TDMA Achieves the Optimal Diversity Gain in Relay-Assisted Cellular Networks

TDMA Achieves the Optimal Diversity Ga in in Relay-Assisted Cellular Networks Suzhi Bi, Ying Jun (Angela) Zhang, Senior Member, IEEE Department of Information Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong. Email: {bsz009, yjzhang}@ie.cuhk.edu.hk Abstract In multi-access wireless networks, transmission scheduling is a key co mponent that determines the efficiency and fairness of wireless spectrum allocation. At one extreme, greedy opportunistic scheduling that allocates airtim e to the user with the largest instantaneou s channel gain achieves the optimal spectrum efficiency and transmission reliability but the poorest user -level fairness. At the other extreme, fixed TDMA scheduling achieves the fairest airti me allocation but the lowest spectrum efficiency and transmission reliability. To balance the two competing objectives, extensive research efforts have been spent on designing opportunistic scheduling schemes that reach certain tradeoff points between the two extremes. In this paper and in contrast to the conventional wisdom, we find that in relay-assisted cellular networks, fixed TDMA achieves the same optimal diversity gain as greedy opportunistic scheduling. In addition, by incorporating very limited opp ortunism, a simple relaxed-TDMA scheme asymptotically achieves the same optimal system reliability in terms of outage probability as gre edy opportunistic scheduling. This reveals a surprising fact: transmission reliability and user fairness are no longer c ontradicting each other in relay-assisted syst ems. They can be both achieved by the simple TDMA schemes. For pr actical implementations, we further propose a fully distributed algorithm to implement the relaxed -TDMA scheme. Our results here may find applica tions in the design of next -generation wireless communication syst ems with relay architectures such as LTE -advanced and WiMAX. Index Terms Scheduling, diversity techniques, user fairness, relay systems. This work was supported in part by the Competitive Earmarked Research Grant (Project Number 419509) established under the University Grant C ommittee of Hong Kong. I. I NTRODUCTION A. Motivations and Summary of Contributions Relay-assisted transmission techniques are known to be effective in combating path loss and enhancing link quality in wireless communications systems [1,2,3]. Such techniques are already adopted in the  G wireless comm unications standards, such as LTE -Advanced and IEEE WiMAX [4,5]. In these systems, fixed relays are deployed as inte rmediate nodes to forward data between mobile users and base stations (BS), thus extending the service coverage of a cell and enhancing the overall throughput performance of the system. In multi-access wireless systems, transmission scheduling is a key co mponent that determines the efficiency and fairness of spectrum resource allocation. In particular, opportunistic scheduling that takes advantage of independent time-varying channels across different mobile users effectively exploits multiuser diversity through scheduling the transmission of users according to their instantaneous channel conditions. Depending on their greediness, different opportunistic scheduling schemes achieve different tradeoffs between user-level fairness and high system performance in te rms of throughput or transmission reliability. At one extreme, greedy opportunistic scheduli ng, which selects the user with the largest instantaneous channel capacity, achieves the highest spectrum efficiency and tr ansmission reliability but the worst fairness among users. On the othe r extreme, being oblivious to channel states, fixed TDMA achieves fairest airtime allocati on but lowest transmission reliability. To balance the two competing objectives, different opportunistic sche duling policies have been designed to reach certain tradeof f between the two extremes [7,8,9,10,11]. One such example is proportional fair scheduling, which schedules transmissions according to the users’ “relative" channel strengths [8]. Less greedier than greedy opportunistic scheduling, proportional fair scheduling achieves equal airtim e allocation among user s in the long run. Another interesting work uses an  -Rule scheduling to achieve a flexible tradeoff between spectrum efficiency and u ser fairness by tuning the variable  in its scheduling policy [9]. In this paper and in contrast to conventional bel ief, we find that the transmission reliability and user fairness are no longer contradicting each other in relay -assisted cellular systems. With optimal relay selection, fixed TDMA achieves the same opti mal diversity gain as greedy opportunistic scheduling. In addition, by incorporating very limited opportunism, a si mple relaxed -TDMA sch eme asymptotically achieves the same optimal outage proba bility as greedy opportunistic sche duling. In other words, we can fully enjoy the multiuser divers ity gain achievable by greedy opportunistic scheduling without suffering its disadvantages such as poor user fairness and high implementation cost. Our contributions are detailed below.  