Opportunistic Multicasting Scheduling using Erasure-Correction Coding over Wireless Channels

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.

copy to all users in the multicast group at a selected transmission threshold on time/frequency slots and using erasure-correction coding to recover erased packets when the instantaneous signal-to-noise ratio (SNR) of the link between the Base Station and a user is insufficient.
On flat fading channel, an analytical framework is developed to establish the optimum selection of transmission threshold and erasure-correction code rate to achieve the best multicast throughput on different fading conditions. Numerical results show that the proposed scheme outperforms both the Worst-User and Best-User schemes for a wide range of SNR. The results also show that multiuser diversity is superior in the low SNR region while multicast gain is most significant at high SNR region. Moreover, 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. Our study indicates that full channel knowledge is beneficial for small multicast groups but at large group size it is sufficient to have partial channel knowledge as the difference in achievable throughput between the two cases is just marginal.
The proposed scheme is further extended for applications to Orthogonal Frequency Division Multiplexing systems to take advantage of frequency diversity in a frequency-selective fading environment. Our study on the effect of frequency correlation on multicast throughput shows that by making use iv of frequency diversity, significant delay reduction can be achieved with minimal penalty on multicast throughput. v ABRÉGÉ Dans les communications sans-fil, la diffusion se prête naturellement aux services efficaces de multidiffusion tandis que les variations des gains de canal, parmi les liens avec les différents utilisateurs, permettent de promouvoir la diversité multiutilisateur qui peut être utilisée pour améliorer  Channel gain on subcarrier .

Number of information packets within Reed-Solomon code.
Resolvable path index.
Number of resolvable paths.
min minimum.
Number of subcarriers of considered OFDM system.
Size of Reed-Solomon code.
Number of multicast users. Additive white Gaussian thermal noise at user .
Number of bits for each Reed-Solomon code symbol.
Average multicast transmission rate.

Number of transmission time-slots in EECOM.
Timeslot index.
Subgroup selection of Threshold-.
Counting index.

Motivation
Multicast is the method of delivering the same information to a subset of network nodes [18]. It has been widely used in many standards such as IP In multicasting, a set of intended receivers (users) is called multicast group and is represented as a single multicast address. Users need to join a multicast group to receive multicast data for that multicast group. The Multicast sender considers the whole multicast group as a single destination and sends only one copy to the multicast group address. The network infrastructure then intelligently replicates the copy only when needed to direct data to the intended users. As an example, Fig. 1-1 shows a multicast transaction for a group of 5 users via 3 network nodes in a network. Node N1 just forwards one copy to node N2 while node N2 needs to send two copies: one for users A and B over a broadcast channel and the other to node N3 that finally forwards one copy to users C, D and E over another broadcast channel.
The example indicates that whenever the connection to multicast users has broadcast nature, (i.e., one signal transmitted through the network can be received by many users, such as Ethernet), multicast information can be transferred by just only one copy to all connected multicast users instead of sending one copy for each user. In this way, in supporting multicast services, broadcast nature of the transmission medium can be exploited to provide resource (bandwidth) saving, which can be effectively represented by the multicast gain -a gain in bandwidth efficiency as compared to a unicast (host-to-host) service of the same amount of information. It can be seen that the maximum achievable multicast gain is if all users of the multicast group are connected to a broadcast medium with the same transmission quality such as Ethernet.
As point-to-point wireless channel is inherently broadcast in nature, it is quite natural to think that the source (e.g., base-station) can exploit full multicast gain by sending each packet once in order to reach all the intended users of the multicast group. This would substantially reduce bandwidth and power consumptions in wireless multicast. The transmission rate is selected in such a way that all the intended users can reliably receive the information.
In other words, the selected transmission rate is dictated by the worst-user responses is assumed at the transmitter. This assumption could be prohibitively expensive, especially for large multicast groups.

