Role of a Relay in Bursty Networks with Correlated Transmissions

Transcription

1 206 IEEE Internatial Symposium Informati Theory Role of a in Bursty etworks with Correlated Transmissis Sunghyun im ETRI, South orea koishkim@etrirekr Soheil Mohajer University of Minnesota, USA soheil@umnedu Changho Suh AIST, South orea chsuh@kaistackr Abstract We explore the role of a relay in multiuser networks where some physical perturbati shared around the users may generate data traffic for them simultaneously, hence cause their transmissi patterns to be correlated We investigate how the gain from the help of a relay varies with correlatis across the users transmissi patterns in a bursty multiple access channel where the users send signals intermittently As our main results, we show that in most cases a relay can provide a greater degreesof-freedom ) gain when the users transmissi patterns are more correlated Furthermore, we demstrate that the gain can scale with the number of users I ITRODUCTIO s have been csidered unable to provide gains in standard informati-theoretic channels where transmitters are usually assumed to send signals at all times [] But recent studies have found that relays can provide gains in bursty channels where transmitters send signals intermittently [2], [3] Further fings therein shed new light the significant role of relays in various bursty channels, making it look promising to employ relays in wireless networks The main results of prior work [2] well emphasize practical advantages of employing a relay in bursty multiple access channels MACs) One is to improve performance in emerging networks, such as device-to-device systems in which mobile devices directly exchange data with little help of base statis When multiple devices cvey bursts of data to a device in such systems, as shown in [2], the assistance of a relay can be useful to achieve higher data throughput, as the relay can provide gains scalable with extra relay antennas, and to enable collisi-free communicati Another advantage is to simplify random access protocols to reduce ctrol signaling overhead When multiple users wish to deliver data to a comm destinati, some protocols are needed to manage signal collisis for reliable data delivery It has been demstrated in [2] that a relay can take a burden the users when it comes to coping with such collisis It turns out that the relay can resolve the collisis behalf of the users Hence, the users are allowed to send signals at random intervals without extra effort such as retransmissis of collided signals In this work, we set out to explore the practical aspects of employing a relay in detail To that end, we look into bursty MACs where endencies across the users intermittent data availabilities cause correlated transmissis across the users to occur Csider a sensor network in which multiple sensors spread gather temperature measurements and cvey them to a central hub which computes the average Alternatively, csider a safety network in which nearby vehicles equipped with sensors detect a possible risk and share the informati to prevent the accident to happen In both, some physical perturbati around objects in close proximity may cause the objects to collect and exchange bursts of data simultaneously A natural questi that arises in this ctext is: will employing a relay be still useful when multiple users tend to send signals simultaneously, and causing severe collisis? To answer the questi, we csider two extreme yet representative cases In e case, the users data availabilities are fully endent; hence all users send signals simultaneously In the other case, the users data availabilities are inendent; hence a user sends signals regardless of the others For each case, we measure the gain from the assistance of a relay And we compare the two gains to see when employing a relay can be more beneficial As our main ctributi, we provide insight into how the benefit of employing a relay varies according to correlatis across the users bursty transmissi patterns The most interesting fings are perhaps those in the case where the relay has sufficiently many antennas to achieve the maximum : The gain from the assistance of a relay is greater when transmissi patterns across the users are more correlated The gain can scale with the number of users We note, however, that observatis in other cases where the relay does not have enough antennas suggest that employing a relay could sometimes be more beneficial when transmissi patterns across the users are less correlated The antenna settings in such cases represent scenarios in which the relay has very limited numbers