- Research Article
- Open Access

# Opportunistic Multicasting Scheduling Using Erasure-Correction Coding over Wireless Channels

- Quang Le-Dang
^{1}and - Tho Le-Ngoc
^{1}Email author

**2010**:595431

https://doi.org/10.1155/2010/595431

© Q. Le-Dang and T. Le-Ngoc. 2010

**Received:**9 August 2010**Accepted:**3 November 2010**Published:**10 November 2010

## Abstract

This paper proposes an opportunistic multicast scheduling scheme using erasure-correction coding to jointly explore the multicast gain and multiuser diversity. For each transmission, the proposed scheme sends only one copy to all users in the multicast group at a transmission rate determined by a SNR threshold. Analytical framework is developed to establish the optimum selection of the SNR threshold and coding rate for given channel conditions to achieve the best throughput in both cases of full channel knowledge and only partial channel knowledge of the average SNR and fading type. Numerical results show that the proposed scheme outperforms both the worst-user and best-user schemes for a wide range of average SNR and multicast group size. Our study indicates that full channel knowledge is only significantly beneficial at small multicast group size. For a large multicast group, partial channel knowledge is sufficient to closely approach the achievable throughput in the case of full channel knowledge while it can significantly reduce the overhead required for channel information feedback. Further extension of the proposed scheme applied to OFDM system to exploit frequency diversity in a frequency-selective fading environment illustrates that a considerable delay reduction can be achieved with negligible degradation in multicast throughput.

## Keywords

- Orthogonal Frequency Division Multiplex
- Channel Gain
- Orthogonal Frequency Division Multiplex System
- Multicast Group
- Multiuser Diversity

## 1. Introduction

Multicast services over wireless communications have recently become more and more popular. Multicast gain has been explored in a worst-user (WU) approach by transmitting only one copy to all users in the multicast group, for example, [1]. In wireline networks, since user channels are fixed, the multicast throughput increases linearly with the multicast group size,
. However, due to a possible large difference in the instantaneous channel gains of the various links from the base-station (BS) to users in a wireless fading environment, the BS may have to apply the lowest supportable rate (corresponding to the worst BS-user link), which results in very low bandwidth efficiency. Based on this fact, opportunistic scheduling for *unicast* transmission to explore the multiuser diversity by sending one copy to the user with the highest instantaneous channel gain has been extensively researched. Unfortunately, such a best-user (BU) approach does not make use of the multicast gain, which can yield low utility of resources, especially for a large multicast group size. While opportunistic scheduling for unicast transmission has been extensively researched, the idea of exploiting multiuser diversity into multicast has not been studied in an equivalent extent yet. In [2], a threshold-
multicast scheduling scheme at MAC layer is proposed in which in each time-slot the BS sends multicast packet if there are at least
users that have sufficiently good channel to receive the packet. Another opportunistic multicast approach is proposed in [3–6], in which, in each time-slot, one copy is sent to only
users with the best channel quality of the multicast group. The transmission rate is selected as the supportable rate of the worst user in these
best users. In this way, in each transmission, only
user can reliably receive the packet while the other (
) users with insufficient channel gains cannot. To cover the whole multicast group, the use of retransmission has been discussed and proposed in [4–6]; however, those schemes are either inefficient or too complex to implement. In [7–9], proportional fair schemes have been studied aiming to maximize throughput while maintaining the fairness between multicast users and multicast group. These studies assume perfect knowledge of the channel responses of all users in the multicast group at the BS.