Contributions of this thesis
Inspired by the above observations, the main focus of this thesis is on opportunistic multicast scheduling that can jointly explore multicast gain, multiuser diversity and time/frequency diversity in a wireless fading environment. In particular, we propose an opportunistic multicast scheduling scheme using erasure-correction coding, in which each packet is sent only once to all users in the multicast group at a transmission rate determined by a selected channel gain threshold. When the instantaneous channel gain of a given user happens to be inadequate, i.e., below the selected threshold, the packet is considered to be erased. 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, which is ( − ). As each packet can be transmitted in a time or a frequency slot, erasurecorrection coding to a block of transmitted packets effectively exploits the time/frequency diversity in a wireless fading environment. Since both the selected channel gain threshold and erasure-correction code parameters contribute to the multicast throughput, they are jointly optimized to achieve the 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. Besides, using optimization methods, we developed a mechanism to find the optimal threshold and code rate in the case the BS possesses partial channel knowledge of BS-user links.
Taking into account the change of the optimal transmission threshold in different fading conditions, the trade-off between multicast gain and multiuser diversity is studied. Our numerical results illustrate that multicast gain is most pronounced at high SNR while multiuser diversity is superior at low SNR region. Throughput comparison shows that with the ability of combining both gains, the proposed scheme outperforms both BU and WU for a wide range of SNRs. Regarding the role of channel knowledge, our study shows 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. This makes the proposed ECOM using partial knowledge more attractive for its low complexity. indicate that by exploring frequency diversity, we can significantly reduce the delay with negligible degradation in multicast throughput.

Thesis outline
The rest of this thesis is organized as follows.

Background and Literature Review
In this chapter, background information on multicast transmission over wireless communications systems is covered. In particular, at first, a brief overview of the characteristics and model of radio propagation is presented to introduce and explain the effects of path loss, shadowing and multipath fading on transmitted signal when travelling through a wireless channel.
Then, the system model is described and the wireless propagation in the context of multicast is discussed to present the benefit of multicast gain and multiuser diversity. Trying to explore both gains, many approaches for opportunistic multicast in the literature are reviewed. Motivated by these works, the comments on open and promising questions to be explored are given, and the issues to be studied in this thesis are proposed.

Radio propagation characteristics and model: a brief overview
Consider a wireless communication link between a transmitter and a receiver. While travelling through this wireless link, depending on the distance between the transmitter and receiver and due to obstacles, reflections and diffractions along the traveling path, the transmitted signal is attenuated. Generally speaking, this signal attenuation is considered to be a product of three main components: path loss, shadowing and multipath fading.
Path loss is the attenuation in power of the transmitted signal, which can be expressed as 0 − where 0 depends on the operating frequency, atmospheric and terrain conditions, and is the distance between the transmitter and receiver, and 2 ≤ ≤ 8 is the path loss exponent. Shadowing is the random variation in the signal power due to blockage from objects in the signal paths. Since path loss and shadowing occur over relatively large

System model and wireless propagation in the context of multicast.
Consider a wireless point-to-multipoint downlink system supporting multicast service for a group of users. this scenario, it can be seen that by using WU approach 3 , the full multicast gain can be achieved.
However, when taking into account small-scale fading, instantaneous channel gains of various user links at a given time can be largely different.
Hence, min =1,2,…, { ( )} and accordingly, WU ( ) is likely to be very low when N is large 4 , which may lead to inefficient use of available resource (bandwidth) although multicast gain is exploited.
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), i.e., 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 N times. Since each packet requires at least N transmissions to cover the whole multicast group, the effective multicast rate that each user receives can be expressed as 5 ). (2.5) As shown in equation (2.5), this effective multicast rate of the BU opportunistic approach is likely to be reduced when N increases 6 .
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.
Which one of the two approaches (i.e., WU and BU) gives better effective multicast rate? While the detailed throughput analysis to address this question will be given in Section 3.2 (of Chapter 3), here we can make the following observations. First, both equation (2.3) for WU and (2.5) for BU indicate that their achievable effective multicast rates reduce as the multicast group size N increases. This implies that both WU and BU approaches may not be efficient to support multicast services for large group size. Second, the rate function, as shown in (2.3) or (2.5), is logarithmically increasing with : Its increasing rate is large at low values of , and greatly compressed at sufficiently high . Therefore, it can be conjectured that at sufficiently high , the variation in the instantaneous user channel gain, ( ) does not make a large difference in the corresponding rates, and, accordingly, the worst-user (WU) approach can offer better effective multicast rate than the best-user (BU) approach, while the BU can outperform the WU at low .
The above observations raise a legitimate question: can we explore both multicast gain and multiuser diversity in order to achieve a better effective multicast rate than both WU and BU approaches? One possible strategy is to select a transmission rate ( ) which can support T of N multicast users in each transmission where 1≤T≤N, in order to enhance the effective multicast rate mu l eff = . (2.6) Following this general strategy, there have been several proposed schemes that will be reviewed in the next section.