of antennas, so it fail to assist well either the transmitters or the receiver More technical details and discussis to follow will make our fings clear Related work: Amg many studies relay networks, the most related are [4] and [5] We obtain our main results by extending noisy network coding for multimessage multicast networks [4] in which relays use compress-forward strategies [5] To the best of our knowledge, there has been little work de multiuser networks where correlatis across the users transmissi patterns are taken into account ote that it is the availabilities of data at a given time that may be correlated across the users The messages of the users are assumed inendent In this work, by any terms regarding correlatis or endencies, we mean the data availabilities across the users, not the messages /6/$ IEEE 2639

2 206 IEEE Internatial Symposium Informati Theory II MODEL Fig describes the -user bursty MIMO Gaussian MAC with a relay The transmitters, receiver and relay have M, and L antennas, respectively Transmitter k wishes to reliably deliver a message W k to the receiver, k =, 2,, Let X kt C M be transmitter k s encoded signal and X Rt C L be the relay s encoded signal at time t The encoded signals are power-cstrained: E[ Xkt 2 ] P and E[ XRt 2 ] P Traffic states S kt are assumed to follow Bernp) to govern bursty transmissis over time Joint distributi 2 PS, S 2,, S k ) captures the bursty transmissi patterns of the users Unlike the transmitters, the relay is not subject to bursty transmissis Let us note the ratiale beh our modeling This work is motivated by sensor networks where transmitters are simple devices, thus less capable; they can neither have a large buffer to store ample data, nor employ advanced scheduling Intermittent data traffic forces the users to send signals whenever they have available data, leading to bursty behaviors In ctrast, a relay can be equipped with rich capabilities such as a large buffer and channel state sensing; it can store sufficient past received signals and use them later according to channel states for better assistance Hence, unlike the transmitters, the relay can send signals at all times As we csider intermittent data traffic to be a cause of bursty transmissis, it might be more practically relevant to model such burstiness in higher layers of the communicati protocol stack However, to simplify the model greatly while capturing the bursty nature to some extent, we incorporate random states into the physical channel Additive noise terms Z t at the receiver and Z Rt at the relay are assumed to be inendent, distributed according to C 0, I ) and C 0, I L ), and iid over time Let Y t C be the received signal at the receiver and Y Rt C L be the received signal at the relay at time t Y t = k H k S kt X kt + H R X Rt + Z t, Y Rt = H Rk S kt X kt + Z Rt k The matrices H k and H Rk describe the time-invariant channels from transmitter k to the receiver and to the relay, respectively The matrix H R describes the time-invariant channel from the relay to the receiver All channel matrices are assumed to be full-rank We assume current traffic states are available at the receiver and the relay Each transmitter knows its current traffic state, as it has access to the availability of data for transmissi Transmitter k encodes its signal at time t based its own message and its own current traffic state; we assume uncoordinated traffic states across all transmitters, which means each transmitter has access to its own traffic state ly The relay encodes its signal at time t based its past received signals, and both past and current traffic states of all transmitters 2 We let joint distributis capture the users transmissi patterns that stem from correlatis across the data availabilities, while for simplicity we assume marginal distributis to be identical W W 2 W Tx X t S t X 2t S 2t X t S t Y Rt Z Rt S t,,s t X Rt Y t Z t S t,,s t Fig -user bursty MIMO Gaussian MAC with a relay Finally, we define the regi D := {d, d 2,, d ) : R R, R 2,, R ) CP ) and d k = lim k P log P }, where CP ) is the capacity regi with power cstraint P III MAI RESULTS For ccreteness, we first state a theorem of prior work [2] that we use to obtain our results of this work The theorem is obtained under the assumpti that the traffic states follow Bernp) over time and are inendent across the users Theorem : The regi of the -user bursty MIMO Gaussian MAC with a relay is characterized as follows S B S i) min im, + L), d k min, S k S B S i) min im + L, ) where S {,, } and B S i) := ) S i p i p) S i Proof: We give a sketch of the proof See [2] for details Extending