission powers.
This is interesting when we remember that the superiority of the INR over the RTD is at the cost of complexity; in the INR, the codewords are changed in each retransmission which results in more complex encoders and decoders. Therefore, compared to the INR, the RTD is preferable at low transmission powers, because the same throughput and outage probability are achieved in both schemes while the RTD leads to less implementation complexity.
As demonstrated in [58], the gain of the INR scheme over the RTD increases with the initial transmission rate R. Also, [58] has previously shown that the difference between the performance of the RTD and INR protocols decreases with the SINR variation between the retransmissions. Thus, compared to the nonCoMP setup, the gain of the INR over RTD decreases in the CoMP scenario.
Finally, it is worth noting that the considered HARQ protocols, RTD and INR, lead to better system performance compared to (1) the openloop communication model, (2) basic ARQ protocols, and (3) the case when repetition codes are implemented in a cooperative fashion; the openloop communication model is a special case of the HARQbased schemes with no retransmissions, i.e., M=0. Hence, setting M=0 in, e.g., (8), the openloop system throughput is
which is clearly less than the throughput obtained in (7) and (11). In basic ARQ, on the other hand, the same codeword is transmitted in different retransmission rounds and the users decode the data in each round independently of the previously received signals. In this way, with some manipulations, the throughput of the basic ARQ approach is found as
Then, as less information is exploited by the basic ARQ decoder, compared to the RTD, the throughput in the RTD model is obviously higher than the throughput in the basic ARQ (the superiority of the HARQ protocols over the basic ARQ has been previously shown in the literature, e.g., [36,38]). Here, it is interesting to note that with a slowfading condition, e.g., [38] has shown that there are no performance gains with basic ARQ and the optimal throughput/outage probability achieved by the basic ARQ is the same as the one in the openloop communication setup if the channel does not change in the retransmissions. However, the proposed CoMPHARQ approach makes it possible to utilize the SINR variations and, depending on the channel pdf, increase the throughput by implementation of basic ARQ.
Finally, using repetition codes the same codeword is transmitted by the switching BSs in different rounds, the same as in RTD. However, as opposed to RTD, the MRCbased decoder is not implemented in each round but only when all possible repetition rounds are used. In this case, removing the mth, m=1…M, probability terms of (7), the system throughput for the repetition codes is obtained by
The same arguments can be used when comparing the outage probability of these schemes. Here, the only important difference is that the RTD HARQ and the repetition code schemes lead to the same outage probability. This is because in RTD, the outage occurs if and only if the data is not decodable in the last retransmission round, the same as in the repetition codes.
6 Simulation results
In Figures 1, 2, 3, 4, and 5, we set λ_{i,j}=1, ∀i,j, i.e., the channel is assumed to be homogenous. Figures 6 and 7 study the heterogenous networks. The throughput is given in npcu and transmission power is presented in decibels which is given by 10 log10P (as the noise variance is set to 1, the transmission power is the same as the transmission SNR). For the outage probability, the results are obtained for fixed initial transmission rates, specified in the figures. For the throughput and in each transmission power, the initial transmission rate is optimized by exhaustive search, to maximize the throughput for that power.
Figure 1. System throughput vs the transmission SNR 10 log10 P. Homogenous network λ=1, N=4 BSs, a maximum of M=1 retransmission rounds for the HARQ protocols.
Figure 2. User outage probability vs the initial transmission rate R. Homogenous network λ=1, N=2 BSs, SNR=5 dB, a maximum of M=1 retransmission rounds for the HARQ protocols.
Figure 3. Difference between the throughput of CoMPHARQ and singleuser channel model introduced in Theorem 2, that is, Δη = η^{INR }− η^{INR,SU}. Homogenous network, INR HARQ protocol with a maximum of M=1 retransmission round. N represents the number of BSs/users.
Figure 4. Throughput of the CoMP setup for different number of retransmissions M. Homogenous network, N=5, transmission SNR=10 dB.
Figure 5. Outage probability for different number of retransmissions M. Homogenous network, N=5, transmission SNR=10 dB, initial transmission rate R=0.5.
Figure 6. Outage probability vs the transmission SNR 10 log10 P. Heterogenous network N=2, a maximum of M=1 retransmission rounds. The fading parameters are set to λ_{1,1}=λ_{2,2}=1, λ_{1,2}=λ_{2,1}=0.1. The same outage probability as in the RTD scheme is achieved by repetition codes with repetition order of 2.
Figure 7. System throughput vs the second BS fading parameters λ_{2,1 }= λ_{2,2}. Heterogenous network λ_{1,i} = 1,i = 1,2, a maximum of M = 1 retransmission round, INR protocol.