Approaches for opportunistic multicast over wireless fading channels
In [2], the authors proposed a threshold-multicast scheduling to maximize multicast performance subject to the stability condition. In this work, the authors considered the multicast scheduling problem at MAC layer by using a queue with a fixed service rate, which represents the selected transmission rate to the multicast users. The channel conditions and receiver of each multicast user are represented by a two-state model. A user in its ready state is considered to have sufficiently good channel that can accommodate the transmission rate and hence can receive the packet.
Otherwise, its not-ready state indicates that the user cannot receive the packet. This two-state model is defined with predetermined transition probabilities and is assumed to be identical for all the users but independent from each other. In each time slot, the BS sends multicast packet if there are at least users ready to receive it; otherwise, the BS will back off by a random duration, and then resend the packet. The authors then derive the optimum value of to maximize the effective multicast rate while maintaining the system stable. The stability region is defined as the maximum value of the arrival rate at which the mean queue length is still bounded. The performance of the optimum scheme is compared with that of two reference schemes: threshold-0 (i.e., in which the BS sends multicast packets regardless of the readiness of users) and unicast-based multicast (i.e., the BS transmits each packet separately to each receiver in a roundrobin manner). Their numerical results show that since threshold-0 scheme allows transmission without caring about the readiness of multicast users, there are lost packets, which result in very low effective multicast rate, but has achieved the largest stability region. On the other hand, in the unicastbased multicast scheme, each packet needs to transmit times, which yields the smallest stability region but since there is no packet loss, the scheme offers the highest effective multicast rate in its stability region. With the ability of adjusting according to the packet arrival rate, the achievable multicast rate of the optimal threshold-scheme outperforms both threshold-0 and unicast-based multicast. As the two-state model cannot fully describe fading channels, the work in [2] could not give relevant details on the behaviour of the scheme such as the selection of or at different fading conditions (e.g., SNRs, different fading types).
Another approach is to select a threshold for multicast transmission in each time slot following a predetermined criterion. This approach is different from threshold-in the sense that the BS always transmits packets at the beginning of each time slot but the transmission rate is adaptive according to some criterion. One of the criteria in the literature is to enhance the effective multicast rate. Specifically, in this criterion, of multicast users with the best link quality are selected and the transmission rate is determined as the lowest transmission rate corresponding to the worst-user in these best users in each time slot. In this way, in each time slot, only users can reliably receive the packet while the other − users with insufficient channel gains cannot. Taking into account this loss, in [3], the optimization problem for finding optimal to achieve the best effective multicast rate as in equation (2.6)  increases linearly with . The authors also investigated the proposed scheme over three different SNR levels and it is shown that as the SNR increase, the improvement of the proposed scheme over WU reduces and the optimal selection tends to increase. Following this trend it can be expected that at a very high SNR, the proposed scheme becomes WU; however, since its gain in throughput as compared to WU is reduced from above, the achievable throughput of the proposed scheme is always better than WU. Also in this work, the author suggests that reliability of the scheme could be improved by using erasure-correction code, and the optimal code rate is conjectured as * where * is the optimal value of . However, the authors did not propose the structure of the scheme using erasure-correction code, nor analyze its throughput performance.