noisy network coding for multimessage multicast networks [4] can achieve the cut-set bound The relay takes a receive-forward strategy The distributi of the traffic states, reflected in B S i), specifies how often a certain number of transmitters are active, which determines the number of s that can be cveyed from them to the receiver In this paper, we focus additive sum gains, and the differences between the sum with a relay and that without a relay, to investigate the number of additial obtained with the assistance of a relay We now state our main theorem Theorem 2: The additive sum gain obtained by adding a relay in the -user bursty MIMO Gaussian MAC is PA) min A M, + L), = min A Ω PA) min A M + L, ) A Ω A Ω PA) min A M, ), where Ω is the set that includes all subsets of {, 2,, }, A is a set that icates which transmitters are active, and PA) is a joint distributi that describes the probabilities of the transmitters icated by A being active Proof: See Secti V-A Ŵ Ŵ 2 Ŵ 2640

3 206 IEEE Internatial Symposium Informati Theory e peak p Fig 2 A relay provides greater gains when the users data availabilities are endent Antenna setting:, M,, L) = 2,,, ) Although it might be comprehensive to explore how gains from employing a relay change with varying levels of correlatis across the users data availabilities, we csider two extreme ends of the spectrum for simplicity On the e end, the users data are available in a fully endent manner; thus all transmitters are either active or inactive at the same time On the other end, they are available in an inendent manner; thus a transmitter is active regardless of the others Corollary : The additive sum gains obtained by adding a relay in the -user bursty MIMO Gaussian MAC with fully endent and inendent data availabilities are [ ] p minm, L), = min, ) p) minl, ) B i) minim, L), i= M + = min M Proof: See Secti V-A B i) minl, im) IV RELAY GAIS 2) The role of a relay and its functiality may vary ending the antenna settings In this secti, we examine three cases A L max M, ): s with enough antennas The cditi implies that the relay can get additial signals that the receiver needs to resolve the worst-case collisis that occur when all transmitters are active, and also forward the maximum number of signals that the receiver can get at a time For this case, the relay can help achieving the maximum for a given bursty MAC: minpm, ), that is collisi-free in the low traffic regime and saturated in the high traffic regime Intuitively, the relay receives additial signals when too many transmitters are active, and later forwards them when ly a few are active, to achieve the maximum We have a few interesting fings, illustrated in Figs 2 and 3 The first fing is that the gain obtained from the assistance of a relay is greater with endent data availabilities than that with inendent data availabilities: >, p 0, ) See Secti V-B for the proof For this case, in the presence of a relay, the same sum minpm, ) can be achieved regardless of endencies across the users data availabilities Fig 3 The growing peak gains with both endent and inendent data availabilities, and the expanding gap between them Antenna settings: M = = and L Let us see what happens in its absence With endent data availabilities, too many s are sent simultaneously compared to the number of s that the receiver can get at a time Without a relay, there would be a big loss due to the severe collisis With inendent data availabilities, however, such severe collisis occur less often given the same level of data traffic Hence, there would be a relatively smaller loss The absence of a relay costs bursty MACs with endent data availabilities more, that is, its presence is more beneficial, ering greater gains The other fing is regarding the variati of the relay gains with respect to the number of users To see this variati, and for the sake of simplicity, we assume M = = Also, we focus two specific values of the relay gains: := max p, := max p We justify the use of these peak values as a fair means of comparing the relay gains for two reass As previously shown, the relay gain is greater with endent data availabilities for all p Moreover, as will be shown in Secti V-C, both relay gains are maximized at the same value of p = 3 M We obtain the peak relay gains as a functi of under the assumpti M = = : =, peak = ) One can easily verify that both peak gains grow as increases, and cverge to and e, respectively This fing shows that the beneficial role of a relay does not deteriorate as the number of users increases To the ctrary, the presence of a relay is more advantageous with more users regardless