Probabilistic Handshake in All-to-all Broadcast Coded Slotted ALOHA

Transcription

1 Probabilistic Hanshake in All-to-all Broacast Coe Slotte ALOHA Mikhail Ivanov, Petar Popovski, Frerik Brännström, Alexanre Graell i Amat, an Čeomir Stefanović Department of Signals an Systems, Chalmers University of Technology, Gothenburg, Sween Department of Electronic Systems, Aalborg University, Aalborg, Denmark {mikhail.ivanov, frerik.brannstrom, alexanre.graell}@chalmers.se, {petarp, cs}@es.aau.k arxiv: v1 [cs.it] 17 Apr 2015 Abstract We propose a probabilistic hanshake mechanism for all-to-all broacast coe slotte ALOHA. We consier a fully connecte network where each user acts as both transmitter an receiver in a half-uplex moe. Users attempt to exchange messages with each other an to establish one-to-one hanshakes, in the sense that each user ecies whether its packet was successfully receive by the other users: After performing ecoing, each user estimates in which slots the resolve users transmitte their packets an, base on that, ecies if these users successfully receive its packet. The simulation results show that the propose hanshake algorithm allows the users to reliably perform the hanshake. The paper also provies some analytical bouns on the performance of the propose algorithm which are in goo agreement with the simulation results. I. INTRODUCTION Vehicular communications (VCs) is presently one of the most challenging problems of communication engineering. Its eployment will enable numerous applications, such as intelligent transportation systems, autonomous riving, an, most importantly, traffic safety. VCs entails a number of challenges, such as all-to-all communication, high mobility networks with rapily changing topologies an a large number of users, an poor channel quality. These challenges require new ieas an esigns at the physical an the meium access control (MAC) layers. The main requirements for VCs are high reliability an low latency. Furthermore, the aforementione challenges prohibit the classical use of acknowlegements in the form of aitional signaling. A novel MAC protocol calle all-to-all broacast coe slotte ALOHA (B-CSA) was propose by the authors in [1], which was shown to be able to satisfy the reliability an latency requirements in rough conitions of vehicular networks uner a set of iealize assumptions, such as perfect interference cancellation. Originally propose for a unicast scenario, coe slotte ALOHA (CSA) can provie large throughputs close to those of coorinate schemes [2], [3]. Different versions of CSA have been propose (see [4] for the most recent review). All of them share a slotte structure borrowe from the original slotte ALOHA [5] an the use of This research was supporte by the Sweish Research Council, Sween, uner Grant No , the Ericsson s Research Founation, Sween, Chalmers Antenna Systems Excellence Center in the project Antenna Systems for V2X Communication, an the Danish Council for Inepenent Research, uner Grants No an No successive interference cancellation. The contening users introuce reunancy by encoing their messages into multiple packets, which are transmitte in ranomly chosen slots. In the unicast scenario, the base station (BS) buffers the receive signal, ecoes the packets from the slots with no collision an attempts to reconstruct the packets in collision exploiting the introuce reunancy. A packet that is reconstructe is subtracte from the buffere signal an the BS procees with another ecoing roun. In contrast to classical CSA, where a BS is the intene recipient of the messages, in B-CSA each user acts as both receiver an transmitter. Every user is equippe with a halfuplex transceiver, so that a user cannot receive packets in the slots it uses for transmission. This can be moele as a packet erasure channel [6] an it affects the esign an the performance analysis as compare to classical CSA. Whereas B-CSA can provie high reliability, the rear communication failure events may be extremely costly in safety applications. Since proviing error-free communication uner the escribe conitions of VCs is not possible, one may attempt to etect communication failure events to use this information in the application level. For instance, consier the scenario where two users A an B are heaing towars each other. If A receives a message from B an obtains the information that B faile to receive its message, A can take extra precautions to avoi collision with B. In this paper, we propose an algorithm to obtain this information base on the by-prouct of ecoing, i.e., no extra signaling is use by the users. In particular, A uses the knowlege of B s transmissions to etect that B faile to resolve A. The problem stuie in this paper resembles the one of hanshake use for establishing connections in connectionoriente protocols. In transmission control protocol (TCP), a three-way hanshake is use by a pair of users to exchange their messages an to acknowlege that the messages were receive [7]. If the messages in the TCP level are exchange, then the hanshake is always performe successfully. In the propose algorithm, however, the ecision about successful hanshake can be in error, which inicates its probabilistic nature.