Following the same approach in [3], various schemes have been proposed to make sure all N users can reliably receive the multicast packet by retransmission. In the scheme proposed in [4] the BS also transmits to the group of best users using the lowest supportable rate of this group, and keeps transmitting each multicast packet in this manner until the entire group receives the packet. Let be the number of required transmissions, the effective multicast rate for this scheme can be written as mu l eff = The scheme is then analyzed in a Rayleigh fading environment at different SNRs. The study on the changes of user selection according to SNR levels from -30dB to 40dB illustrates that the optimal increases from around 0.5 to 1 as the SNR increases and hence the scheme tends to converge to WU at high SNR. In this work the authors have also compared the throughput performance of the proposed scheme and the BU and WU approaches at different SNR levels to show the superiority of the proposed scheme.
Numerical results illustrate that the proposed scheme can provide significant improvement at low SNR region but as the SNR increases, this improvement reduces and at 25dB, its performance converges to that of WU. It can be seen that the scheme proposed in [4] is very simple in implementation as the BS needs to maintain only one queue for the multicast service. However, since the BS always transmits to users with best channel conditions regardless of their receipt of the packet in the previous time slots, there may be some users that receive the same packet many times. This duplication is a waste of resources (bandwidth and power) and makes the scheme inefficient. To avoid this, in [5] and [6], a different approach to retransmission has been taken to provide reliability while keeping multicast efficiency at the same time. In these works, the BS maintains a queuing system consisting of queues, each for a combination of users. These queues are further divided into sets such that the combinations of users served by queues in each set are mutually exclusive and collectively exhaustive. An example of such a queuing system for the case of = 4, = 2 is illustrated in Fig. 2   Another criterion for choosing the multicast rate is based on the fairness among the users as in [7]- [9]. In these works proportional fair scheduling schemes have been proposed for multicast to not only take advantage of temporal variations of users' channel responses but also to guarantee fairness among the users by looking at the long-term average rate of users rather than the instantaneous rate in selection of transmission rate.
Let us denote the transmission rate supported by user i at time slot k as . Assume that channel responses of users are available at BS before each transmission. At time slot , the BS schedules the packet to user with = arg max , where is the average data rate of user observed over a predefined sliding window of length time slots until time slot . At time slot + 1 , is updated through 7 7 In [9], many ways of updating ( ) are proposed in which the author integrates not only average throughput but also delay and in this case fairness can be understood as the combination of throughput and delay.
From (2.7), it can be seen that users are not ranked according to their instantaneous channel gains but rather to their instantaneous channel gains relative to the average of their own channel conditions. Therefore, users with high instantaneous channel gain do not necessarily have advantage over the others and in this way fairness is preserved while channel variations of users are also considered. Since the transmission rate is chosen based on fairness, it is expected that the average rate of proportional fairness cannot compete with the schemes using the criteria of maximizing multicast rate at the optimal selection * ; however, in [7], the authors have shown that proportional fairness can provide improvement over the median scheme which can support half of the users in each transmission. It can be seen that the selection of transmission rate as in (2.7) provides an indication of fairness; however, it is difficult to say that the scheme can guarantee fairness for all the users, as the user with lower priority in (2.7) may have higher supportable rate than the chosen multicast rate and can still receive the packet.