We derive the optim al outage probability in relay-assisted cellular ne tworks. In particular, we show that the optimal outage probability is achieved by greedy opportunistic scheduling, which fully exploits multiuser diversity. By letting the number of users go t o infinity, a lower bound on the outage probability is obta ined. Interestingly, thi s lower bound is independent of the user -side parameters including the transmission powers and channel conditions of users.  We find that fixed TDMA scheduling achi eves the same diversity gain as greedy opportunistic scheduling in relay -assisted networks. In addit ion, we show that the diversity order is solely determined by the number of relays. This implies that the greediness of the scheduli ng policy is irrelevant when it comes to diversity order in relay-assisted cellular network. This is quite different from most wirele ss systems where the diversity orde r is closely related to the scheduli ng policy.  A power gap is observed between fixed TDMA and gre edy opportunistic sched uling. We quantify this power gap in high SNR region and find that the gap depends on the power allocation ratio between mobile users and relay set. Interestingly, we show that this power gap is closed when minimum oppor tunism is introduced to TDMA. In particular, a relaxed-TDMA scheme asymptotically achieves the optimal outage probability that is otherwise achievable by greedy opportunistic sche duling. This reveals an encouraging fact: through the use of TDMA, optimal outage probability can be achieved in relay- assisted networks without compromising the user-level fairness.  We propose a fully distributed algorithm to implement the relaxed-TDMA scheme, where the scheduling decision is made in a distributed manner at the relays based on their own local chann el conditions. We show t hat, the proposed distributed algorithm achieves the optimal outage probability while generating very little signaling overhead. The rest of this paper is orga nized as follows. We describe the system model and the optimal relay se lection method in Section II . In Section III, we derive the explicit expression of optimal outage probability in re lay-assisted networks. It is proved in Section IV that fixed TDMA yields the same diversity gain as greedy opportunistic scheduling. A relaxe d-TDMA scheme is int roduced in Section V, where we show that it achieves the optimal outage probability and high user - level fairness at the same time. Simulations results are given in Section VI. Finally, the paper is concluded in Section VII. B. Related Works Like in traditional cellular networks, channel-aware opportunistic user scheduling is of great interests in relay-assisted cellular networks. A common objective shared by most recent work is to maximize the total throughput [12,13,14,15]. This, howev er, le ads to poor user-level fairness, for some users may experience excessively long access delay, if they are stuck in deep fading channels. With this in mind, some work aims to strike a balance between spectrum efficiency and fairness by applying scheduling schemes that are less aggressive [3,16,17]. For example, [3] incorporated the queue-size information into its scheduling protocol design, where the downlink throughput is maximized under the constraint that queue -length at all nodes are finite. [17] extended proportional fair scheduling to relay- assisted systems to achieve long -term user-level fairness at the cost of throughput reduction. In contrast to conventional wisdom, our work shows that with optimal relay selection, a simple relaxed -TDMA scheduling scheme obta ins the optimal outage probability and excellent fairness among users at the sam e time. This indicates that the conventional tr adeoff between transmission reliability and user fairness is not necessary in relay-assisted cellular networks. Opportunistic scheduling in relay-assisted networks normally requires strong centralized control at the BS to coordinate the transmissions of both mobile users and relays. In [3,12,13], the channel between every two nodes is estimated and fed back to the BS for cent ralized processing. The cost of either time or bandwidth on transmitting large am ount of pilot signals ine vitably decreases the overall spectrum efficiency. On the other hand, [18,19] proposed distributed scheduling schemes that allow the relays to par ticipate in scheduling decisions based on the limited local channel state information (CSI) at the relays. Compared with centralized schemes, distributed implementations effectively reduce signaling overhead and processing delay. Nevertheless, the pro posed distributed schemes decouple user scheduling and relay selection p rocess, thus are suboptim al co mpared to centralized scheduling schemes. By contrast, our distributed scheduling protocol achieves optimal outage probability through jointly scheduling user and selecting the relay. When a user is scheduled, a proper set of relays needs to be ass igned to ass ist its transm ission. The optimal strategy that maximizes the received SNR at the BS is for all available relays to form a virtual antenna array and jointly transmit the source information using beamforming techniques [1]. However, beamforming is costly to implement in distributed relay networks, since it requires the knowledge of global CSI and strong centralized coordination. Similarly, a distributed space-time code (DSTC) scheme tha t makes use of all available re lays is proposed by Laneman [20]. Although DSTC achieves full diversity gain, practic