of endencies across the users data availabilities More importantly, it turns out that the difference between both peak gains peak ) grows as increases This fing shows that the difference in the amounts of additial attained with the aid of a relay expands as the number of users increases Csidering all fings, we can cclude that employing a relay in bursty MACs is more beneficial with endencies across the users data availabilities, and more favorable when more users are involved We note that the previous fings are valid for the case where the relay has sufficiently many antennas to help achieving the maximum In the rest of this secti, we examine 3 We assume M > Otherwise, there is no point in discussing relay gains, as the receiver is able to decode all sent s instantaneously 264

4 206 IEEE Internatial Symposium Informati Theory forwarded many wasted forwards a) correlated transmissis forwarded s few wasted forwards b) uncorrelated transmissis Fig 4 Antenna setting, M,, L) = 4, 2, 7, ): M L Limited capability of the relay in transmissi mode stands out with endent user data availabilities See Fig 6a) reserves reserved many lost a) correlated transmissis reserves reserved few lost b) uncorrelated transmissis Fig 5 Antenna setting, M,, L) = 4, 2,, ): L M Limited capability of the relay in recepti mode stands out with endent user data availabilities See Fig 6b) other cases where the relay does not have enough number of antennas to help achieving the maximum, and hence ers limited gains B M L < A relay ers gains operating in two different modes It receives additial s in recepti mode, and forwards them in transmissi mode For this case, particularly when L, the relay may reveal its drawback in transmissi mode In this mode, the relay forwards additial s to the receiver, to provide gains by utilizing the receive antennas otherwise unused In that sense, L icates its limited capability to utilize all receive antennas The limitati stands out with endent data availabilities, and makes the relay less beneficial with such endencies With endent data availabilities, all transmitters are either active or inactive When they are inactive, the relay operates in transmissi mode to provide a gain However, the relay can utilize a very small fracti of receive antennas due to its drawback L, leaving a large fracti of them wasted Fig 4a)) With inendent data availabilities, the other hand, these undesired incidents do not occur as often, given the same data traffic level Fig 4b)) Fig 6a) illustrates the relay providing greater gains with inendent data availabilities in high p regimes This is because in those regimes, the relay is guaranteed to receive enough additial s from the transmitters, hence the number of additial s it can forward in transmissi mode is what determines the amount of gain it ers In short, the limitati in transmissi mode L ), which appears in high p regimes, affects the relay s capability more adversely with endent data availabilities, hence makes the relay less beneficial with correlated user transmissis C L < M For this case, particularly when L M, the relay may reveal its drawback in recepti mode In this mode, the relay receives and reserves the additial s from active transmitters otherwise lost, to provide gains by forwarding them later to the receiver In that sense, L M icates its limited capability to reserve all surplus s The limitati stands out with endent data availabilities, and makes the relay less beneficial with such endencies With endent data availabilities, when all transmitters are active, the relay operates in recepti mode to provide a 084 p 06 p a), M,, L) =4, 2, 7, ) b), M,, L) =4, 2,, ) Fig 6 gains with inendent user data availabilities are greater than those with endent user data availabilities: when M L <, with high traffic left) and when L < M, with low traffic right) gain However, the relay can reserve a very small fracti of surplus s due to its drawback L M, leaving a large fracti of them lost Fig 5a)) With inendent data availabilities, the other hand, these undesired incidents do not occur as often, given the same data traffic level Fig 5b)) Fig 6b) illustrates the relay providing greater gains with inendent data availabilities in low p regimes This is because in those regimes, the relay is guaranteed to f enough idle moments of the transmitters to forward additial s to the receiver, hence the number of additial s it can receive in recepti mode is what determines the amount of gain it ers In short, the limitati in recepti mode L M ), which appears in low p regimes, affects the relay s capability more adversely with endent data availabilities, hence