Proposed studies
Inspired by the works that have been done in the area, first we reckon that the approach of transmitting to best users using a transmission rate threshold in each time slot is a good approach for getting better multicast throughput by exploring both the multicast gain and multiuser diversity. We also notice that the idea of using erasure-correction code for better reliability is very promising and can be improved.  in which − additional packets contain parity symbols as overhead. 9 A similar packet-level coding structure used for a different purpose has been proposed for DVB-S2, e.g., see [11].  The transmission rate (in b/s/Hz) to send packets is selected as where * is the predetermined channel gain threshold. The choice of * for certain criterion will be discussed later. Taking into account the overhead of the parity packets, the effective transmission rate in the proposed ECOM scheme is ECOM .
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 a RS , code can correct up to − erased symbols, e.g., [10]. Therefore, in the proposed ECOM scheme, user can correctly decode all packets when the number of events that ( ) < * , is Interestingly, WU and BU can be considered as two specific cases of ECOMF, i.e., WU is ECOMF with ′ = (all users), = (no coding) while BU is ECOMF with ′ = 1 (best user), = 1 (repetition code).
The choice of the subgroup size ′ , and code rate / is crucial in optimizing the required transmission rate and will be discussed in the next sections.

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 * = th that maximizes the average multicast rate based on the partial knowledge of the channel stochastic properties of the BS-user links. The throughput analysis of ECOMP is to be discussed in the next sections.

Throughput analysis
We consider a quasi-static i.i.d. fading environment so that 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 ≜ .

Worst-user (WU) scheme:
In this scheme, only one copy is sent to all users using the transmission rate corresponding to the channel gain of the worst user. Under the assumption of a quasi-static i.i.d. fading environment, 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, i.e., (3.6) The expected transmission rate for the best user in any given time-slot is given by where the pdf BU = −1 ( ).
As one copy is sent to each user, effectively, the average achievable It is noted that since = 1 , the probability that a given user can receive the packet after consecutive transmissions according to binary probability law is not 1. Hence, further implementation is needed for BU to achieve (3.10).
One of such implementations is illustrated in [6] with a separated queue for each user. Equations (3.11)-(3.12) 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. and the corresponding pdf is As can be shown in (3.19), when the number of users is very large, ECOMF can approach BU.

Proposed ECOM schemes
For a given channel fading type denoted by ( ), the average achievable multicast rate of ECOMF can be optimized by selecting ′ and / .

ECOMP:
In this approach, for a selected channel gain threshold th , the probability ECOMP that channel gain of a certain user is greater than channel gain threshold * is (3.21) For a given channel fading type denoted by ( ), * and / can be selected to maximize the above average achievable multicast rate of the ECOMP scheme.
It is straight forward to see that ECOMP does not depend on the multicast group size ; there always exist k and ℎ so that ECOMP is bounded from zero regardless of . However, the previous statement may be misleading without taking into account the effect of the average SNR. Equation (3.21) shows that at a given SNR, ECOMP is always bounded from zero but as the average SNR reduces, the multicast rate of ECOMP also reduces. In other words, if the SNR is sufficient high, the achievable multicast rate for each user offered by ECOMP is unchanged and bounded from zero regardless of the multicast group size.

Comparison between ECOMF and ECOMP:
(3.25) It follows that (3.26) Using the relation + 1 = + − 1 , we can write with , 0 = 1/ . Using the above recursive relation, we obtain For i=3,...,N it can be verified that It is interesting to see that the right-hand side of inequality (3.29) is equivalent to the multicast rate of ECOMP as in (3.21) with ′ ≡ th . 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 th = ln ′ .

Illustrative results
In this section, the numerical results and discussions on the behavior and

Effect of * on throughput
We first analyze the effect of the selected * on the achievable throughput of ECOM schemes over different Rayleigh fading conditions and code rates. In this case, as an illustrative example, a multicast group size of = 100 and RS (255, ) defined over GF (2 8 )  Consider a Rayleigh fading environment with average SNR of 20dB, the effect of subgroup size and cut-off threshold selection is depicted in Fig. 3-3.
It is shown that for a given value of , there is an optimum value of * that Optimal bound maximizes the multicast rate. It is noted that from the derived relationship * = th = ln ′ in the last section, increasing the subgroup size ' in ECOMF is equivalent to decreasing the cut-off threshold th in ECOMP.
Keeping this inverse relationship in mind, the selection of * has similar effect on both ECOM schemes. It is shown that for a given value of , there is an optimum * that maximizes the effective multicast throughput of ECOM. It is observed that these optimum * 's decrease as increases, which indicates that multicast gain is preferred over multiuser diversity as more users can receive the packet. It is noted that the normalized throughput of ECOM's schemes drops sharply after this optimal point when * increases (accordingly with the increase of th and decrease of ′ ). 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 each scheme. The results in Fig.   3-3 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 timeslot, 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 Fig. 3 We are now extending our observation of the optimal throughput versus * for different SNR's as shown in Fig. 3-4. 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(255, ) 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 (3.1), 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. The effect of multicast group size on multicast throughput on Rayleigh fading channel at 20dB is shown in Fig. 3-5a for WU, BU and ECOM schemes.