al distributed space -time code design is very difficult, since the set of available relays is time var ying due to channel fading. Besides, its implementation requires strict symbol level synchronization, which is also considered difficult in distributed networks. Alternatively, single- relay selection, which employs only one “best" relay to assist transmission, greatly reduces system implementation complexity and saves significant signaling overhead. Besides, recent studies also show that single-relay selection schemes achieve co mparable system reliability as m ulti- relay transmission schemes. F or example, [22,23] showed that single-relay selection schemes yield near optimal outage performance. [21] proved that single-relay selection achieves lowest outage probability under aggregate relay power constraint. [24] further showed that single-relay selection outperforms DSTC scheme when the number of relays is greater than three. In this paper, we will also demonstrate the optimality of the single-relay selection method. Before leaving the session, we would like to emphasize that relay selection is viewed as par t of t he operations at the network infrastructure side. For each scheduled user, there exists a mechanism that assigns a proper set of relays to assist the transmission of the user. In this paper, our focus is on the scheduling of the users, given that an optimal relay selection mechanism is used. II . S YSTEM M ODEL AND O PTIM AL R ELAY S ELECTION M ETHOD A. System Model We consider the uplink of a cellular network with a base station (BS),  stationary relays and  mobile users communicating to the BS thr ough the relays. The direct user- to -BS links are assumed to be non-existent, so tha t all user- to -BS communicationstake place in a two-hop manner through the relays. Each relay works in a half-duplex mode using decode and forward (DaF) scheme. Suppose that all users and relays transmit with a fixed data rate. The received message can be correctly decoded only when the received signal- to -noise ratio (SNR) exceeds a prescribed threshold  . Suppose that channel fading is independent across different links. Moreover, channel fading is assumed to remain unchanged during each two -hop transmission period. In the first hop, a mobile user, say user  , is scheduled to tr ansmit its signal   with power   , where  󰇟  󰇠    . Then, the received signal at the   relay is                      (1) Here,         is the instantaneous channel fading coeff icient between the   user and the   relay. Denoted by        , the instantaneous channel gain follows an exponential distribution with mean  󰇟  󰇠    . The noise   is assumed to be i.i.d and     󰇛    󰇜 . Then the received SNR at the   relay is       . Let   be the set of relays that successfully decode the   mobile user’s message. That is, relay     if         . The relays in   are referred to as decoding relays. In the second hop, all or a subset of decoding relays forward   to the BS using different orthogonal channels, e.g. in separate frequency channels, to avoid mutual interference. At first glance, this orthogonal transmission scheme requires excessive bandwidth to accommodate numerous relays. Fortunately, as we will show in subsection II -B, it is optimal to allow only one decodi ng relay to transmit in the second hop. Some careful readers may suggest using multiple relays to form an optimal distributed antenna array that maxim izes the received SNR at the BS . It is, however, costly to implement thi s method. This is mainly because to form the tr ansmission beamforming vector, each relay will have to know the indices of all the decoding relays as well as the channel conditions from these relays to the BS 1 . In contrast, as we w ill show in a later section, our proposed method can be implemented in a fully distributed manner where each relay only needs to know the channel fad ing of its own link. Suppose that the   relay transmits with power   , and          , where   is the total transmission power allocated to the relay period. Then, the received signal at the B S is                    (2) To maximize the output SNR, the BS combines the received signal using maximum ratio com bination as follows,                 󰇭          󰇮                  (3) where         are the instantaneous channel gains of the relay -BS (R-B) channels. The corresponding maximum received SNR is calculated as                  󰇟    󰇠                         (4) An outage event occurs when      . The probability of such an event, referred to as outage probability, is a key metric to measure the transm ission reliability of the system. B. Relay Selection method The following proposition shows that selecting a single relay obtains the highest    and hence the lowest outage probability. Proposition 1 : Selec ting the relay      with the largest R-B channel gain yields the lowest outage probability. Proof: We can infer from (4) that 1 The optimal beamforming is t he maximum ratio transm ission (MRT) bea mforming, where the relay     transmits                  to the BS. The calculati on of the beamforming vector cle arly requires the knowled ge of both indices and channel state informat ion of all decoding relays.                                              (5) This shows that the highest SNR, and hence the lowest outage probability, is obtained by allocating all transmission power to the relay with the highest instantaneous R -B channel gain. That is, it is optim al to let only one “best" relay to relay the message, where the “best" relay is chosen as             (6) 󰆢 Based on the above “single -relay selection" ar gument, we propose in Corollary  the optimal relay selection scheme for relay- assisted cellular networks. This scheme is convenient for both anal ysis and distributed implementation. Note that the the scheme in Corollary  is optimal in the sense that it yields the same opti mal outage probability as the one in Propositi on  . Nonetheless, it is possible that the two schemes end up selecting different but equally optimal relays. Corollary 1 : For the   mobile user, allocating full transmission power to the “best" relay    , given by                    (7) yields lowest outage probability. Remark 1 : (7) is indeed selecting the relay that provides the best end- to -end channel. Proof of Co rollary 1: Let us refer to the optimal relay selection method in (5) as Method  , and in (7) as Method  . Define a set  where each entry     is a    channel statevector for the   user, i.e.           󰆒 and          . We say       , if selecting a relay using Method  under channel state   yields an outage. The complement of    , denoted as    contains the channel states that result in a successful transm ission. Similarly, we define    and    for Method  . Accordi ng to (6),we note that an outage occurs in Method  when                    (8) To prove Corollary  , all we need to show is that        . Suppose that a channel state vector                  . In this case, for those relays     , we have           . And for th e relays  that are not in   ,          holds by definition. Therefore, the following inequality holds for all relays                          (9) This is equivalent to                      (10) Note that when the above ine quality holds, an outage also occurs with Method  , i.e.       . Therefore, we have        . Next, we show that        . Let us consider a channel sta te vector                  . Then there must be at least one relay  󰆒 that satisfies both     󰆓      and     󰆓       , or     󰆓       󰆓         (11) That is to say                      (12) which means       too. Therefore we have        , or equivalently        . Now that        and        hold simultaneously, it is sufficient to claim that        . That is, both methods yield the same outage probability. Since Method  yie lds the lowest outage probability acc ording to Proposition  , selecting the relay acc ording to (7) is also outage optimal. 󰆢 So far, we have described the way to opti mally select a single relay that minimizes the outage probability for any m obile user that is being scheduled to transmit. A distributed algorithm tha t implements this optimal relay se lection scheme will be introduced in the Appendix. In the remainder of thi s paper, we will focus on transmission scheduling at the user side, which is the main concern of this paper. III . O PTIMAL O UTAGE P ROBABILITY OF R ELAY - ASSISTED N ETWORKS A. Optimality of Greedy Opportunistic Scheduling The key idea of greedy opportunistic scheduling is to allocate airtime to the mobile user with the largest instantaneous channel gain. With the optimal rela y selection method, greedy opport unistic scheduling in a relay assisted network selects the “best” u ser   according to       󰇝               󰇞 (13) where the optimal relay    for the   mobile user is given in (7). The following theorem prove s that greedy opportunistic sche duling yields the lowest outage probability, which is intuitive. Theorem 1 : In multi-access DaF relay-assisted networks with an aggregate relay po wer constraint, greedy user scheduling in (13) is outage optimal. Proof: Denote by 󰇛        󰇜 the user-relay pair selected according to ( 13). Consider another user-relay pair 󰇛   󰇜  󰇛        󰇜 , where the inequalit y means the two equalities     and       do not hold simultaneously. As proved in Corollary  , selecting    according to (7) yields optimal outage probability for the particular  under aggregate relay power constraint. That is   󰇟 󰇛     󰇜 󰇠    󰇟 󰇛   󰇜 󰇠    (14) Meanwhile, the selection in (1) guarantees that  󰇥                  󰇦                    (15) Hence, 󰇛        󰇜 yields lower outage probability than 󰇛     󰇜 . As a result, we have   󰇟 󰇛       󰇜 󰇠    󰇟 󰇛     󰇜 󰇠    󰇟 󰇛   󰇜 󰇠      (16) This completes the proof. 󰆢 3.2. Outage Probability Analysis In this subsection, we compute the optimal outage proba bility achi eved by greedy opportunistic scheduling. By definition, we have     󰇥      󰇝           󰇞     󰇦 (17) To simplify the notations, we define a decision parameter  as follows        󰇝           󰇞      󰇝           󰇞    󰇝             󰇞 (18) Then,   can be written as     󰇟      󰇠  (19) Let           . The probability distribution of   is  󰇟     󰇠                            󰇧   󰇫      󰇬󰇨    (20) Likewise, let             . Since    is independent of   (user- to -relay (U-R) and R-B channels are independent), the probability distribution of   can be calculated as  󰇟     󰇠     󰇟     󰇠     󰇟     󰇠   󰇩        󰇪        󰇧   󰇫      󰇬󰇨      󰇫      󰇬  (21) Since   󰆒  are independent for different  , we can obtain the system outage probability     󰇟      󰇠   󰇣          󰇤        󰇟       󰇠         