makes the relay less beneficial with correlated user transmissis We have not examined the case L < min M, ) The cditi implies that the relay s limited capabilities may be revealed in either recepti or transmissi mode, leading to its worse performance with endent data availabilities Regimes in which it underperforms end the antenna settings: if L then it has limitatis in transmissi mode hence underperforms in high p regimes, and if L M then in low p regimes V PROOFS A Proof of Theorem 2 and Corollary Proof of Theorem 2: As briefly noted in the proof sketch of Theorem, we extend noisy network coding for multimessage multicast networks [4] to achieve the cut-set bound Unlike Theorem, to csider the case where endencies across the users transmissis exist, we replace the distributi B S i) in Theorem which represents the users mutually inendent transmissis by the distributi PA) in Theorem 2 which can describe the users endent transmissis as well 2642

5 206 IEEE Internatial Symposium Informati Theory Proof of Corollary : To get, we set the distributi PA) in Theorem 2 as p if A = Ω and p if A = By some computati, for M we get = 0, and for M > we get the claimed gain ) To get, we set the distributi PA) in Theorem 2 as B i) = ) i p i p) i By some computati, we get the claimed gain 2) B Proof of > 0 With L M and L, from Corollary and some manipulati, we get = p + B i) minim, ) We can verify that > 0 by showing the secd term is always greater than p: B i) minim, ) = B i) minim, ) = p j=0 i= B j) min M, j + ) > p The secd equality holds since B i) = p i B i ) and by the change of variables j = i The inequality holds since M > and j+ > for 0 j < C Proof of increasing, peak With L M and L, from Corollary and some manipulati, we get = min pm ), p) ), = minpm, ) B i) minim, ) It is straightforward to verify that is maximized at p = M To verify that is also maximized at p, we csider two cases: 0 < p < M and M p < For 0 < p < M, we get = B i)im ), i= M + where the equality holds since B i) im = pm By taking the derivative of the equality, we can verify that is increasing = i= M + B i) i p)im ) 0, p p) where the inequality holds since if 0 < p < M and i M +, then i p 0 and im 0 For M p <, similarly we can verify that is decreasing Both and increase for 0 < p < p and decrease for p p <, thus maximized at p = M With M = = assumed, from Corollary, we get ) =, peak = We can readily verify that grows as increases We can also verify that grows as increases by showing that the ratio of with + users to that + )+ ) = with users is greater than e: ) + > ) + + ) ) ) =, ) + where the inequality holds due to Bernoulli s inequality Thus, both and peak grow as increases Similarly, by applying Bernoulli s inequality to the forward difference of sequence peak, we can verify that peak grows as increases VI COCLUSIO We investigated the role of a relay in correlated bursty MACs where endencies across the users intermittent data availabilities lead to correlated transmissis across the users We showed that the relay in most cases can provide greater gains with the endencies Also, we demstrated that the gap between the gain with correlated transmissis and that with uncorrelated es can grow with more users We found, however, that in some rare cases where the relay has very few antennas, severe collisis from the endencies can make the relay er less gains Csidering all fings, we cclude that in general the relay s assistance is more beneficial with correlated transmissis across the users ACOWLEDGMET This work was supported by an Institute for Informati & communicatis Technology Promoti IITP) grant funded by the orean government MSIP) o R , Development of 5G Mobile Communicati Technologies for Hyper-cnected Smart Services) and the orea Space Launch Vehicle SLV-II) program, funded by the Ministry of Science, ICT and Future Planning MSIP) of the orean Government REFERECES [] V R Cadambe and S A Jafar, Degrees of freedom of wireless networks with relays, feedback, cooperati, and full duplex operati, IEEE Trans Inf Theory, vol 55, no 5, pp , May 2009 [2] S im and C Suh, Degrees of freedom of bursty multiple access channels with a relay, 53rd Annual Allert Cference Communicait, Ctrol, and Computing, Oct 205 [3] S im, I-H Wang, and C Suh, A relay can increase degrees of freedom in bursty interference networks, IEEE Int Symp Inf Theory, June 205 [4] S H Lim, Y-H im, A El Gamal, and S-Y Chung, oisy network coding, IEEE Trans Inf Theory, vol 57, no 5, pp , May 20 [5] S Avestimehr, S Diggavi, and D Tse, Wireless network informati flow: A deterministic approach, IEEE Trans Inf Theory, vol 57, no 4, pp , Apr