Effect of group size on multicast throughput
As defined at the beginning of Section 3.3, the effective multicast throughput  Fig. 3-5b. Fig. 3-5a  However, for the case of BU, as discussed in Section 3.2, a complicated queuing system is needed to guarantee loss free transmission to achieve (3.10). Some additional results on the maximum multicast rate of ECOMF at different SNR are provided in Appendix 3B.

Performance comparison and the trade-off between multicast gain and multiuser diversity
In this part, the performance of the two proposed ECOM schemes will be evaluated and compared with the WU and BU schemes.  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 5dB, 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 5dB in Fig. 3-7), the instantaneous SNR's in various BS-user links are likely more different, i.e., some users may be in deep fades while the others have adequate SNR's. This suggests a more pronounced role of multiuser diversity, and hence the BU scheme outperforms the WU scheme as confirmed in Fig. 3-7. It is interesting to note that, by optimizing the subgroup size ′ or the threshold value, th , 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, e.g., 18 times better than that of the BU and WU schemes at an average SNR of 5dB. 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. Fig. 3-7, 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 enough to have only the 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 ℎ ( )'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.

Effect of different Nakagami-m fading environments on ECOM.
Consider a quasi-static i.i.d. Nakagami-fading environment with pdf  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 higher transmission rate, , and a higher code rate / as shown in Equation (3.21) for multicast transmission.

Chapter summary
In this chapter, an opportunistic multicast scheduler with erasure-    The above equation gives the relationship between and th at the peak rate of ECOMP . Plugging (3A.13) back to (3A.10), subject to (3A.4)-(3A.6), the optimal pairs of and th can be found numerically, since in (3.21) the code rate is integer number, the nearest integer of is the result code rate 10 .
Another way of finding this optimal pair of and th is using the relationship in (3A.13), do the search on th to find the peak multicast rate and use the constraints on (3A.4)-(3A.6) to limit the search. 10 Results computed using this approximation approach are found to be in a very good agreement with those in Fig. 3.7 obtained by exhaustive search using the exact binomial distribution, i.e., (3A.1). As a result, the simplified method in this Appendix can be used to find the optimal operational pairs of ( th , ) for ECOMP.

Motivation
In this chapter we want study the performance of ECOM scheme on environment with correlated channels to see the effect of correlation on the achievable multicast rate. In particular, we are interested in extending ECOM scheme onto OFDM system to explore frequency diversity in a frequency- where , is the fast Fourier transform (FFT) of , and , denotes the channel response in frequency domain, which is the FFT of (4.1), i.e., 4-3 that, for a given level of correlation, when the number of resolvable paths increases the frequency separation decreases. For example, when the number of resolvable paths increases from 2 to 4, the minimum frequency separation to achieve a correlation of 0.3 decreases from 105 to 52 subcarriers, which is approximately two times. The same observation applies when increases from 4 to 8 taps. However, when is larger than 8 this observation is no longer valid as shown in Fig. 4-3. Hence, multipath fading channel introduces frequency diversity that can be used, especially for L from 2 to 8.

ECOMP for OFDM
Using the system model as described in Section 4.1 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.
Applying  where Pr { ≥ } is the probability that a given user can receive at least non-erased packets on all the subcarriers. At the first look, this probability is similar to that in the last chapter; however, it is noted that, in the scenario of OFDM, channel gains on subcarriers are correlated and a simple expression using Binomial distribution as in the last chapter is not applicable. Also, to the best of our knowledge, there is even no closed-form expression for To study the throughput performance of ECOMP for OFDM at first the effects of multipath on multicast throughput is illustrated and then the throughput comparison of the proposed scheme with BU and WU is presented. In our simulations, multicasting is done on a multicast group of   Fig. 4-5a show that the effective multicast throughput depends on the number of resolvable paths. When increases, the correlation among the subcarriers reduces, i.e., less chance that packets are erased at the same time, and hence the achievable multicast rate increase as increases. This is not the case for unicast where the ergodic capacity is independent of the number of resolvable paths as shown in Appendix 4A and [13]- [15]. However, as shown in Fig. 4-5b, this gain comes with a cost: at the same code rate, as the number of taps increases, the throughput curve becomes more sensitive to the channel gain threshold th . This effect can be explained as follows. At lower multipath tap gains, the channel responses of OFDM subcarriers are highly correlated; hence, they follow similar trend and changes in th , which results in smaller change in the multicast throughput as compared to the weakly correlated case.  Similar to the performance comparison in the Chapter 3, BU is better than WU in the low SNR region while WU outperforms BU after the crossover point of around 5dB. Combining both multicast gain and multiuser diversity, the throughput performance of ECOMP is superior to both BU and WU in the considered SNR range and asymptotically converges to WU at high SNR.

