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.
. 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 [
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 [
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 [
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.
users. For simplicity, without loss of generality, downlink transmission from the BS to users is assumed to consist of nonoverlapping time-slots; each slot can accommodate one equal-length packet. Let 
be the transmitted signal in the time-slot 
; 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 BS-user links, (As our main focus in this paper is to study opportunistic multicast schemes for wireless communications in presence of small-scale 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.) small-scale fading. The analytical framework can be extended for the nonhomogeneous case.) The received signal 
at user 
is then given by
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.
's, of all BS-user 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 time-slot 
as
. 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).
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
increases. (As to be shown in Section 2.3.2, 
asymptotically converges to zero as 
increases.)
) code.
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.
packets is selected as
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.
, 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, [
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.
, 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.
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 
.
(all users), 
(no coding), while BU is ECOMF with 
(best user), 
(repetition code).
and code rate 
is crucial in optimizing the required transmission rate and will be discussed in Section 2.3.3 (1).
, 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).
can be represented by a random variable 
with the probability density function (pdf) 
and the instantaneous SNR is denoted by the random variable 
.
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
is the cdf of 
.
users, effectively, the average achievable multicast rate of the WU scheme is 
times the average transmission rate, that is,
.
and, therefore, according to Jensen's inequality
, 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.
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
.
time-slots can be expressed as
being the probability that a given user can receive the packet, (23) becomes
, 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 [
, we have
grows large; therefore, for large multicast group, exploiting only multiuser diversity is also not an efficient way for multicasting.
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 time-slots, and the probability that each user can receive at least 
nonerased packets can be expressed as
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
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
is given by
, and 
.
, 
, (23) becomes
and 
, (23) becomes
approaches BU.
, the average achievable multicast rate of ECOMF, 
, can be optimized by selecting 
and
.
, the probability 
that the channel gain of a certain user exceeds the threshold 
is
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
, 
and 
can be selected to maximize the above average achievable multicast rate of the ECOMP scheme.
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.
in (23) can be approximated as
, we can write
. Using the above recursive relation, we obtain
it can be verified that 
, and hence 
.
then becomes
. 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
.
. 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%.
on Throughput
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.
, 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 cut-off 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 
until reaching its peak and decreases afterwards, which implies that if we try to increase a short-term rate in each time-slot, the payoff will be the long-term 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 
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. 
for ECOM schemes using RS(
) in a Rayleigh fading channel with an average SNR of 20 dB. ECOMFECOMP
for different SNRs as shown in Figure 
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.
derived in the previous Section is confirmed in Figure 
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).
to 
(Rayleigh fading channel with average SNR of 0 dB).
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 BS-user 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 
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 
'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.
Fading Environments on ECOM
fading environment with pdf
is the Gamma function, 
, and 
is the lower incomplete Gamma function, 
.
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.
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 
), 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.
for ECOMP scheme in different Nakagami-
channels with average SNR of 20 dB.
subcarriers over a bandwidth 
,where 
is the subcarrier bandwidth, as shown in Figure 
, 
is a block of complex signals in frequency domain where 
is the transmitted signal on frequency 
. 
is assumed to have zero mean and power 
. These transmitted signals are transformed into samples 
in time domain through the discrete inverse Fourier transform with sampling time 
. These time samples are serially transmitted through a multipath fading channel and are contaminated by additive white Gaussian noise 
with 
noise power. The received signal 
can be expressed as
be the number of resolvable paths in the multipath fading channel, with 
being the tap gain of the 
th resolvable path. Similar to Section 2, 
's are assumed to remain unchanged in a given time-slot and vary independently from one time-slot to another. The impulse response of the multipath channel in time domain can be written as
's are assumed to
follow power delay profile in COST 259 [
with 
.
in frequency domain (after FFT block in Figure 
is the fast Fourier transform (FFT) of 
and 
denotes the channel response in frequency domain, which is the FFT of (39), that is,
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.
and 
can be calculated as follows:
,
. Figure 
and 
is small; hence if one subcarrier is in deep fade, the other is also likely to be in deep fade which may result in packet loss on both subcarriers if we use these two for transmission. On the other hand, when this correlation factor is small, deep fade on one subcarrier is less likely to affect the other subcarrier. Moreover, it is observed from Figure 
increases from 4 to 8. However, when 
is larger than 8, this observation is no longer valid as shown in Figure 
from 2 to 8.