󰇧    󰇫        󰇬󰇨      󰇫        󰇬     (22) Define SNR by       , where          is the total power needed to transmit one symbol. Let       and    󰇛    󰇜   , where   󰇛  󰇜 . From (22), we can express the outage probability as a function of  ,  and SNR   󰇛     󰇜         󰇧   󰇫     󰇬󰇨      󰇫  󰇛    󰇜     󰇬     (23) It can be seen from (23) that the outage probability decr eases as we incre ase either  or  . In the extreme case when the number of users becom es ver y large, we can obtain the lower bound on the outage probability as follows    󰇛   󰇜    󰇛     󰇜   󰇫   󰇫  󰇛    󰇜    󰇬󰇬      󰇫   󰇫        󰇬󰇬     (24) Note that the lower bound only depends on the parameters related to the relays, including the relay transmission power   and R-B mean channel gains   . It is, however, not dependent on the user-side parameters such as   and   . Due to the optimality of greedy opportunistic scheduling,   is also the lowest outage probability in a DaF relay-assisted network. C. Diversity Order Analysis It is not surprisi ng that   decreases as we increase either  or  in ( 23), as more users or relays yield a higher order of diversity. However, in what follows, we show that diversity order only depends on the relay number  . In other words, applying greedy opportunistic scheduling among a large number of users, i.e. having large  , is immaterial in improving the outage performance at high SNR region. Define diversity order as the negative slope of the outage probability as a func tion of SNR in a log-log plot, i.e.,         󰇛      󰇜  󰇛  󰇜  (25) Noting that  󰇝  󰇞     , as    and    , we can approximate the right hand side (RHS) of (23) in high SNR region by   󰇛    󰇜     󰇛    󰇜                              󰇛    󰇜               (26) When    , (26) can be simplified as     󰇛     󰇜        󰇧  󰇛    󰇜         󰇨               󰇛    󰇜                (27) (27) shows that the diversity order  is         󰇛      󰇜  󰇛  󰇜    (28) which implies that diversity order of outage probability here is dominated by the number of re lays, regardless of the nu mber of user s involved in the greedy opportunistic scheduling. This is in contras t to opportunistic scheduling in conventional wireless systems where a larger number of user s yields a higher diversity order. The result here provides a guideline in the system design: to decrease the outage probability, it is much m ore effective to increase the number of rela ys than to do large scale greedy opportunistic scheduling. Note that the variable  in the above equations is not necessarily the number of users in the cell. It is indeed the num ber of users am ong which opportunistic scheduling is applied, if opportunistic scheduling is performed in a smaller scale. For example, if the users are partitioned into groups and opportunistic sche duling is applied within each group, then this  should be replaced by the group size. IV . F IXED TDMA A CHIEVES F ULL D IVERSITY G AIN It is commonly believed that to achieve the best system reliability, greedy opportunistic scheduling should be applied to select the “best" user to transmit every time. Indeed, this is consistent with our analysis above, which shows that optimal outage probability is obt ained by greedy opportunistic scheduling. However, this optimal transmission reliability comes at the cost of the poor fairness among users. The greedy opportunistic scheduling are strongly biased to schedule the user with the best average U-R channel conditions. On the other hand, fixed TDMA achieves the fairest a irtime allocation among user s. However, it fails to exploit multiuser diversity and is commonly believed to achieve the lowest diversity order. In this section and in contrast to th e common belief , we show that fixed TDMA achi eves the same full diversity order as greedy opportunistic scheduling in relay-assisted networks, as long as the relay is selected properly. A. Diversity order of fixed TDMA The outage probability of fixed TDMA can be calculated by letting    in (23). The   user, for example, transmits with outage probability equal to  󰇫    󰇫   󰇧       󰇛    󰇜   󰇨󰇬󰇬     (29) The outage probability of the system is an average of that of the  individual users. Thus,    󰇛      󰇜          󰇫   󰇧       󰇛    󰇜   󰇨󰇬        (30) In the high SNR region, we can approximate    as    󰇛      󰇜       󰇫   󰇧       󰇛    󰇜   󰇨󰇬              󰇫       󰇛    󰇜   󰇬              (31) which leads to a diversity order of           󰇛      󰇜  󰇛  󰇜    (32) The above analysis shows that full diversity order  that is achieved by greedy opportunistic scheduling is also achievable by fixed TDMA. Considering that the two scheduling schemes are two extremes of all scheduling policies, we can infer that full diversity order can always be achieved in relay-assisted networks, as long as that the relay is selected optimally according to ( 7). This is a good indication that being greedy does not provide much gain in relay-assisted systems. The simple fixed TDMA scheme can achieve good system reliability and user fairness at the sam e time. B. Power gap between fixed TDMA and greedy opportunistic scheduling Although fixed TDMA achieves the same diversity order as greedy opportunistic scheduling, there may exist a power gap between the two scheduling schemes. Here, we quantify this power gap under symmetric channel condition, where        . It will be shown