Performance comparison:
However, it is noted that the improvement in throughput of ECOMP on OFDM system compared with BU and WU is smaller than in the case of single carrier. For instance, at 5dB, ECOMP offers an improvement in multicast rate of 10 times higher than BU and WU in the case of multicarrier while, as shown in Chapter 3 ( Fig. 3-7), its improvement in multicast rate in the case of single carrier is 18 times higher. This can be explained by the fact that, due to the high correlation in frequency responses, channel gains of adjacent subcarriers are similar and consequently, changes in the SNR threshold th 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.

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 that the correlation among the subcarriers be as low as possible. As shown in Fig. 4 Let be the number of time-slots for transmission, Fig. 4-7 illustrates the transmission mechanism of EECOMP for the case = 2. First, the BS encodes data packets using RS erasure code to form a RS block of packets in the same way as in Chapter 3. Since = 2, these coded packets are sent in 2 timeslots on equally separated subcarriers with the frequency separation of one subcarrier. In Fig. 4 where Pr ≥ is the probability that a given user can receive at least non-erased packets over time-slots. For the same reason as in the last part, the probability Pr ≥ does not have close form mathematical expression and throughput of EECOM is analyzed by means of simulations. Similar to Section 4.3.1, our simulation results are based on a group of 100 users on an OFDM system with =256 subcarriers.  in this case is the same as in Fig. 3-3 b, i.e., around 3.34 b/s/Hz which is about 25% higher than in the case of =1. Moreover, when is large enough, the effective throughput for EECOMP is approximately equal to that of =256.
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 =256. This indicates that there is a significant delay reduction with virtually no penalty in multicast rate at  Fig. 4-3, the correlation between two subcarriers is similar. Hence, for larger , the delay cannot be reduced further. By calculating the frequency separations in these cases and referring to Fig. 4-3 for the correlation values, it is shown that the correlation levels in these cases are less than 0.3.
In addition, it is observed that for points in Fig. 4-8 with the same multicast rate, they yield the same correlation level as shown in Fig. 4-3. For instance, at =32 for =4 and =64 for =2, the multicast rate is about 3.3 b/s/Hz/user (as shown in Fig. 4-8). At these points, the subcarrier separations are 32 and 64 subcarriers when =4 and =2 respectively, for the same correlation factor of about 0.7 (as shown in Fig. 4-3). The same observation applies for other points with approximately the same effective multicast throughput in

Chapter summary
In this chapter, we consider to extend ECOMP to OFDM system, aiming at exploring frequency diversity in the frequency-selective fading environment.
System throughput of the extended scheme is investigated by simulations to study the effect of correlation on multicast rate. The results show that the proposed scheme can make use of multipath fading channel to enhance multicast rate.
Moreover, the proposed scheme is extended to take advantage of both time and frequency diversity and the trade-off between delay and throughput is studied. Simulation results illustrate that when the correlation between two consecutive subcarriers is less than 0.3, the extended scheme can achieve the same multicast rate with significant reduction in delay as compared to the case of purely time diversity. 2 's follow the same distribution with the distribution functions as the following

Conclusion
In this work, 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 depending 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 thesis, 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 is 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.

Suggested future work
In this thesis the throughput performance of the proposed scheme is studied in i.i.d. and correlated channels. However, in the case of correlated channel, the study is based on simulation results only. Developing a mathematical framework to analyze throughput performance would give better understanding about the proposed scheme on correlated channels.
Furthermore, the proposed technique assumes the use of Reed-Solomon erasure-correction codes. One can go further by analyzing the multicast throughput with other codes and come up with the optimal bound that opportunistic multicast can achieve.