Considering N=4 BSs and a maximum of M=1 retransmission round, i.e., a maximum of two (re)transmissions, Figure 1 demonstrates the system throughput for different schemes. As seen in the figure, the difference between the INR and RTD protocols is negligible at low transmission powers. Also, the system throughput starts to become bounded at high powers, in harmony with Theorem 1. Therefore, implementation of HARQ in CoMP networks is more appropriate at low transmission powers. With even a single (re)transmission round, substantial performance improvement is achieved by the HARQ schemes, compared to the case with no CSI at the BSs. Also, the CoMP data transmission scheme results in considerable throughput increment, compared to the nonCoMP models. Interestingly, the cooperation gain, i.e., the difference between the throughput of the CoMP and nonCoMP schemes, is observed to be (almost) constant, at medium and high powers. Finally, the repetition codes are simple but perform poorly in terms of throughput, in comparison to the other schemes.
The effect of CoMP transmission on the user outage probability is studied in Figure 2 where, setting N=2, the user outage probability is obtained for different initial transmission rates. As shown in the figure, CoMP transmission leads to considerable outage probability reduction, particularly at low rates. Moreover, the difference between the performance of the RTD and INR protocols increases with the initial transmission rate.
The validity of Theorem 2 is verified in Figure 3. Here, the throughput difference Δη=η^{INR}−η^{INR,SU} is plotted as a function of the transmission power where η^{INR} and η^{INR,SU} are obtained by (11) and (20), respectively. As demonstrated in the figure, the difference between the throughput of the two models considered in Theorem 2 becomes small as the number of users increases. Also, the throughput difference increases with the transmission power. However, using (4) and (18), it can be shown that the difference becomes bounded as the power increases.
Setting N=5, Figures 4 and 5 study the system throughput and outage probability as a function of the maximum number of retransmissions. As demonstrated, considerable performance improvement is achieved by increasing the number of retransmissions. Also, the difference between the performance of INR and RTD protocols and the difference between the CoMP and nonCoMP data transmission models increase with M substantially.
Figures 6 and 7 focus on heterogenous networks. With N=2 BSs and a maximum of M=1 retransmissions, Figure 6 shows a user outage probability for different transmission powers and initial transmission rates. Here, the fading parameters are set to λ_{i,i}=1,i=1,2, and λ_{j,i}=0.1,i≠j,i,j=1,2. Again, the results show that the INR protocol outperforms the RTD, particularly at high transmission rates. However, as stated before, the superiority of the INR over RTD is at the cost of complexity at the encoders and decoders. Moreover, it is worth noting that, with the same argument as in Theorem 2, the user outage probability becomes bounded at high powers, as the system becomes interferencelimited.
Finally, setting N=2 and M=1, Figure 7 investigates the system throughput in a heterogenous network. Here, while we set λ_{1,i}=1,i=1,2, the throughput is plotted as a function of the fading coefficients associated with the second BS, i.e., λ_{2,1}=λ_{2,2}. As it can be seen, the system throughput decreases when the distance between the second BS and the users, modeled by λ_{2,i},i=1,2, increases. At asymptotic condition λ_{2,i}→∞,i=1,2, the network is mapped to a 1×2 SIMO channel working in a TDMA fashion.
7 Conclusions
This paper studied the performance of a CoMPHARQ network. The proposed approach exploits the advantages of coordinated data transmission schemes and, meanwhile, solves some of the important problems that may limit the practical implementation of CoMP networks. As demonstrated both theoretically and numerically, considerable performance improvement can be achieved by implementation of CoMPHARQ systems. However, the system performance becomes bounded if the number of HARQbased retransmission rounds does not scale with the transmission power. The proposed CoMP network can be modeled by a collection of interferencefree singleuser channels experiencing a specific SNR. Finally, the diversity combining HARQ schemes outperform the codecombining HARQ protocols at low transmission powers. This is because the same outage probability and throughput are achieved by these protocols, while the diversity combining schemes lead to less implementation complexity.
Endnotes
^{a} A packet is defined as the transmission of a codeword along with all its possible retransmission rounds.
^{b} This is the best scheme for scheduling N users with no CSIT at the BSs [56] as well as when, due to scheduling delay and complexity, the users are selected for transmission based upon queue lengths instead of on channel conditions.
^{c} For heterogenous Rayleigh fading channels, λ_{j,i}e^{−λ}_{j,i}g,g≥0, (4) is rephrased as
^{d} Straightforward modifications can be applied to (13) and (14) to prove the theorem for heterogenous networks.
^{e} The random variable X dominates the random variable Y if F_{X}(x)≥F_{Y}(x), ∀x[59].
Competing interests
The authors declare that they have no competing interests.
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