2 users A C B D E F slots Fig. 1: Users transmissions in a B-CSA system within one frame. Shae rectangles represent transmitte packets. (a) Original unicast graph G. II. SYSTEM MODEL In this section, we first escribe how encoing an ecoing are performe in B-CSA. Base on that, we escribe the propose hanshake algorithm. A. Coe Slotte ALOHA We consier a fully connecte network with m users that want to communicate between each other over a share meium. We focus on the exchange of cooperative awareness messages (CAMs) [8] use for safety applications, which are transmitte perioically by each user. The transmission perio is calle frame an it is ivie into n slots of equal uration. Users are assume to be frame-synchronize by means of Global Positioning System (GPS). Each user maps its message to a physical layer packet an repeats it l times (l is a ranom number chosen base on a preefine istribution) in ranomly chosen slots, as shown in Fig. 1 for a system with 6 users an 7 slots. Such a user is calle a egree-l user. Every packet contains pointers to its copies, so that, once a packet is successfully ecoe, full information about the location of the copies is available. Uner the assumptions escribe in the following, the system can be analyze using the theory of coes on graphs on the binary erasure channel (BEC). Each user correspons to a variable noe (VN) in the bipartite graph an represents a repetition coe, whereas slots correspon to check noes (CNs) an can be seen as single parity-check coes. In the following, users an VNs are use interchangeably. An ege connects the jth VN to the ith CN if the jth user transmits in the ith slot. For the example in Fig. 1, the corresponing bipartite graph is shown in Fig. 2(a). A bipartite graph is efine as G = {V,C,E}, where V, C, an E represent the sets of VNs, CNs, an eges, respectively. The performance of the system epens on the istribution that users use to choose the egree l or, using graph terminology, on the VN egree istribution q λ(x) = λ l x l, (1) l=0 where x is a ummy variable,λ l is the probability of choosing egreel, anq is the maximum egree, which is often boune ue to implementation constraints (b) Inuce graph G A for user A. G A (B) = 1. (c) Intermeiate graph A reconstructe by user A an slots with resiual interference (marke with gray). () Intermeiate graph A reconstructe by user A excluing slots with resiual interference. (e) Graph A reconstructe by user A (incluing user A s slots). A C B D E F (f) Inuce graph A B for user B base on A. A B (A) = 1. (g) Inuce graph G B for user B. G B (A) = 0. Fig. 2: The graph evolution over the course of the hanshake algorithm. Circles represent users (VNs) an squares represent slots (CNs). The example correspons to the outcome g = [1, 1, 0] (see Section II-C).

3 Users buffer the receive signal whenever they are not transmitting. The ifference between classical CSA an B- CSA is illustrate in the example of Fig. 2. The entire graph G is available to the BS in CSA (Fig. 2(a)). For a generic user A in B-CSA, the part of the graph that correspons to user A s transmissions is not available to it ue to the half-uplex moe. Thus, this part of the graph shown with gray in Fig. 2(b). The available part of the graph is calle the inuce graph an is enote by G A. The receive signal buffere by user A in slot i is y i = j U i h i,j a j, (2) where U i U is the set of users that transmit in the ith slot, U is the set of all users, a j is a packet of the jth user in U i, an h i,j > 0 is the channel coefficient. A slot is calle a singleton slot if it contains only one packet. If it contains more packets, we say that a collision occurs. When ecoing, user A first ecoes the packets in singleton slots an obtains the location of their copies. Using ata-aie methos, the channel coefficients corresponing to the copies are then estimate. After subtracting the interference cause by the ientifie copies, ecoing procees until no further singleton slots are foun. The performance parameters of B-CSA are efine as follows. The channel loa is efine as g = m/n. The average number of users that are not successfully resolve by user A, terme unresolve users, is enote by w. The reliability is measure by means of the average packet loss rate (PLR), p = w/(m 1), which is the probability of a user to be unresolve by user A. B. Probabilistic Hanshake One of the main