Reliable Safety Broadcasting in Vehicular Ad hoc Networks using Network Coding. Behnam Hassanabadi

Transcription

1 Reliable Safety Broadcasting in Vehicular Ad hoc Networks using Network Coding by Behnam Hassanabadi A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical And Computer Engineering University of Toronto c Copyright 2013 by Behnam Hassanabadi

2 Abstract Reliable Safety Broadcasting in Vehicular Ad hoc Networks using Network Coding Behnam Hassanabadi Doctor of Philosophy Graduate Department of Electrical And Computer Engineering University of Toronto 2013 We study the application of network coding in periodic safety broadcasting in Vehicular Ad hoc Networks. We design a sub-layer in the application layer of the WAVE architecture. Our design uses rebroadcasting of network coded safety messages, which considerably improves the overall reliability. It also tackles the synchronized collision problem stated in the IEEE standard as well as congestion problem and vehicleto-vehicle channel loss. We study how massage repetition can be used to optimize the reliability in combination with a simple congestion control algorithm. We analytically evaluate the application of network coding using a sequence of discrete phase-type distributions. Based on this model, a tight safety message loss probability upper bound is derived. Completion delay is defined as the delay that a node receives the messages of its neighbour nodes. We provide asymptotic delay analysis and prove a general and a restricted tighter asymptotic upper bound for the completion delay of random linear network coding. For some safety applications, average vehicle to vehicle reception delay is of interest. An instantly decodable network coding based on heuristics of index coding problem is proposed. Each node at each transmission opportunity tries to XOR some of its received original messages. The decision is made in a greedy manner and based on the side information provided by the feedback matrix. A distributed feedback mechanism is also introduced to piggyback the side information in the safety messages. We also construct a ii

3 Tanner graph based on the feedback information and use the Belief Propagation algorithm asanefficient heuristic similar to LDPCdecoding. Layered BPisshown to beaneffective algorithm for our application. Lastly, we present a simple experimental framework to evaluate the performance of repetition based MAC protocols. We conduct an experiment to compare the POC-based MAC protocol with a random repetition-based MAC. iii

4 Dedication I dedicate this thesis to my lovely family. iv

5 Acknowledgements First and foremost I would like to express my gratitude to Professor Shahrokh Valaee, without his guidance, inspirations and mentorship this work could not be possible. I would like to appreciate his scientific and financial support which greatly helped me with my research. I would also like to thank my colleagues in WIRLAB for their support. Specially I would like to thank Le Zhang for all his efforts in our collaborative research and helping me with simulations. I would like to deeply thank University of Toronto for providing financial support and most importantly for giving me the opportunity. v

6 Contents 1 Introduction Why Vehicular Networks? Communications Service Standardization Frequency Spectrum and Channel Allocation Modulation Network Topology Medium Access Control Applications Contributions Related Work Network Coding in Vehicular Networks Cooperative Collision Warning and Driver Assistance: Periodic Safety Messages Contention based MAC: IEEE broadcast mode Broadcast Support Multiple Access (BSMA) Broadcast Medium Window (BMW) Batch Mode Multicast MAC (BMMM) Location Aware Multicast MAC (LAMM) vi

7 2.2.2 Time Reservation Based MAC MAC with Directional Antennas Random Repetition-based MAC Using POCs in Repetition-based MAC Network Coded Safety Broadcasting System Model and Performance Metrics Safety Message Reliability Control Congestion Control Message Retransmission Message Coding Coding Overhead Analysis Numerical Results Asymptotic Delay Bound Phase-type Model Delay Analysis Tighter Delay Upper Bound Opportunistic Network Coding Opportunistic Network Coded Broadcasting System Model and Performance Metrics Index Coded MAC Distributed Feedback Index Coding Algorithm Simulation Results Comparison to Random Linear Network Coding System Model and Performance Metrics vii