in the next section that this power gap can be closed by introducing limited opportunism to TDMA. With        , the outage probability of fixed TDMA becomes    󰇛    󰇜              󰇛    󰇜              󰇛    󰇜      (33) Meanwhile, the outage probability lower bound in (24) obtained by opportunistic scheduling becomes   󰇛    󰇜               (34) Comparing (33) and (34), we notice tha t there exists a power gap of      dB between fixed TDMA and greedy opportunistic scheduling. Recall that   󰇛  󰇜 is defined in Section III.B as the portion of total powe r allocated to users to transmit one symbol. The power gap diminishes as we allocate more power to the users. In particular, when    , the power gap is  dB. V. R ELAXED -TDMA S CHEDULI NG In this section, we show that the power gap shown in the last paragraph can be closed by a simple relaxed TDMA scheduling scheme. That is, the simple scheme can achieve the opti mal outage probability without suffering the drawbacks of gre edy opportunistic scheduling. In other words, there is no longer a tradeoff between efficiency, fairness, and implementation complexity in relay assisted networks. A. Relaxed-TDMA Scheduling While greedy opportunistic scheduling has full freedom to swap the transmission order of mobile users, fixed TDMA has zero. In between, we define a  -user relaxed-TDMA scheme, where  users are divided into groups of  users, i.e.   groups. Then, fixed TDMA is adopted to allocate different time slots to different groups in a static manner. Within each slot, we are free to choose, among the  users that are pre-assigned to the slot, the one with the highest instant aneous channel gain totransm it. By doing so, we only have small- scale opportunistic scheduling within each group. Note that greedy opportunistic sche duling and fixed TDMA are the special cases with    and    , respectively. The following theorem proves that the outage probability lower b ound in (24) achieved by greedy opportunistic sche duling can also be achieved by introducing only very little opportunism to fixed TDMA, i.e. increasing  from  to  . Theorem 2 : A two-user relaxed-TDMA scheme, i.e.    , achieves the optimal outage probability in (24) at the high SNR region. Proof: For a group of two user s, the outage probability at the high SNR region can be obtained by letting    in (27). The equat ion shows that this outage probability only depends on the R- B channels regardless of which users are in the group. Hence, it is the same for all groups, and consequently is equal to the system outage probability. That is, the outage probability of the two -user relaxed TDMA scheme, denoted by    , is given by        󰇫  󰇛    󰇜      󰇬     (35) Meanwhile, at the high SNR region,   in (24) becomes   󰇛   󰇜   󰇫   󰇫  󰇛    󰇜    󰇬󰇬      󰇫  󰇛    󰇜      󰇬     (36) which is exactly the same as    in (1). This completes the proof. 󰆢 Theorem  proves that the two- user relaxed-TDMA scheme suffers no p erformance loss compared with greedy opportunistic scheduling at high SNR region. At the same time, it largely decreases computational complexity and signaling overhead, as the system only needs to estimate the U- R channels of two users and select the “bet ter" user between the two in each ti me slot. We also note that the proof in Theorem  does not depend on a spec ific way of grouping users. Indeed, the theorem holds regardless of how we group the users. To implem ent the relaxed-TDMA scheme, a fully distributed protocol is presented in the Appendix. B. Relaxed-TDMA enhances fairness Intuitively, the two -user relaxed-TDMA scheme also yields better fairness among the users compared with greedy opportunistic scheduling. This is illustrated in this subsection by the variance of channel access delay and the Jain’s fairness index. 1) Variance of channel access delay: The delay a user experiences between two consecutive transmissions, referred to as channel access delay, is a direct reflection of quality of service received by the users. While average channel access delay is a good indicator of throughput performance, t he variance of channel access delay reflects the dispersion among transmission opportunities perceived by different users. For simplicity, suppose that all users are ho mogeneous, i.e.,   are th e same for all  . By symmetry, each user transmits with probability   in its designated time slot. Let  be the the channel access delay and  be thelength of a relay cycle. Then, the average channel access delay  󰇟  󰇠               󰇛    󰇜     (37) and its second moment is  󰇟   󰇠                󰇛    󰇜               (38) Hence, the delay variance is  󰇟  󰇠   󰇟   󰇠   󰇟  󰇠                    (39) From (37) and ( 39), we can see that while the average channel access delay re mains constant, the variance increases with the increase of group size  . For fixed TDMA where    , delay variance is zero. On the other hand, the largest delayvariance occurs when greedy opport unistic scheduling is adopted, i.e.,    . This implies that al though homogeneous users equally share the wireless resource in the long run, some o f them may temporarily be in severe starvation, which leads to poor short term fairness. In contrast, with    , two-user relaxed-TDMA largely reduces the delay variance and enhances the short term fairnes s among users com pared with greedy opportunistic scheduling. 