ifferences of CSA compare to actual coes on graphs is that, in CSA, the graph is not known to the ecoer a priori an the ecoer reconstructs it while ecoing. We use this reconstructe graph to perform the hanshake. Without loss of generality, we concentrate on the hanshake between users A an B. We first introuce the necessary notation to escribe the propose hanshake algorithm. Given a particular realization of the graph G with m VNs an n CNs generate ranomly using the istribution λ(x), the reconstruction of the graph G obtaine by user A after ecoing is enote by A. We recall that the inuce graph for user A is enote by G A. With a slight abuse of notation, if user B is resolvable by user A base on G A, we write G A (B) = 1 an we write G A (B) = 0 otherwise. Using this notation, the PLR can be written as p = Pr{G A (B) = 0}. A B enotes the graph that user A obtains after removing user B s slots from the reconstructe graph A. Hence, we write A B (A) = 1 if user A conclues that user B resolves it using the reconstructe graph A. For the example in Fig. 2, user A uses the inuce graph G A shown in Fig. 2(b) for ecoing. However, user A may not be able to fully reconstruct G. In fact, user A can only resolve users B an E. Users D an F cannot be resolve because they form a so-calle stopping set, a harmful graph structure that makes ecoing fail. A stopping set is a subset of VNs of non-zero egrees S V where all neighboring CNs of S are connecte to S at least twice [9]. After ecoing, user A reconstructs the graph A shown in Fig. 2(c). Aitionally, user A obtains the knowlege that slots 5 an 7 belong to a stopping set. It is worth noting that this is a stopping set from user A s perspective an not necessarily a stopping set in the original graphg. Nonetheless, user A assumes that these slots cannot be use for ecoing by any other user. Therefore, user A removes these slots, as well as all the eges connecte to them, to obtain the grapha shown in Fig. 2() (the remove slots an eges are shae). As the the last step to reconstruct G, user A as itself as a VN to the graph A an connects it to the corresponing CNs. The reconstruction of the graph A is shown in Fig. 2(e). Since user A oes not know who exactly transmitte in slots 1 3, these slots are shown with gray. User A uses this graph to run the ecoing on behalf of other users, e.g., Fig. 2(f) shows the graph A B that user A uses for ecoing on behalf of user B. In this case, A B (A) = 1. However, in reality user B uses G B for ecoing an its true outcome is G B (A) = 0. Therefore, user A makes an erroneous ecision about user B s awareness of user A, which happens ue to partial knowlege about the slots user A uses for transmission. C. Hanshake Outcomes If user A is in a stopping set containe in the original graph G, then it will not be resolvable by any other user in the network. User A has no means to learn about this since this information is containe in the slots that it uses for transmission. Hence, user A can never be sure about its successful hanshake ecision. To escribe the possible outcomes of the hanshake algorithm an analyze their probabilities, we introuce the vector g = [G A (B), A B (A), G B (A)]. If user A fails to resolve user B, i.e., G A (B) = 0, with a slight abuse of notation we say that A B (A) = x, meaning that A cannot perform a hanshake with B. All possible outcomes with the corresponing probabilities are summarize in Table I. In the table, p 1 is the probability that user A successfully etects communication failure at user B s sie. p 2 is the probability that user A fails to etect communication failure at user B s sie an erroneously assumes that user B successfully receives its packet. p 5 is the probability of correct hanshake.p 3 an p 4 are auxiliary probabilities, where p 3 = Pr{G B (A) = 0,G A (B) = 0} is the probability that users A anbo resolve each other simultaneously. The sump 3 +p 4 equals the probability that user A oes not resolve user B, i.e., p 3 +p 4 = Pr{G A (B) = 0} = p. Interestingly, the outcome g = [1, 0, 1] can not occur, which is formally proven in the following theorem. Theorem 1. Pr{g = [1, 0, 1]} = 0. Proof: G A (B) = 1 an A B (A) = 0 imply that user A is in a stopping set S of the graph A B. This stopping set S has to be present in the graph G B as well since A B is a subgraph of G B. Hence, G B (A) = 0, which completes the proof.