8 5.2.2 Index Coded MAC using perfect feedback Random Linear Network Coding Simulation Results Heuristics for XOR-based Broadcasting Greedy Algorithm Tanner Graph Representation and Belief Propagation Algorithm. 101 Belief Propagation Variants Simulation Results Experimental Evaluation Experiment Setup Disabling the CSMA/CA functionality Experiment Results Conclusion Future Works Maximum Weighted Clique Multi-hop, Hidden terminals and Capture Carrier sensing Performance in Vehicular Networks: Experimental Evaluation Bibliography 120 viii

9 List of Tables 3.1 Physical and MAC layer parameters: IEEE p Minimum reception threshold for IEEE p Table of Notations Physical and MAC layer parameters ix

10 List of Figures 1.1 DSRC Channel allocation Using IEEE PCF mode for VANETs The D-MAC Protocol IEEE Periodic Channel Switching Network Coding Example Reliability Sub-layer: Interfaces with WAVE Short Message Protocol(WSMP), Sensors, and Safety Applications Success probability vs. Number of nodes: IEEE p Broadcast mode Congestion Control SPR in an active subframe SPCR vs. SPR Markov chain model Simulation results vs. theoretical upper bound: n = Simulation results vs. theoretical upper bound: n = Simulation results vs. theoretical upper bound: n = Simulation results vs. theoretical upper bound: n = Reception Probability vs. Distance Loss probability comparison for different rates E(MTM) vs. w (number of active subframes in 10 subframes): n = x

11 3.16 E(MTM) vs. w (number of active subframes in 10 subframes): n = Markov chain model Modified Markov chain model Delay Bound based on Modified Markov Chain: n = Delay Bound based on Modified Markov Chain: n = Delay Bound Performance for Small n: n = Delay Bound Performance for Small n: n = Delay Bound II Performance for Small n: n = Delay Bound II Performance for Small n: n = Graph G(V,E) for node A, with two cliques of size three Message loss probability versus number of retransmissions. L = 128, p e = 0.2 and N = Message loss probability versus number of retransmissions. L = 128, p e = 0.5 and N = Average delay versus number of retransmissions. L = 128, p e = 0.2 and N = Average delay versus number of retransmissions. L = 128, p e = 0.5 and N = Maximum delay versus number of retransmissions. L = 128, p e = 0.2 and N = Maximum delay versus number of retransmissions. L = 128, p e = 0.5 and N = Average feedback header size versus number of retransmissions. L = 128, p e = 0.2 and N = Average feedback header size versus number of retransmissions. L = 128, p e = 0.5 and N = xi

12 5.10 Reception Probability vs. Distance Simulation Topology Loss Probability vs. Number of repetition (µ p = 0.2) Average Delay vs. Number of repetition (µ p = 0.2) Loss Probability vs. Number of repetition (µ p = 0.4,0.6) Average Delay vs. Number of repetition (µ p = 0.4) Average Delay vs. Number of repetition (µ p = 0.6) Tanner Graph representation for the feedback matrix Update Rules LBP Example Performance Comparison: n = Performance Comparison: n = Simple experiment to confirm CSMA/CA is disabled Experiment Result Example xii

13 List of Acronyms ACK ACKnowledgement AFR Asynchronous Fixed Repetition AGC Automatic Gain Control AIFS Arbitration Inter-Frame Spacing AP Access Point API Application Programming Interface APR Asynchronous p-persistent Repetition AODV Ad hoc On-Demand Distance Vector Routing ASTM American Society for Testing and Materials BCH Basic CHannel BMMM Batch Mode Multicast MAC BMW Broadcast Medium Window BP Belief Propagation BPSK Binary Phase-Shift Keying BSMA Broadcast Support Multiple Access CCH Control CHannel CRB Conditional Reception Probability CSMA/CA Carrier Sense Multiple Access with Collision Avoidance CTS Clear To Send xiii