In this paper, we propose an opportunistic multicast scheduling scheme that can jointly explore multicast gain, multiuser diversity, and time/frequency diversity in a wireless fading environment. In the proposed scheme, each packet is sent only once to all users in the multicast group at a transmission rate determined by a selected channel gain threshold and an erasure-correction coding is used to deal with possible erasures when the instantaneous signal-to-noise ratio (SNR) of a BS-user link happens to be inadequate. Reed-Solomon erasure-correction code is applied to a block of transmitted packets such that erased packets can be recovered as long as the number of erased packets in a block does not exceed the erasure correction capability, that is, . As each packet can be transmitted in a time or a frequency slot, erasure-correction coding to a block of transmitted packets effectively explores the time/frequency diversity in a wireless fading environment. The selection of channel gain threshold and erasure correction code parameters are jointly optimized for best multicast throughput. Furthermore, to study the role of channel knowledge, the proposed scheme is considered in two cases: (i) with full channel gain knowledge and (ii) with only partial knowledge of fading type and average SNR. An analytical framework has been developed to evaluate the multicast throughput of the proposed erasure-correction coding opportunistic multicast scheduling (ECOM) scheme as well as the BU and WU approaches. We prove that the effective multicast throughput (i.e., the multicast rate that each user can receive) of WU and BU asymptotically converges to zero as the group size increases while that of our proposed scheme is bounded from zero depending on the SNR. Numerical results illustrate that for small multicast group size, full channel gain knowledge can offer better multicast throughput than partial channel knowledge; however, for large group size, the difference in multicast rates of these two cases is just negligible. Besides, performance evaluation shows that with the ability of combining both gains, the proposed scheme outperforms both BU and WU for a wide range of SNRs.

Furthermore we consider extending ECOM for applications to Orthogonal Frequency Division Multiplexing (OFDM) systems. In particular, we explore frequency diversity in a frequency-selective fading environment by sending coded packets over subcarriers. However, since there is a correlation in channel gains among the subcarriers, deep fade on one subcarrier may result in insufficient instantaneous SNR on neighbouring subcarriers. Hence, we have investigated the effects of correlation in subcarrier channel gains on the achievable multicast throughput of the proposed scheme. Numerical results indicate that by exploring frequency diversity, we can significantly reduce the delay with negligible degradation in multicast throughput.

The rest of this paper is organized as follows. In Section 2 the proposed ECOM schemes are described and the analytical framework on multicast throughput of the proposed schemes and BU and WU is provided. Then performance evaluation and comparisons are discussed to illustrate the trade-off between multicast gain and multiuser diversity and the significance of full and partial channel knowledge. In Section 3, ECOM scheme is extended for applications to OFDM systems. The effects of correlation in subcarrier channel responses in a frequency-selective fading environment on the multicast throughput and the trade-off between throughput and delay are discussed. Finally, Section 4 provides concluding remarks.

## 2. ECOM Scheme over Block Flat Fading Channels

### 2.1. System Model

where is the instantaneous channel gain in the time-slot on the link from the BS to user . represents the instantaneous channel gain on wireless link from the BS to user with normalized power of . Further, fades over BS-user links in each time-slot are assumed to be block frequency-flat fading channels; that is, the channel impulse response can be expressed as , where is assumed to be independent and identically distributed (i.i.d.) and quasistatic; that is, any BS-user link fade remain unchanged during a given time-slot and varies independently from one time-slot to another.

This is known as the worst-user (WU) approach. In a line-of-sight (LOS) environment, the wireless links only suffered from path loss and shadowing, which result in small difference among the channel gains, that is,
. In this scenario, it can be seen that by using WU approach, the *full* multicast gain can be achieved. However, when taking into account small-scale multipath fading, instantaneous channel gains of various user links at a given time can be largely different. Hence,
and accordingly,
is likely to be very low when
is large, which may lead to inefficient use of available resource (bandwidth) although multicast gain is exploited (As to be shown in Section 2.3.1,
asymptotically converges to zero as
increases).

*effective multicast rate*that each user receives can be expressed as

As shown in (5), this effective multicast rate of the BU opportunistic approach is likely to be reduced when increases. (As to be shown in Section 2.3.2, asymptotically converges to zero as increases.)

From the previous discussion, it can be seen that if we try to take advantage of multicast gain by using WU approach, the BS needs to send multicast packets only once but the consequence is that the transmission rate must be chosen as the lowest rate of all the users. On the other hand, if we try to make use of multiuser diversity by using BU approach, the BS can maximize its transmission rate at each time slot; however, each packet needs to be sent many times.

### 2.2. Proposed ECOM Schemes

Each *information* packet is partitioned into
symbols; each symbol has
bits. Organizing the
*information* equal-length packets (to be sent) in a rowwise manner, they are encoded in a columnwise manner by using a Reed-Solomon code RS(
) defined over the Galois field GF
, as follows. Each RS codeword contains
information
-bit symbols and
*parity*
-bit symbols. The
information symbols of the RS codeword
,
, are the
th symbols of the
*information* packets and are used to generate the
*parity* symbols of the RS codeword
. Each of these
parity symbols forms the
th symbol of one of
parity packets. In other words, for
*information* packets, the proposed ECOM scheme sends
packets, in which
additional packets contain parity symbols as overhead.

where
is the predetermined channel gain threshold. Taking into account the overhead of the *parity* packets, the *effective* transmission rate in the proposed ECOM scheme is
. The choice of
for certain criterion will be discussed later.

It can be seen that, in the time-slot , users with can correctly receive the packet. For other users with , the packet is likely in error due to insufficient instantaneous SNR. In this case, the erroneous packets can be assumed to be erased and this event can be denoted at the receiver. It is well known that an RS code can correct up to erased symbols, for example, [11]. Therefore, in the proposed ECOM scheme, user can correctly decode all packets when the number of events that is not exceeding -within the time-slots. It can be seen that the proposed ECOM schemes explore multicasting gain by sending only one copy to all users while making use of both multiuser diversity (by selecting ) and time diversity (with erasure-correcting codes). Although RS code is used as an illustrative example in this paper, other erasure-correcting codes can be applied in the proposed ECOM schemes.

Regarding the choice of , interesting questions are raised: whether possessing exact channel gain knowledge of all users can help to increase multicast throughput? And if it can, in which case channel gain knowledge is most pronounced and in which case the gain provided by this side information is negligible. Motivated by these questions, the selection of is considered for two following scenarios.

#### 2.2.1. ECOM with Full Channel Knowledge (ECOMF)

Inspired by WU and BU as extreme cases of multicast gain and multiuser diversity and threshold- scheme, if the base-station transmitter has full knowledge of the instantaneous channel gains, 's, of all users in every times-lot, the BS can sort users in the descending order of their instantaneous channel gains, that is, , and selects a subgroup of users that have the highest channel gains and as .

Interestingly, WU and BU can be considered as two specific cases of ECOMF; that is, WU is ECOMF with (all users), (no coding), while BU is ECOMF with (best user), (repetition code).

The choice of the subgroup size and code rate is crucial in optimizing the required transmission rate and will be discussed in Section 2.3.3 (1).

#### 2.2.2. ECOM with Partial Channel Knowledge (ECOMP)

As the full knowledge of the instantaneous channel gains,
, of all users at any time-slot
comes at the costs of required fast and accurate channel measurements and signalling between the BS and users, it is interesting to consider the case without perfect channel information at transmitter. In particular, we investigate an approach called ECOMP to select
that maximizes the average multicast rate based on the partial knowledge of the channel stochastic properties of the BS-user links, for example, the fading *type* and average SNR
. The throughput analysis of ECOMP is to be discussed in Section 2.3.3 (2).

### 2.3. Throughput Analysis

For the considered quasistatic i.i.d. fading environment, the channel gain can be represented by a random variable with the probability density function (pdf) and the instantaneous SNR is denoted by the random variable .

#### 2.3.1. Throughput of Worst-User (WU) Scheme

*multicast*rate of the WU scheme is times the average transmission rate, that is,

Since , the effective throughput of WU approaches zero as the multicast group size grows large; therefore, for large multicast group, exploiting only multicast gain is not an efficient way to do multicast.

#### 2.3.2. Throughput of Best-User (BU) Scheme

*multicast*rate of BU scheme over time-slots can be expressed as

It is noted that since , the probability that a given user can receive the packet after consecutive transmissions is not 1. Hence, further implementation is needed for BU to achieve (15). One of such implementations is illustrated in [6] with a separated queue for each user.

Equations (17)-(18) prove that the effective throughput of BU approaches zero as the multicast group size grows large; therefore, for large multicast group, exploiting only multiuser diversity is also not an efficient way for multicasting.

#### 2.3.3. Throughput of Proposed ECOM Schemes

- (1)
Throughput of ECOMF

and ECOMF becomes WU.

As shown in (25), for a very large number of users, ECOMF with approaches BU.

- (2)
Throughput of ECOMP

*multicast*rate of the ECOM scheme with RS code is given by

For a given channel fading
,
and
can be selected to maximize the above average achievable *multicast* rate of the ECOMP scheme.

- (3)
Comparison between ECOMF and ECOMP

In this part, an analytical derivation is given to compare the average achievable multicast rates of ECOMF and ECOMP.

For it can be verified that , and hence .