4 TABLE I: Possible outcomes of the hanshake algorithm. g = [G A (B), A B (A), G B (A)]. G A (B) A B (A) G B (A) Pr{g} Event p 1 Failure etecte p 2 False hanshake 0 x 0 p 3 Auxiliary 0 x 1 p 4 Auxiliary p 5 Successful hanshake A. Inuce Distribution III. PERFORMANCE ANALYSIS The performance of CSA exhibits a threshol behavior, i.e., all users are successfully resolve if the channel loa is below a certain threshol value when n. The threshol epens only on the egree istribution an is obtaine via ensity evolution [10]. A finite number of slots, however, gives rise to an error floor in the PLR performance. In [1] it was shown that the error floor can be accurately preicte base on the inuce istribution observe by the receiver. The inuce istribution for a egree-k receiver can be expresse similarly to (1), where min{q,k+} λ (k) = l= ( n k )( k ) l ( n l) λ l (3) is the fraction of users of egree as observe by user A if it chooses egree k. We efine the PLR for a egree- user as observe by a egree-k receiver as p (k) = w(k) m (k) = w(k) mλ (k), (4) where m (k) an w (k) are the average number of all an unresolve egree- users for a egree-k receiver, respectively. For egree-0 users, p (k) 0 = 1 for all k. B. Analytical Results In this section, we are intereste in escribing the probabilities p 1 an p 2. First, we observe that p 1 +p 2 +p 3 = Pr{G B (A) = 0} = p. (5) From (5), we can immeiately write that p 1 +p 2 p. (6) In the following, we tighten the boun in (6). The probability p 3 can be written as p 3 = Pr{G A (B) = 0}Pr{G B (A) = 0 G A (B) = 0}. (7) The fact that the inuce graphs for all users arise from the same original graphg gives rise to epenency between users performance. This is expresse as the conitional probability in the right-han sie of (7). Examining (7), we conjecture that p 3 p 2. (8) For the asymptotic case, when n, it is easy to show that p 3 = p 2 since the probability for users A an B to use the same slots for transmission is zero. For finite frame lengths, the rationale behin this conjecture is as follows. Let us take a closer look at the conitional probability in (7). We conjecture that Pr{G B (A) = 0 G A (B) = 0} Pr{G B (A) = 0}. (9) Using the law of total probability, we can write Pr{G B (A) = 0 G A (B) = 0}Pr{G A (B) = 0} +Pr{G B (A) = 0 G A (B) = 1}Pr{G A (B) = 1} =Pr{G B (A) = 0}. Exploiting Pr{G B (A) = 0} = Pr{G A (B) = 0} = p gives Pr{G B (A) = 0 G A (B) = 1} = p 1 p (1 Pr{G B(A) = 0 G A (B) = 0}). (10) Therefore, showing (9) is equivalent to showing Pr{G B (A) = 0 G A (B) = 1} Pr{G B (A) = 0}. (11) Assuming an unequal error protection (UEP) property [6], i.e., p (k) l+1 < p(k) l for a given k, it can be shown that (11) an, hence, (8) (9) hol. The erivations are omitte ue to lack of space. The UEP property oes hol when n. It is also easy to fin a counterexample for extremely short frame lengths, when it oes not hol. For instance, consier a unicast system (k = 0) with two users, two slots, an the istribution λ(x) = 0.5x+0.5x 2. For this toy example, the PLR for users of ifferent egrees can be foun by han, yieling p (0) 1 = 0.25 an p (0) 2 = 0.5. However, from our extensive simulations, we conjecture that the UEP property oes hol for sufficiently large values of n. Proving it rigorously an characterizing sufficient frame lengths is subject of ongoing work. Using the conjecture in (8) together with (5), we can write a tighter version of (6) as p 1 +p 2 p(1 p). (12) Next section presents numerical results an confirms the conjectures mae in this section. IV. NUMERICAL RESULTS In Fig. 3, we show simulation results for two ifferent istributions an frame length n = 200. The istribution λ(x) = 0.25x x x 8 is taken from [10], where it was optimize for classical CSA base on the threshol obtaine via ensity evolution. The fraction of egree-2 users was limite to 0.25 to yiel low error floor. The secon istribution, λ(x) = 0.86x x 8, was optimize in [1] base on error floor approximations to provie low error floor for B-CSA. The re an the green curves show the probabilities p 1 an p 2, respectively, an characterize the hanshake performance.