14 CW Contention Window CWC Constant Weight Code DCF Distributed Coordination Function DCU DFS Control Unit DIFS DCF Inter Frame Space DSRC Dedicated Short Range Communications EIRP Effective Isotropic Radiated Power FCC Federal Communications Commission FI Frame Information GDP Gross Domestic Product GI Guard Interval GPS Global Positioning System HAL Hardware Abstract Layer IEEE Institute of Electrical and Electronics Engineers IP Internet Protocol ISI Inter Symbol Interference ITS America Intelligent Transportation Society of America LAMM Location Aware Multicast MAC LBP Layered Belief Propagation LDPC Low-density parity check xiv

15 LFSR Linear Feedback Shift Register LLR Log-Likelihood Ratio MAC Medium Access Control MANETs Mobile Ad hoc NETworks MTM Map Throughput Metric NACK Negative-ACKnowledgement NAV Network Allocation Vector OBU On Board Unit OFDM Orthogonal Frequency-Division Multiplexing PDF Probability Density Function PHY PHYsical layer PLCP PHY Layer Convergence Procedure POC Positive Orthogonal Code PRNG Pseudo Random Number Generator QoS Quality of Service RSU Road Side Unit RTS Request To Send SBP Shuffled Belief Propagation SCH Service CHannel SFR Synchronous Fixed Repetition xv

16 SLNC Symbol-Level Network Coding SNR Signal-to-Noise Ratio SPCR Synchronized p-persistent Coded Repetition SPR Synchronous p-persistent Repetition TDMA Time Division Multiple Access UDP User Datagram Protocol USDOT United States Department of Transportation V2V Vehicle to Vehicle V2I Vehicle to Infrastructure VANET Vehicular Ad hoc NETwork WAVE Wireless Access in Vehicular Environment WLAN Wireless Local Area Network WSM Wave Short Message xvi

17 Chapter 1 Introduction 1.1 Why Vehicular Networks? About road accidents occur every year in Canada, which results in approximately 3000 deaths each year [1]. Transport Canada has estimated that car collisions cost 62.7 billion each year which is about 4.9 percent of Canada s 2004 Gross Domestic Product (GDP) [1]. Most of traffic accidents and car collisions are avoidable using Intelligent Transportation Systems and safety communications. Statistics show that at least 60 percent of chain accidents could be avoided if drivers are informed about an accident and collision at least 500ms beforehand [73]. The driver reaction to the braking light typically ranges from 0.7 to 1.5 second [73, 28] which delays the collision information propagation. Vehicular Ad hoc NETwork (VANET) is a network of vehicles communicating with their neighbours through a wireless channel without a need for an access point. Unlike the Mobile Ad hoc NETworks (MANETs) in which the assumed mobility pattern is random and nodes might have power constraint, in VANETs, mobility is over deterministic paths determined by highways and roads and nodes do not have any energy constraint. Vehicular networks provide a reliable and fast system for active safety communications. A local neighbourhood map and early notification of an accident or collision can 1

18 Chapter 1. Introduction 2 greatly help the driver to choose other driving strategies and potentially avoid upcoming dangerous situations. Although the primary objective of vehicular networks is safety related applications, data communications and internet access are other interesting applications. This can potentially provide cheaper and faster internet access than cellular networks. 1.2 Communications Service According to [4, 8, 26] Dedicated Short Range Communications (DSRC) is a short to medium range communications service to provide a reliable communication link for Vehicle to Vehicle (V2V) and Vehicle to Infrastructure (V2I) communications. In response to the Intelligent Transportation Society of America (ITS America) request, in 1999, the Federal Communications Commission (FCC) [26] allocated the frequency range of GHz to DSRC, to accommodate the emergent need for safer roads and intelligent transportation systems. On December 2003, the FCC established and publicly announced the licensing and service rules for the DSRC 5.9 GHz band. The efficiency of IEEE (Institute of Electrical and Electronics Engineers) a link in the vehicular scenario has been assessed in [54] with experimental setups. Sub-urban, urban and freeway (both following and crossing traffic) scenario have been considered and different performance metrics have been measured. It has been discovered that a connectivity range of 1000m is possible under certain conditions and the sub-urban and urban link qualities are respectively the best and the worse link qualities while the freeway lies in between. For urban areas the increase in packet size increases the throughput while in the freeway scenario for higher distances(more loss) the smaller packet length increases the throughput.