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.
on subcarrier 
is given by
be the transmitted signal on subcarrier 
at time-slot 
with normalized power of 1, the average SNR on subcarrier 
of user 
at time-slot 
can be expressed as
of user 
at time-slot 
is given by
.
as follows:
, and hence, the whole RS-coded packet block is sent in one time-slot with each coded packet transmitted on one subcarrier as illustrated in Figure 
.
is the probability that a given user can receive at least 
nonerased packets on all the subcarriers. At the first look, this probability is similar to that in the previous section; however, it is noted that, in the scenario of OFDM, channel gains in subcarriers are correlated and a simple expression using binomial distribution as in the previous section is not applicable. For this, we use simulation to study the throughput performance of ECOMP for OFDM. In particular, we investigate the effects of the number of resolvable paths on the effective multicast throughput and compare the performance of the proposed scheme with that of BU and WU. In our simulations, multicasting is done on a multicast group of 
users. For simplicity, an OFDM system with 
subcarriers and RS
is considered. 
, and 11. The results in Figure 
increases, the correlation among the subcarriers reduces, that is, less chance that packets are erased at the same time, and hence the achievable multicast rate increases as 
increases. This is not the case for unicast where the ergodic capacity is independent of the number of resolvable paths as shown in [
.
for ECOMP scheme in different numbers of resolvable paths with average SNR of 20 dB. Optimal effective multicast throughputEffective multicast throughput with RS(256,180)
to adjust multicast gain and multiuser diversity give less effect on the multicast throughput than in the case of independent time slots. In other words, correlation in channel responses will decrease the benefits of combining multicast gain and multiuser diversity.
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 
data packets using RS erasure code to form a block of 
-coded packets in the same way as in Section 2, but, instead of sending all 
-coded packets on all 
subcarriers in one time-slot (as in Figure 
-coded packets are sent over 
time-slots on only a subset of 
OFDM subcarriers equally spaced with a frequency separation of 
. For the same assumption of 
, 
. Figure 
. 
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 
scale. It is observed that as the number of time-slot 
increases, the multicast throughput monotonically increases, which illustrates the trade-off between throughput and delay. When 
, EECOMP exploits only frequency diversity by sending all coded packets of one RS block in one time-slot on all subcarriers. The effective multicast throughput for EECOMP is the same as in Figure 
, EECOMP exploits only time diversity by sending only one coded packet in each time-slot on one subcarrier and the multicast rate in this case is the same as in Figure 
. Moreover, when 
is large enough, the effective throughput for EECOMP is approximately equal to that of 
. For example, when 
is larger than 32 for 8-tap channels, 64 for 4-tap channels, or 128 for 2-tap channels, the achievable multicast rates for EECOMP are roughly the same as the case of 
, which correspond to a correlation factor of 0.3 or less (by calculating the frequency separations in these cases and referring to Figure 
is reduced when the number of resolvable paths increases; for example, 
is reduced by 2, 4, 8 times when 
, respectively. However, when 
is very large, as shown in Figure 
. Hence, for larger
, the delay cannot be reduced further. 
for 
and 
for 
, the multicast rate is about 3.3 b/s/Hz/user (Figure 
and 
, respectively, for the same correlation factor of about 0.7 (Figure 
) 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.
in (27), the effective multicast rate of ECOMP can be rewritten as
; therefore, the derivative of the Larangian is given by
, we then solve 
for the peak rate
gives us
gives us
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.