5 Probability p 1 p 2 p p(1 p) p 1 +p 2 can further simplify (13) an write p 1 +p 2 p. (15) In other wors, the probability Pr{G B (A) = 0} = p consists of p 1 an p 2, so that user A manages to etect communication failure events G B (A) = 0 in p 1 / p 30% of the cases for both istributions. We remark, however, that this ratio may change epening on the istribution. This fact suggests a new esign criterion for optimizing the egree istribution, i.e., the minimization of the probability of false hanshakep 2. We also remark that the sum of p 1 + p 2, shown with ashe purple curves, is strictly smaller than p(1 p) (black ashe curves), which is in agreement with the boun in (12). Probability g [user/slot] (a) λ(x) = 0.25x x x 8. p 1 p 2 p p(1 p) p 1 +p g [user/slot] (b) λ(x) = 0.86x x 8. Fig. 3: Hanshake performance of user A in B-CSA for two ifferent istributions an the frame length of n = 200 slots. The blue curves show the PLR. From Fig. 3 we observe that for low to moerate channel loa p 1 +p 2 p(1 p). (13) Using (5), we conclue that p 3 p 2, which, using (7), leas to Pr{G B (A) = 0 G A (B) = 0} p. (14) Therefore, for low to moerate channel loa, the probability for users to overlap in some slots is very small, which explains that the correlation between users performance is negligible. Bearing in min that p 1 in the error floor region, we V. CONCLUSIONS AND FUTURE WORK In this paper, we propose a probabilistic hanshake algorithm for vehicular communications base on B-CSA. In the rare cases of communication failure between two users, this event can be etecte by one of the users. The simulation results show that aroun 30% of such events can be etecte for the consiere istributions. We also propose analytical bouns on the performance of the hanshake algorithm, which match well the simulation results. The analytical bouns rely on the UEP property of CSA, for which a rigorous proof for finite frame lengths is left for future work. REFERENCES [1] M. Ivanov, F. Brännström, A. Graell i Amat, an P. Popovski, All-to-all broacast for vehicular networks base on coe slotte ALOHA, in Proc. IEEE Int. Conf. Commun. Workshop, Lonon, UK, June [2] E. Paolini, G. Liva, an M. Chiani, High throughput ranom access via coes on graphs: Coe slotte ALOHA, in Proc. IEEE Int. Conf. Commun., Kyoto, Japan, June [3] C. Stefanovic an P. Popovski, ALOHA ranom access that operates as a rateless coe, IEEE Trans. Commun., vol. 61, no. 11, pp , Nov [4] E. Paolini, C. Stefanovic, G. Liva, an P. Popovski, Coe ranom access: Applying coes on graphs to esign ranom access protocols, IEEE Commun. Mag., 2015 (to appear), available at [5] L. G. Roberts, ALOHA packet system with an without slot an capture, SIGCOMM Comput. Commun. Rev., vol. 5, no. 2, pp , Apr [6] M. Ivanov, F. Brännström, A. Graell i Amat, an P. Popovski, Error floor analysis of coe slotte ALOHA over packet erasure channels, IEEE Commun. Lett., vol. 19, no. 3, pp , Mar [7] C. A. Sunshine an Y. K. Dalal, Connection management in transport protocols, Computer Networks, vol. 2, no. 6, pp , Dec [8] ETSI EN : Draft V.0.0.5, Intelligent transport systems (ITS); vehicular communications; basic set of applications; part 2: Specification of cooperative awareness basic service, Tech. Rep., June [9] C. Di, D. Proietti, I. E. Telatar, T. J. Richarson, an R. L. Urbanke, Finite-length analysis of low-ensity parity-check coes on the binary erasure channel, IEEE Trans. Inf. Theory, vol. 48, no. 6, pp , June [10] G. Liva, Graph-base analysis an optimization of contention resolution iversity slotte ALOHA, IEEE Trans. Commun., vol. 59, no. 2, pp , Feb