19 Chapter 1. Introduction Standardization DSRC is a developing technology that provides a crucial communication link for future intelligent transportation systems. DSRC will provide a short to medium range, reliable, secure, and robust communication link between mobile vehicles or between the vehicles and the roadside infrastructure. In July 2003, a new standard for DSRC applications was approved by the American Society for Testing and Materials (ASTM) which was based on the IEEE a Physical and (Medium Access Control layer) MAC layer. A new task group in the IEEE organization, TGp, began to develop a new standard based on the published ASTM-E standard. Exploiting existing IEEE devices in vehicles requires modifications with respect to non-vehicular nature of IEEE The safety related application of vehicular networks imposes some changes in the IEEE to support reliablity and low latency. For instance the normal association process of IEEE can not be used when the total connection set up phase should be less than 100ms. Two classes of standards are currently being developed to address the design of different layers of the communication stack. IEEE p Wireless Access in Vehicular Environment (WAVE) is an extension to IEEE It defines amendments to MAC and Physical layer (PHY) specification of the IEEE in order to support several applications defined by the ITS. The final version of the p standard was approved and published in July, IEEE 1609 is a series of standards defining the security, management and application layer of vehicular networks [6]: Resource Manager Security Services Networking Services Multi-channel Operation

20 Chapter 1. Introduction 4 According to the standards there are two types of nodes in vehicular networks: Road Side Units (RSUs) and On Board Units (OBUs). RSU functionality is similar to the access point functionality in the IEEE a standard and provides infrastructurebased services. There are seven 10 MHz channels available for DSRC communication in North America. There are two channel types in the standard: control channel and service channels. Control channel is used for broadcasting and communication setup in service channels. All DSRC units continuously monitor the control channel. Scanning and association is done through the periodic beacon broadcasting by RSUs in the control channel. If a single radio is used, a fast channel switching (less than one millisecond) is necessary to switch between the service and control channels Frequency Spectrum and Channel Allocation As we mentioned, in 1999, 75 MHz of DSRC spectrum at 5.9 GHz was allocated by the United States Federal Communication Commission (FCC) to be used solely for vehicleto-vehicle and infrastructure-to-vehicle communications. This DSRC spectrum is divided into seven 10 MHz non-overlapping channels (Fig. 1.1) Service Service Frequency (GHz) Ch Ch Ch Ch Ch Ch Ch Control Public Safety: Veh-Veh Public Safety: Intersections Figure 1.1: DSRC Channel allocation In 2004, FCC assigned one control and six service channels to the DSRC. As shown in Fig. 1.1, the DSRC spectrum consists of seven 10 MHz wide orthogonal channels. Channel 178 is the control channel and is used for safety communication. Channels

21 Chapter 1. Introduction and 184 are reserved for future accident avoidance applications such as intersections and vehicle-to-vehicle safety. The four remaining channels are used for both safety and non-safety applications. Vehicles monitor the control channel all the time. The control channel is usually used for broadcasting while the service channels are used for a twoway communication between RSUs and OBUs and among OBUs. In IEEE p operation, any device can broadcast in the control channel, but only RSUs can broadcast Beacon frames in the control channel. At the beginning of each IEEE p physical frame there are ten identical short training symbols (t1-t10) which are used for coarse symbol timing estimation, coarse frequency offset estimation, Automatic Gain Control (AGC) and different diversity combining schemes. After the short training symbols, there are two identical long training symbols (T1 and T2) which are used for the channel estimation, fine frequency offset estimation and synchronization purposes. After the training sequences the modulated data symbols begins. The first data symbol is BPSK (Binary Phase-Shift Keying) modulated and specifies the modulation scheme used in the following OFDM (Orthogonal Frequency-Division Multiplexing) data symbols. Before each data symbol there is a Guard Interval (GI) which is used for eliminating the Inter Symbol Interference (ISI) caused by the multi-path propagation. In order to reduce the effect of multi-path propagation and fading, the data bits are coded and interleaved before they are modulated on subcarriers Modulation The modulation technique of IEEE p is a variation of the OFDM based IEEE a standard [2, 80, 78]. Both of them have the same encoder and decoder, the same preamble type and similar frame structure but the distinction is the channel bandwidth which is 20 MHz for the IEEE a and 10 MHz for the IEEE p. The usage of the OFDM technique has been recently increased in the wireless communications due to its robustness to the multi-path fading, high spectral efficiency and simple re-