2) Jain’s Fairness Index : Now, let us remove the hom ogeneity assumption and consider a general case where   ’s can be different for di fferent  . Here, we use Jain’s fairness index (  ) [25] to quantify the fairness of airtime allocation as a result of user scheduling. We show that the fairness improves as the group size  decreases. Suppose that  mobile user s contend for a total airtime of length  . Let   󰇛󰇜 denote the portion of airtime received by the   user during the time period  . The  of an airtime allocation vector 󰇛󰇜 is defined as   󰇛  󰇜   󰇛     󰇛 󰇜 󰇜         󰇛󰇜  (40) For example, when 󰇛󰇜  󰇟  󰇠 ,   󰇛  󰇜     . Jain’s  is continuous and bounded between 󰇟󰇠 . The higher the index, the fairer the airtime allocation. If m obile users equally share the airtime, i.e.,   󰇛 󰇜 ’s are equal, then the fairness index is  . On the contrary, the fairness index value tends to be low if the airtime allocation is in favor of few users. Here, we examine the upper and lower bounds of the Jain’s  of the relaxed-TDMA scheme. In particular, the upper bound occurs when the users in each group equally share the airtime, le ading to    regardless of the group size. Meanwhile,  reaches the lower bound when there exists a dominant user in each group that takes up all the airtim e allocated to the group. The resource allocation v ector becomes  󰇛  󰇜  󰇟       󰆄 󰆈 󰆈 󰆈 󰆅 󰆈 󰆈 󰆈 󰆆  ′       󰆄 󰆅 󰆆   ′  󰇠 . This extremely unfair airti me allocation yields the lowest  given by  󰇛 󰇜        󰇡   󰇢      (41) where      . We can see that the  lower bound is a function of the group size k. It overlaps with the upper bound when    , which corresponds to fixed TDMA. Meanwhile, the minimum  lower bound occurs when    , which corresponds to greedy opportunistic scheduling. In general, the lo wer bounds improves as we decrease  . In particular, by letting    , the two -user relaxed-TDMA guarantees that  is no less than   . The increase in the lower bound is a good indicator that relaxed-TDMA that limits the scale of opportunism helps to enhance user fairness, as it reduces the potential disparity of airtime allocation among users especially when  is lar ge. VI. S IMULATION R ES ULTS In this section, we verify the analysis in the pre vious sections thr ough numerical simulations. In our simulations,     and decoding threshold    . The length of a relay cycle is   and Doppler spread is   . Unless otherwise specified, power is equally allocated between users and the relay set, i.e.    . A. Outage probability We consider a system with  users and  rela ys in Fig.  . The mean channel gains of R-B links are      󰇟  󰇠 and the mean channel gains of U-R links are listed in Table I. Outage probability is plotted against     for both fixed TDMA schedul ing (i.e.,    ) and greedy opportunistic scheduling (i.e.,    ). Both analytical and simulation results are presented. The curves that represent analytical resu lts are calculated according to ( 23). It can be seen that the analytical and sim ulation results are on top of each other. Besides, we observe that fixed TDMA and greedy opportunistic scheduling have the same diversity gain, which validates our clai m that fixed TDMA achieves optimal diversity gain in Section IV.A. Assuming that the channels are symmetric, i.e.        , we show in Fig.  that the power gap between fixed TDMA and greedy opportunistic scheduling unde r different power allocation parameter  . Same as we observe in Fig.  , fixed TDMA achi eves the same diversity order as greed y opportunistic scheduling. When    , there is a  dB power gap. Moreover, the power gap decreases from  dB to  dB as we al locate more power to the users (  change s from  to  ). This verifies our analysis in Section IV.B that the power gap is     󰇡   󰇢 dB. Fig.  depicts the system optimal outage probability as a function of  and  . The mean channel gains are uniformly distributed in 󰇛󰇜 for both U-R and R-B links. We notice that when the number of relays remains unchanged, the diversity orders remains constant, despite that the outage probability decreases as  increases. On the other hand, the diversity order is increased when the number of relays increases from  to  . This validates our analysis that the diversity order depends onlyon the number of relays. In Fig.  , we examine the system outage proba bility of  -user relaxed-TDMA as a function of  . There are  relays and  mobile users in the system, where the R -B mean channel g ains are within 󰇛 󰇜 . Besides,  out of  users are loc ated closer to the relays. Considered as “fortunate" users, their U-R mean channel gains are within 󰇛󰇜 . The other  “unfortunate" users are located at the edge of the cell. Their U -R mean channel gains ar e within 󰇛 󰇜 . We set the group siz e to be   󰇝    󰇞 , re spectively, and the grouping pattern is random. The curves in the figure ar e the average perf ormance of  independent grouping patterns. The figure shows that the two-user relaxed-TDMA scheme significantly decreases the system average outage proba bility compared with fixed TDMA scheduling. In fact, it over laps with the outage probability lower bound at high SNR region, which verifies our analysis in Theorem  . However, the improvement becomes marginal if we further increase thegroup size. B. Fairness improvement of relaxed-TDMA scheduling Fig.  shows the improvement of short term fairness by  -user relaxed-TDMA when users are homogeneous. Here Jain’s fairness index is plot ted against normalized Doppler frequency. The normalized Doppler frequency is Doppler frequency (   ) normalized with respect to the symbol rate (   ). In our case, one unit in X-axis represents the length of      relay cycl es. By rule of thumb, the fairness performance within a sliding window t hat is less than  units can be regarded as the measurement of short term fairness, otherwise it is long te rm fairness. Due to homogeneity of user s, each user has an equal chance to transmit in its designated time slot. Thus,    in the long run regardless of the group size. Nonetheless, the figure shows that short term fairness is enhanced by having a smaller  in the  -user relaxed-TDMA scheme (    ) compared with greedy opportunistic scheduling (    ). This verifies our access delay analysis that relaxed-TDMA enhances short term fairness in Section V.B. When users ar e no longer hom ogeneous, Fig.  compares the fairness performance of greedy opportunistic scheduling and two -user relaxed-TDMA, where the grouping is random. The system model is the same as tha t in Fig.  and SNR is fixed at   . Compared with opportunistic scheduling, two-user relaxed-TDMA enhances long term fairness from about  to  . The improvement is especially significant when the sliding window is short. Intuitively, airtime allocation between the two users in a group is closely related to their U - R channel statistics. Fig.  examines the impact of grouping pattern to fairness. Three kinds of grouping patterns are compared. The “good" grouping pattern puts users with simila r channel g ains in a group.. The “bad" grouping patter n, however, groups users with very distinct channel gains together. The third grouping method is random grouping. We can see that grouping users who are statistically similar obtains m uch better fairnes s. Its Jain’s fairness index is about   higher than that of random grouping. VII. C ONCLUSIONS In this paper, we found that the conventional tradeoff between transm ission reliability and user fairness is not necessary in relay-assisted systems. With optimal relay selection, fixed TDMA achieves the optimal diversity gain that is typically only obtainable by greedy opportunistic scheduling in conventional wireless systems. In addition, by introducing limited opportunism into fixed TDMA, a simple two-user relaxed-TDMA scheduling scheme asymptotically achieves the optimal outage probability in high SNR re gion. Compared with greedy opportunistic scheduling, the two-user relaxed-TDMA scheme not only achieves the same transmission reliability, but also significantly improves user -level fairness and decreases computational complexity. As such, we can safely enjoy the advantages of both greedy opportunistic scheduling and fixed TDMA at the sam e time. For practical implem entations, we have proposed a simple and fully distributed algorithm for relaxed-TDMA scheduling. VIII. A PPENDIX D ISTRIBUTED RELAXED -TDMA S CHEDULING P ROTOCOL Without loss of generality, consider a particular time slot that is assigned to a group of  users, say user  to user  . The distributed relaxed-TDMA scheduling protocol operates in the following steps, which is also illustrated in Fig.  . 1). The  users and the BS send pilot signals so that each relay can estimate its local channel information based on the pilots received. The   relay, for instance, has knowledge about   for        and    . Then, the   relay selects the user    according to        󰇝          󰇞 (42) 2). Relay  attempts to access the wireless m edium by setting a backoff timer according to a decreasing function of parameter   , given by    󰇝              󰇞 (43) 3). Then, the backoff ti mer counts down in a fixed time interval unti l it reaches zeros. Therefore, the relay   with the largest    will expire its backoff timer first. 4). When its backoff timer counts down to zero, relay   sends out a RTS packet, which contains the ID of its selected user     . Upon hearing this RTS packet, other relays will clear their timers and keep silent until the next time slot. Onthe other hand, the  mobile user s will compare their IDs to the received RTS packet. The selected user will identif y itself and transmit its data packet, while the other    users will keep silent. 5). Upon this point, the best user-relay pair has been selected exactly according to ( 13) in a distributed fashion. 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Hawe. “A Quantitative Measure of Fairness and Discrimination For Resource Allocation in Shared Com puter Systems," Technical Report TR-301, DEC Research Report , Sep 1984. TABLE I M EAN C HANNEL G AINS OF THE U-R L INKS IN F IG .                                                             Fig.1. Analytical and sim ulation results of outage proba bility Fig.2. Power gap between fix ed TDMA and greedy opportunisti c scheduling Fig.3. Outage probability of greedy opportunistic scheduli ng as a function of  and  Fig.4. Outage probability of re laxed-TDMA as a function of group size  (    ) Fig.5. relaxed-TDMA im proves short term fairness whe n users are homogeneous Fig.6. Fairness improvem ent of relaxed-TDMA under general U -R channels Fig.7. Impact of grouping pa ttern of relaxed-TDMA to fair ness Fig.8. Distributed implem entation of relaxed-TDMA sche duling