It is interesting to see that the right-hand side of inequality (35) is equivalent to the multicast rate of ECOMP as in (27) with . In other words, the multicast rate of ECOMF is lower-bounded by that of ECOMP and therefore is also bounded away from zero. The relationship further shows that when the multicast group size is sufficient large, ECOMP can converge to ECOMF by setting .

### 2.4. Illustrative Results

As a figure of merit to evaluate and compare the performance of different schemes, we define the *effective multicast throughput* in units of b/s/Hz/user as the ratio of the average achievable *multicast* rate (as shown in (8), (14), (23) and (27)) to the multicast group population,
. Our numerical results are based on (8), (14), (23), and (27) and are confirmed by simulation at a very good agreement with difference of less than 1%.

#### 2.4.1. Effect of on Throughput

We first analyze the effect of the selected on the effective throughput of ECOM schemes over different Rayleigh fading conditions. As an illustrative example, a multicast group size of and RS is considered.

*optimal*throughput versus for different SNRs as shown in Figure 3. It is observed that the peak throughput decreases with SNR as expected. As the average SNR decreases, the optimum channel gain threshold increases which illustrates that erasures occur more often at lower average SNR and has to be reduced to increase the erasure-correction capability of RS at the expense of lower coding rate (and hence lower achievable throughput). The results also show that as the average SNR increases, the proposed ECOM schemes select a lower transmission rate, as shown in (6), implying that the multicast gain becomes more dominant at higher SNR as more users can receive multicast packet in each time-slot.

The above results and discussions confirm that the proposed ECOM schemes can flexibly combine the multicast gain with the multiuser diversity and time diversity via the use of erasure correction coding to achieve optimum achievable throughput in various fading conditions.

#### 2.4.2. Effect of Group Size on Multicast Throughput

*each*user of the multicast group can expect. When the number of users increases, the achievable multicast rate of the WU and BU schemes is quickly reduced to zero, as indicated by (10) and (18), while the proposed ECOM schemes achieve a high multicast rate with the effective multicast throughput of ECOMP unchanged with the multicast group size, shown by (27). This can be explained by the fact that, in the proposed ECOMP scheme, the probability of successful decoding/reception of the multicast copy does not depend on the multicast group size and is the same for every user in the group in an i.i.d. fading environment while the decision for transmission in WU, BU, and ECOMF cases requires the consideration of the whole multicast group for determining transmission rate at each time-slot. Further performance comparisons of the four schemes at low SNR of 0 dB are shown in Figure 4(b). The results confirm the previous observations that WU and accordingly, multicast gain are more favorable at high SNR (Figure 4(a)) while BU or multiuser diversity is superior at low SNR (Figure 4(b)). Furthermore, at any SNR, the performance of both BU and WU quickly decreases as the multicast group size increases, which indicates that for a large group size, neither multicast gain nor multiuser diversity alone can fully exploit multicast capacity and a hybrid treatment is more suitable. Figures 4(a) and 4(b) also indicate that, for a very small multicast group size, ECOMF has the same performance as WU, as shown in (24), which yields the best throughput at high SNR. At very low SNR of 0 dB and for a very small multicast group size (Figure 4(b)), ECOMF has slightly lower performance than BU due to erasure code overhead (i.e., coding rate is less than 1). However, for the case of BU, a complicated queuing system (e.g., in [6]) is needed to guarantee loss-free transmission to achieve (15). The results in Figures 4(a) and 4(b) also confirm that the performance of ECOMF is lower-bounded by that of ECOMP as discussed in Section 2.3.3(3), and as the multicast group size increases, the performance difference between ECOMF and ECOMP is greatly reduced. In other words, the

*full*channel knowledge is beneficial to enhance the throughput of ECOMF, but for a large multicast group size, such performance advantage of ECOMF over ECOMP diminishes, and ECOMP with only required

*partial*channel knowledge can be a better choice for simplicity.

*full*channel knowledge).

#### 2.4.3. Performance Comparison and Trade-off between Multicast Gain and Multiuser Diversity

*average*SNR (e.g., below 5 dB in Figure 6), the

*instantaneous*SNRs in various BS-user links are likely more different; that is, some users may be in deep fades while the others have adequate SNRs. This suggests a more pronounced role of multiuser diversity, and hence the BU scheme outperforms the WU scheme as confirmed in Figure 6. It is interesting to note that, by optimizing the subgroup size or the threshold value, , and code rate according to the

*average*SNR, as well as fading type (e.g., Rayleigh) of the channel, the proposed ECOM schemes can jointly adjust the use of multicast gain and the multiuser diversity (and time diversity) to obtain a much larger achievable throughput over a wide SNR range, for example, 18 times better than that of the BU and WU schemes at an average SNR of 5 dB. At a very high average SNR, the performance of the WU scheme asymptotically approaches that of the proposed ECOM schemes. This implies that at high average SNR, the proposed ECOM schemes will select a very high coding rate (i.e., approaches , or without coding) and essentially explore only the multicast gain. Figure 6 also confirms that for large multicast group size the gain provided by ECOMF is just marginally larger than that provided by ECOMP, and hence, it is sufficiently efficient to have only the

*partial*knowledge of the channel distribution which varies much more slowly than the channel itself and is much easier to estimate than the instantaneous channel. Without the required knowledge of the instantaneous user channel responses 's, the proposed ECOMP scheme can significantly reduce the system complexity and resources for channel estimation and feedback signaling. Furthermore, it can cope with fast time-varying fading channels, especially in mobile wireless communications systems.

#### 2.4.4. Effect of Different Nakagami- Fading Environments on ECOM

where is the Gamma function, , and is the lower incomplete Gamma function, .

In this part, the effect of different Nakagami- fading environments on ECOM is investigated. Since both ECOMP and ECOMF have the same characteristics as shown in the last parts, for simplicity, only the results of ECOMP are illustrated.

## 3. ECOM Scheme over Frequency-Selective Multipath Fading Wireless Channels

In this section, we consider to extend the application of the proposed ECOM scheme to broadband OFDM systems in a frequency-selective fading environment by exploiting frequency diversity. OFDM divides the entire transmission bandwidth into many subchannels, each with sufficiently narrow bandwidth such that the corresponding subchannel response can be regarded as being frequency-flat. To apply ECOM scheme to OFDM systems, one approach could be transmitting coded packets on *one* selected subcarrier in many independent time-slots as in Section 2. Since packets are coded in blocks and sent in time, each user needs to receive the entire block before decoding, and hence, this introduces a delay. One modification to reduce this delay is to send many coded packets simultaneously on many subcarriers, that is, exploring frequency diversity.

Nevertheless, in multicarrier OFDM systems, fading in neighbor subcarriers can be correlated so that deep fade in one subcarrier can also result in deep fade in other nearby subcarriers. As a consequence, for large fading correlation, the multicast rate can be greatly reduced as packets sent on these subcarriers are likely to be erased. At this point, two questions arise: how this correlation factor affects the multicast throughput and how we can take advantage of frequency diversity with as little as possible degradation in the achievable multicast rate. These two questions will be addressed in this section. As discussed in Section 2, for large group size, the throughput performance of ECOMP is as good as ECOMF. Furthermore, ECOMP requires only the knowledge of average SNR and fading type of BS-user links. Therefore, when applied to OFDM the ECOMP scheme is more suitable than ECOMF as it can significantly reduce feedback signaling. Hence, in this section we focus only on ECOMP scheme. First, using only frequency diversity to study the effect of correlation on multicast rate, all the coded packets are sent in only one time slot, each on one subcarrier. ECOMP is then expanded to make use of both time diversity and frequency diversity to investigate the trade-off between throughput and delay.

### 3.1. OFDM System Model and Correlation Factor among the OFDM Subcarriers

In our study, 's are assumed to follow power delay profile in COST 259 [12], that is, with .

indicates the fade level of the signal on subcarrier and hence determines the channel quality at that frequency. Since with , . Since 's remain unchanged in a given time-slot and vary independently from one time-slot to another, 's also remain unchanged in a given time-slot and vary independently from one time-slot to another.

### 3.2. ECOMP for OFDM

Using the system model as described in Section 2.1 to support multicast scenario for users, we first derive the relationship between average SNR and the instantaneous SNR on each subcarrier and then describe the operation of ECOMP in the case of OFDM.

#### 3.2.1. ECOMP for OFDM Using Only Frequency Diversity

- (1)
Effects of the Number of Resolvable Paths on Multicast Throughput

- (2)
Performance Comparison

#### 3.2.2. ECOMP for OFDM Using Both Time and Frequency Diversity

When applying ECOMP to OFDM by sending all coded packets on all subcarriers, it can be seen that we gain times reduction in the delay as each RS block can be sent in only one time-slot. However, as the channel gains of OFDM subcarriers are correlated, if one subcarrier of a given user is in deep fade (i.e., is very low), it is likely that the subcarriers close to it are also in deep fade and the packets that are sent on these subcarriers will likely be erased. To compensate for the erased packets ECOMP has to select a lower transmission rate and lower RS code rate to gain more erasure correction capability and hence this reduces the multicast rate. To enhance this throughput performance, it is necessary to keep the correlation among the subcarriers in use as low as possible by increasing their frequency separation. As shown in Figure 9, this required frequency separation depends on the number of resolvable paths. In other words, by transmitting coded packets on subcarriers far from each other we can achieve lower correlation for higher multicast throughput. However, fewer packets can be transmitted on one time-slot and as a consequence more time-slots are needed for transmitting each RS block, introducing more delay.

where is the probability that a given user can receive at least nonerased packets over time-slots. For the same reason as in the last part, the probability does not have a closed-form mathematical expression and the throughput of EECOM is investigated by simulations. Similar to Section 3.2, our simulation results are based on a group of 100 users in an OFDM system with subcarriers.

In addition, it is observed that for points in Figure 14 with the same multicast rate, they yield the same correlation level shown in Figure 9. For instance, at for and for , the multicast rate is about 3.3 b/s/Hz/user (Figure 14). At these points, the subcarrier separations are 32 and 64 subcarriers when and , respectively, for the same correlation factor of about 0.7 (Figure 9). The same observation applies for other points with approximately the same effective multicast throughput in Figure 14.

## 4. Conclusion

We have proposed and studied an erasure-correction coding-based opportunistic multicast scheduling scheme aiming at exploiting multicast gain, multiuser diversity, and time/frequency diversity to enhance the throughput performance over wireless fading channels. In the proposed scheme, the BS sends each packet only once at a transmission rate determined by a channel gain threshold and using erasure correction capability of RS( ) to recover erased packets due to insufficient instantaneous SNR on BS-user links. RS coding scheme is applied to a block of packets and coded packets are sent in time or frequency slots to effectively explore time/frequency diversity. The channel gain threshold and erasure code rate are jointly optimized for best multicast throughput.

On frequency-flat fading channels, the selection of channel gain threshold is considered in two cases of full channel knowledge and partial knowledge of average SNR and fading type of wireless channel. An analytical framework has been developed to analyze the effective multicast throughput of BU, WU, and of the proposed scheme. In this framework, we prove that while the effective multicast rates of both BU and WU asymptotically converge to zero as the multicast group size increases, this effective multicast rate of the proposed scheme is bounded from zero and depends on the average SNR. We further prove that, for the proposed ECOM scheme, the benefit of full channel knowledge is only pronounced at small multicast group sizes. As the group size increases, partial knowledge of channel response is sufficient in providing approximately the same throughput performance but significantly reducing resources (bandwidth, power) for feedback signalling.

In addition, numerical results illustrate that multiuser diversity is most pronounced at low SNR region since the difference in supportable rates of various users is large while multicast gain is superior at high SNR region where the difference in channel gain is compressed by the log-function that results in small difference in supportable rates among the users. The throughput comparison illustrates that with the ability of combining multicast gain and multiuser diversity, the proposed scheme outperforms both BU and WU for a wide range of SNR.

Furthermore, in this paper, we have extended ECOM for applications to OFDM system aiming at exploiting both time and frequency diversity in a frequency-selective fading environment. The effects of frequency correlation on multicast rate are investigated and our study shows that by exploiting both time and frequency diversity, we can significantly reduce transmission delay with negligible degradation in multicast throughput.

## Appendix

### Rate Optimization for ECOMP

The above equation gives the relationship between and at the peak rate of . Plugging (A.8) back to (A.6), subject to (A.3), the optimal pairs of and can be found numerically; since in (27) the code rate is integer number, the nearest integer of is the result code rate. Another way of finding this optimal pair of and is using the relationship in (A.8), doing the search on to find the peak multicast rate and using the constraints on (A.3) to limit the search.

## Authors’ Affiliations

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