22 Chapter 1. Introduction 6 ceiver/transmitter design. Based on the OFDM technique, a high data rate data stream is split into several low data rate data streams which will be transmitted over orthogonal frequency subcarriers. The physical layer of IEEE a uses an OFDM technique with 64 subcarriers where 52 subcarriers are used for actual data transmission. Four of the subcarriers (-21, -7, 7 and 21) are used as the pilot signals for estimating the frequency offset and the phase noise. The worst delay spread of the vehicle-to-vehicle communication is 400ns in the non-line-of-sight communication, so the coherence bandwidth is approximately equal to 500 khz. On the other hand, the pilot spacing is around 2 MHz which is larger than the coherence bandwidth and causes a frequency selective fading. To eliminate this problem, a pseudo-pilot scheme is suggested in [53]. The IEEE a standard is designed for indoor wireless access when the users have very low mobility. Therefore all of the physical parameters are optimized for this low mobility environment. To change IEEE a to account for high mobilities (up to 120 miles per hour), the signal bandwidth is reduced from20 MHz to 10MHz (all the time domain parameters are doubled) and the transmit power is adjusted according to the IEEE p transmission range. 1.4 Network Topology Multi-hop communication provides enhanced safety communication range as well as low cost inter-vehicular communications by limiting the bandwidth consumption of the cellular networks. The main advantages of VANETs over cellular networks are low delay communication, cost effectiveness and addressing based on the position [30]. Periodic communications among neighbor cars can excessively increase the bandwidth usage of the cellular network while the vehicular network can alleviate this problem by local communications. In contrast to MANETs, fast and predictable topology change is the unique characteristic of VANETs ([11]). According to simulations in [11], in a realistic scenario,

23 Chapter 1. Introduction 7 a fast topological change due to high relative mobility is likely and frequent fragmentation happens due to small penetration rate. Furthermore, the link between vehicles in opposite directions suffers from small life time. Increasing the radio range, increases the link lifetime and network connectivity at the cost of lower network throughput. Discarding the opposite direction nodes improves the average link lifetime but results in lower network connectivity ([11]). Due to fast topology change and high mobility the centralized tree based approach is not reliable and localized algorithm should be utilized in VANETs [68]. 1.5 Medium Access Control Using RTS(Request To Send)/CTS (Clear To Send) inmanets isaway to alleviate the hidden terminal problem but exposed terminal problem is a more challenging problem which limits the network throughput. For a vehicular network increasing the radio range to increase the network connectivity, path and link lifetime, increases the number of exposed nodes linearly and decreases the throughput based on the following formula ([11]): MaximumThroughput = Bitrate HighwayLength 2 InterferenceRange. (1.1) Considering the fast changes in topology, resource allocation schemes such as time, frequency or code division multiple access does not seem to be a logical choice. MANETs are distributed wireless networks (in contrast to the centralized networks) in which communication between any two nodes could exist through multi hop with no demand for infrastructure or base stations. In MANETs, if two adjacent nodes in the transmission range of each other, transmit at the same time, collisions are likely. In order to have a well-organized channel sharing, several MAC protocols have been proposed in recent years. In VANETs, the design of an efficient MAC protocol is more challenging due to high mobilities and volatile topology. One of the main applications of

24 Chapter 1. Introduction 8 Figure 1.2: Using IEEE PCF mode for VANETs VANETs is active safety communication. In the event of an accident, all approaching cars should be informed immediately through multi-hop (or one hop) communication. The amount of transmitted information is small but a low latency and reliable transmission is crucial. To design an efficient MAC protocol for VANETs, the traffic nature and emerging applications of vehicular networks should be carefully taken into account. If vehicles move along the highway with the speed of 80 km/h and RSUs, with a coverage range of 200m, are mounted at the edge of the highway, the maximum time that a vehicle is in the range of a RSU will be s. Since the connection time 22 is short, the system is frequently in the transient state which leads to low efficiency and stability. This simple example validates the shortcomings of IEEE PCF (Point Coordination Function ) mode in vehicular networks [82]. For every data packet there is a RTS/CTS handshake and the association and disassociation times are comparable to link lifetime. The Point Controller (PC) in each RSU should update the list of associated OBUs and since the connection time for each vehicle is short (18s) the updating should be regular. IEEE supports several data rates, depending on the channel quality, and vehicles experience variable data rate based on their distance to RSU. Designing MAC protocol for VANETs is a recent and ongoing research area. Certain

25 Chapter 1. Introduction 9 characteristics of vehicular networks should be considered in MAC protocol design. The node mobility is high and topology changes rapidly. There is no power and storage limitation and vehicles are equipped with a GPS (Global Positioning System) device. GPS provides access to a global clock which can be used for synchronization. The MAC protocol should support a variety of applications and in particular should satisfy the reliability and low latency requirements of safety applications. In Chapter 2, we throughly discuss and review several MAC protocols designed for vehicular networks. 1.6 Applications The safety messages can be categorized into periodic safety messages and event based safety messages. Periodic safety messages are small heart beat messages containing the vehicle state. These messages are transmitted periodically and should be received by one-hop neighbours. The communication of these messages requires low bandwidth and frequent channel access. The event based safety messages are generated when there is an incident on the road and should be propagated as fast as possible. These messages should be received by multi-hop neighbours. Event-based messages require low bandwidth and demand stringent latencies. The following is a list of important applications that we can consider for vehicular networks: Cooperative collision warning and Accident notification (event-based): fast propagation of the hard braking information Cooperative driver assistance (periodic): periodic sensor data exchange among neighbour cars Overtaking assistance Minimum distance warning Optimal driving strategy and dynamic traffic routing

26 Chapter 1. Introduction 10 Platooning Telematics Data Communications and Internet Access Internet Access: Web browsing, , etc. Online games Commercial Vehicles and Marketing along the road 1.7 Contributions In this thesis, we study the application of network coding for optimizing the reliability of periodic safety broadcasting in VANETs. We investigate how random linear network coding performs in periodic safety broadcasting. We consider n vehicles in a cluster. Each vehicle generates a safety message at the beginning of a time frame. The success probability is defined as the probability that a vehicle receives the message of every other node by the end of the time frame. We propose a discrete phase-type distribution to model the state of the network. Using this model we derive a tight upper bound for the loss probability. Safety message repetition has been shown to improve the reliability of IEEE p broadcast mode. On the other hand, channel congestion is recently known to be the bottleneck for implementation of VANETs. We study this tradeoff and investigate when the network coded repetitive broadcasting can be effective. More specifically we observe that using random network coding, more number of vehicles can be accommodated to achieve the same reliability. We study the asymptotic delay performance of random network coding. Using the introduced stochastic model, we prove fundamental asymptotic delay bounds for a binary erasure channel with loss probability p e. First we prove that the completion delay of the network coding algorithm is O( enlog((1 pe)n) 1 p e ) which provides a linear factor improvement

27 Chapter 1. Introduction 11 over the algorithm without network coding. Then for 0 p e 5 1 2, we prove a tighter upper bound of O(enlog(n)+f(p e )n), in which f(.) is a bounded continuous function. For some safety application, pairwise message loss probability is of concern. For these applications, we propose a XOR-based network coding algorithm that is instantly decodable. The algorithm is opportunistic and tries to maximize the number of decoding users using the side information provided by a feedback matrix. We propose a distributed feedback mechanism that piggybacks the local feedback information in the safety messages. We show, through simulations, that this algorithm outperforms the random linear network coding in terms of average reception delay. In order to reduce the complexity of the coding algorithm we provide a low-complexity heuristic using the Belief Propagation (BP) algorithm. We build a Tanner Graph based on the feedback matrix in which the users represent the function nodes and messages represent the variable nodes. The layered BP algorithm is then used to find the coding vector. Lastly, we describe a proof-of-concept experimental evaluation of the POC (Positive Orthogonal Code)-based MAC protocol. Our experiment consists of six wireless nodes using network interface cards with Atheros chipsets and using Madwifi driver. We explain how we disable the CSMA/CA (Carrier Sense Multiple Access with Collision Avoidance) and confirm the functionality through experiments. We then compare, through experiments, the performance of POC (Positive Orthogonal Code)-based MAC protocols to random repetition-based schemes. In Chapter 2, we review some of the notable previous works on the application of networkcodinginvanetsandafewproposedmacprotocols. InChapter3, wepropose a sublayer in the application layer of WAVE and prove a tight reliability bound. Delay analysis of the proposed sublayer can be found in Chapter 4. In Chapter 5, we propose opportunistic network coding algorithms for periodic safety broadcasting application. An experimental framework to compare repetition-based MAC protocols for VANETs is presented in Chapter 6. Finally, we conclude with future works and conclusion remarks

28 Chapter 1. Introduction 12 in Chapter 7.

29 Chapter 2 Related Work In this chapter we review some of the recent and significant works on safety communications in VANETs. Since the main focus of this thesis is optimizing safety broadcasting metrics using network coding, we first review previous works on network coding application in vehicular networks. In the second section, we look into periodic safety communication and application requirements. We discuss the obstacles in designing a MAC protocol and examine previously proposed approaches. 2.1 Network Coding in Vehicular Networks Most of the previous research on application of network coding in vehicular network deals with content distribution from a RSU to multiple OBUs. To the best of our knowledge there are only a few works on network coding application in safety message broadcasting. In the following, we review some of the important previous works. Content distribution using network coding in vehicular networks has been previously studied in several works. In [5], an infrastructure based topology is considered in which the gateways (RSUs) are used to distribute randomly coded file blocks in the first phase. In the second phase an ad-hoc local communication is performed by vehicles to distribute their received coded blocks in the first phase. 13

30 Chapter 2. Related Work 14 A network coding algorithm for p2p applications in VANETs is proposed in [40]. The algorithm is an extension of network coding application in p2p wired networks. All nodes periodically send a request packet to their neighbours indicating a file id. A seed contains all the blocks of a file with a given file id and broadcasts a random linear combination of the blocks if it receives a request. All nodes overhear their onehop neighbours transmissions. Each node, in case of receiving a request and having the required file blocks in its memory, broadcasts a random linear combination of the coded blocks with the required file id. The communication is multi-hop and the network coding is compared to Ad hoc On-Demand Distance Vector Routing (AODV) routing protocol. Distribution of multi-media files and software updates are potential applications for p2p communication in vehicular networks. An in depth analysis and implementation details is presented in [39]. In [49], authors investigate the reliable communication of multimedia safety traffic in vehicular networks. Unlike the small heartbeat messages, because of larger message size, reliability for multimedia traffic can not be provided with repetition. Therefore, a network coding algorithm is proposed to provide reliability. The vehicles in the opposite direction of the highway are used as data mules to act as mobile relays similar to delay tolerant networks. The relay platoon receives coded data from the exposed platoon multimedia stream and delivers it to the intermittent approaching platoon. Analysis in [49] shows the superior performance of network coding compared to repetition-based schemes. The use of XOR network coding for safety message transmission was first proposed in our work in [32]. We proposed an instantly decodable network coding algorithm inspired by index coding problem heuristics. We then compared the performance of this algorithm to random linear network coding in [31] and further investigated the reliability under a more realistic channel model. The algorithm details and simulation results is presented in Chapter 5.

31 Chapter 2. Related Work 15 In [67, 66], authors assume the feedback information can not be easily acquired and propose a XOR-based network coding in which each user blindly picks a number of original messages for encoding. It is assumed that Conditional Reception Probability (CRB), defined as the probability that a packet is received by a neighbour vehicle given it is successfully received by the transmitter, is known. The optimal number of packets to encode is then determined based on CRB. The optimized metric is the average number of pairwise recovery. This means if vehicle A is transmitting, the number of XORed packets is optimized in order to maximize the expected number of recovered packets by vehicle B. However, in practice, after vehicle A transmission, multiple destinations can potentially decode and a more reasonable metric should be the expected number of decoded packets by all neighbours. The assumption that the feedback information is hard to obtain can be reasonable at times, however in [32] we proposed a distributed feedback mechanism in which there is no need for a dedicated feedback channel and the local feedback information can be piggybacked in the coded messages. Furthermore estimating CRB could be as hard as estimating the feedback matrix. Unfortunately no comparison is presented to our earlier work in [32] which we believe will outperform the proposed algorithm in [67, 66]. In [74, 75], Symbol-Level Network Coding (SLNC) is utilized for multimedia streaming from RSUs to vehicles. In [41], SLNC is used for content distribution in order to maximize the download rate from the access points. SLNC is shown to achieve better performance compared to packet level network coding in unicast transmissions. For small safety messages if we decompose each message into smaller symbols, the network coding overhead can be comparable to the symbol size which could defeat the purpose of SLNC. However the use of SLNC for broadcasting small safety messages with stringent delay requirement is not well understood and could be a subject of future research. In[76,77], a1-dvehicularnetworkisconsidered. Asourcenodesendsafile, consisting of M packets, to all other vehicles. An ideal scheduler is assumed. Authors compute the

32 Chapter 2. Related Work 16 Probability Density Function (PDF) of completion delay for a 3-node case scenario. For the 3-nodescenario, node 0,1 and 2 are placed in sequence and node 0 has all the packets. In the first round node 0 starts broadcasting random linear combinations of the packets. Depending on the channel, it takes a random time T 1 for node 1 to collect all M linearly independent coded packets. Meanwhile node 2 has overheard C < M coded packets. In the second round node 1 starts broadcasting till node 2 collects M linearly independent coded packets. The second round takes T 2. The Probability Density Function (PDF) of T 1 +T 2 is derived in [76, 77] and has been compared to the uncoded scenario. 2.2 Cooperative Collision Warning and Driver Assistance: Periodic Safety Messages One of the major driving forces for implementing vehicular ad-hoc networks is developing a reliable and fast active safety system. In a desired active safety system, each car has a local neighbourhood map. This map contains the state of the neighbour cars. The state specifies the position as well as other parameters such as velocity, acceleration, etc. A GPS unit provides position, velocity and clock information. The clock information can be used for a slotted MAC such as slotted ALOHA. Recent vehicles are equipped with sensors and radars. The radars mounted on a car can be used to estimate the distance to neighbour cars. All cars periodically broadcast their state information. The goal of the system and MAC is a reliable and fast delivery of the state information Contention based MAC: IEEE broadcast mode The off-the-shelf IEEE devices and the approved IEEE p standard makes it the first candidate to be considered for periodic safety messages. The IEEE broadcast mode is an asynchronous MAC that uses CSMA/CA without virtual carrier sensing (RTS/CTS). