2.1. System Model
Consider a wireless point-to-multipoint 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 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
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.
In the case of perfect channel knowledge at the transmitter, that is, the BS knows exactly the instantaneous channel gains,
'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
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 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).
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 best-user (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 erasure-correcting code, for example, Reed-Solomon (RS) code, to encode the transmitted packets as shown in Figure 1. (A similar packet-level coding structure used for a different purpose has been proposed for DVB-S2, e.g., see [10].)
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.
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 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
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 Best-User (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 time-slot is given by
where the pdf
.
As one copy is sent to each user, effectively, the average achievable multicast rate of BU scheme over
time-slots 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 time-slots, 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 time-slot, 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 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.
Consider a Rayleigh fading environment with average SNR of 20 dB; the effect of subgroup size and cut-off 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 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 2 show that the optimum throughput increases with
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 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 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
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 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.
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 Trade-off 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 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 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
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.