2.1. System Model
Consider a wireless pointtomultipoint downlink system supporting multicast service for a group of users. For simplicity, without loss of generality, downlink transmission from the BS to users is assumed to consist of nonoverlapping timeslots; each slot can accommodate one equallength packet. Let be the transmitted signal in the timeslot ; let be additive white Gaussian thermal noise with noise power. The average SNR, which is denoted by , represents the average link quality of the channel assumed to be the same for all BSuser links, (As our main focus in this paper is to study opportunistic multicast schemes for wireless communications in presence of smallscale fading, we consider the homogenous case in which users in a multicasting group have similar average SNR and independent and identically distributed (i.i.d.) smallscale fading. The analytical framework can be extended for the nonhomogeneous case.) The received signal at user is then given by
where is the instantaneous channel gain in the timeslot 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 BSuser links in each timeslot are assumed to be block frequencyflat 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 BSuser link fade remain unchanged during a given timeslot and varies independently from one timeslot to another.
In the case of perfect channel knowledge at the transmitter, that is, the BS knows exactly the instantaneous channel gains, 's, of all BSuser links, adaptive modulation and coding (AMC) can be applied to achieve the maximum transmission rate, in terms of bandwidth efficiency, b/s/Hz for user at timeslot as
Since wireless environment is broadcast in its nature, the BS can transmit each multicast packet to the whole multicast group using only one transmission by sending at the supportable rate of the user with lowest channel response, that is,
This is known as the worstuser (WU) approach. In a lineofsight (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 smallscale 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).
In fact, this difference in instantaneous channel responses among the users promotes multiuser diversity that has been explored in unicast services by sending information to the bestuser (BU), that is, the user with the best instantaneous channel gain. This opportunistic approach can be also used to support multicast services with the transmission rate of
In this way, the resource utilization can be maximized in each time slot at the cost of sending each packet times. Since each packet requires at least transmissions to cover the whole multicast group, the 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
Taking into account both multiuser diversity and multicast gain, the proposed ECOM schemes try to maximize the achievable multicast throughput. ECOM schemes make use of an erasurecorrecting code, for example, ReedSolomon (RS) code, to encode the transmitted packets as shown in Figure 1. (A similar packetlevel coding structure used for a different purpose has been proposed for DVBS2, e.g., see [10].)
Each information packet is partitioned into symbols; each symbol has bits. Organizing the information equallength packets (to be sent) in a rowwise manner, they are encoded in a columnwise manner by using a ReedSolomon code RS() defined over the Galois field GF, as follows. Each RS codeword contains information bit symbols and paritybit 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.
The transmission rate (in b/s/Hz) to send packets is selected as
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 timeslot , 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 timeslots. 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 erasurecorrecting codes). Although RS code is used as an illustrative example in this paper, other erasurecorrecting 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 basestation transmitter has full knowledge of the instantaneous channel gains, 's, of all users in every timeslot, 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 timeslot 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 BSuser 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 WorstUser (WU) Scheme
In the WU scheme, only one copy is sent to all users using the transmission rate corresponding to the channel gain of the worst user. The cumulative distribution (cdf) of the channel gain of the worst user is given by
where is the cdf of .
As only one copy is sent to all users, effectively, the average achievable multicast rate of the WU scheme is times the average transmission rate, that is,
where the pdf .
According to (8), the effective average throughput of WU for each user is given by
For Rayleigh fading channel, and, therefore, according to Jensen's inequality
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 BestUser (BU) Scheme
In the BU scheme, each packet is sent times at the rate of the user with best channel condition. Under the assumption of a quasistatic i.i.d. fading environment, the cdf of the instantaneous SNR of the best user is given by
The expected transmission rate for the best user in any given timeslot is given by
where the pdf .
As one copy is sent to each user, effectively, the average achievable multicast rate of BU scheme over timeslots can be expressed as
With being the probability that a given user can receive the packet, (23) becomes
According to (24), the effective average throughput of BU for each user is given by
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.
For Rayleigh fading channel,
and therefore, according to Jensen's inequality,
Using L'Hospital rule for (17) at the limit , we have
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
In the ECOM schemes, a user can correctly decode its information if it can receive or more nonerased packets within transmitted packets. Under the assumption of a quasistatic i.i.d. fading environment, the probability that channel gain of a certain user is greater than channel gain threshold is the same for all users in all timeslots, and the probability that each user can receive at least nonerased packets can be expressed as

(1)
Throughput of ECOMF
As previously discussed, the ECOMF selects a subgroup of users that have the highest channel gains and as the lowest instantaneous channel gain of the th user. Under the assumption of a quasistatic i.i.d. fading environment, according to order statistics, the cdf of is given by
and the corresponding pdf is
It is obvious that, in a given timeslot, the channel gain of a certain user is greater than channel gain threshold if this user belongs to the selected subgroup of users. Since the user channel gains distributions are i.i.d., the probability that a user is in this selected subgroup is. In other words, the probability that the channel gain of a certain user exceeds the threshold is
As a result, the average achievable multicast rate of the ECOMF scheme with RS is given by
where , and .
When , , (23) becomes
and ECOMF becomes WU.
When and , (23) becomes
As shown in (25), for a very large number of users, ECOMF with approaches BU.
For a given channel fading , the average achievable multicast rate of ECOMF, , can be optimized by selecting and.

(2)
Throughput of ECOMP
In ECOMP, for a selected channel gain threshold, the probability that the channel gain of a certain user exceeds the threshold is
For example, for a Rayleigh fading channel. In average, there are only users that can successfully receive the multicast packets at an effective transmission rate of . Therefore, effectively, the average achievable 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.
From (27), it is straightforward to see that does not depend on the multicast group size; that is, at a given SNR, , there always exist and so that is bounded from zero regardless of , and is reduced as reduces.

(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.
Using the Jensen inequality, in (23) can be approximated as
For a Rayleigh fading channel, we have
where
It follows that
Using the relation , we can write
with . Using the above recursive relation, we obtain
For it can be verified that , and hence .
Hence, then becomes
From (23) and (34), the lower bound of ECOMF multicast rate can be expressed as
It is interesting to see that the righthand 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 lowerbounded 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.
Consider a Rayleigh fading environment with average SNR of 20 dB; the effect of subgroup size and cutoff threshold selection is depicted in Figure 2. It is shown that for a given value of, there is an optimum value of that maximizes the effective multicast rate. From the derived relationship , increasing the subgroup size in ECOMF is equivalent to decreasing the cutoff threshold in ECOMP. It is observed that the optimum value of decreases as increases. This indicates that multicast gain is preferred over multiuser diversity as more users can receive the packet. It is noted that the normalized throughput of both ECOMF and ECOMP drops sharply after this optimal point when or increases (or, equivalently, decreases). In this case, a lower with its corresponding is a better choice since it provides better erasure correction capability at the expense of more coding overhead. The optimal bound (dashed line) presents the maximum achievable multicast throughput over all possible values of for ECOMF and ECOMP. The results in Figure 2 show that the optimum throughput increases with until reaching its peak and decreases afterwards, which implies that if we try to increase a shortterm rate in each timeslot, the payoff will be the longterm average throughput as the erasure correction capability has to be high to compensate for packet loss, which makes multicast transmission inefficient after its optimal point. It is shown in Figure 2 that over Rayleigh fading channel at an average SNR of 20 dB, the optimal for best multicast throughput is 190 for ECOMF and 184 for ECOMP with an appropriate optimum threshold value at for ECOMF and for ECOMP. The optimal values of and for ECOMP for each channel condition can be found through optimization method as illustrated in the appendix.
We are now extending our observation of the 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 erasurecorrection 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 timeslot.
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
The effect of multicast group size on multicast throughput on Rayleigh fading channel at 20 dB is shown in Figure 4(a) for WU, BU, and ECOM schemes. As defined at the beginning of Section 2.4, the effective multicast throughput in terms of b/s/Hz/user represents the effective rate 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 timeslot. 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 lossfree transmission to achieve (15). The results in Figures 4(a) and 4(b) also confirm that the performance of ECOMF is lowerbounded 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.
The relationship derived in the previous Section is confirmed in Figure 5. As shown in Figure 5 at an average SNR of 0 dB, the ratio converges to from above as increases or, correspondingly, the ECOMF threshold approaches the ECOMF threshold with . This implies that, in average, ECOMF always obtains higher transmission rate than ECOMP (as the benefit of full channel knowledge).
2.4.3. Performance Comparison and Tradeoff between Multicast Gain and Multiuser Diversity
Figure 6 compares the effective multicast throughput of the WU, BU, ECOMF, and ECOMP in a Rayleigh fading environment for a wide SNR range from −20 dB to 40 dB with a multicast group size of users. It is observed that the BU has higher throughput than the WU in the low SNR region, but as the average SNR increases above the crossover point of 5 dB, the BU scheme has inferior performance with an almost saturating throughput. The results indicate that when the average SNR is sufficiently high, the various BSuser links are sufficiently good, and, as a consequence, it is more likely that all users in the multicast group are able to successfully receive the transmitted packets. Hence, it is better to explore multicast gain (i.e., transmission only one copy for all users) to achieve higher normalized throughput in the case of high SNR. However, at a low average SNR (e.g., below 5 dB in Figure 6), the instantaneous SNRs in various BSuser 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 timevarying fading channels, especially in mobile wireless communications systems.
2.4.4. Effect of Different Nakagami Fading Environments on ECOM
Consider a quasistatic i.i.d. Nakagami fading environment with pdf
and cdf
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.
In Figure 7, performance comparison of ECOMP on different fading type conditions is investigated. Consider Nakagami channels at the same average SNR of 20 dB for different values of : for a Rayleigh channel, for a milder situation, equivalent to a Ricean channel, and for a considerably severe fading channel. The results in Figure 7 illustrate that as the fading becomes less severe (i.e., with larger value of ), the optimum achievable throughput is increased as we can expect. Correspondingly, the optimum value of threshold is increased in a milder fading environment. This can be explained as follows. When increases, the peak of the Nakagami probability density function occurs at a higher value and its variance decreases; in other words, more users have good channels and therefore are less likely to receive erased packets. Hence the proposed ECOMP scheme can select a higher transmission rate,, and a higher code rate as shown in (27) for multicast transmission.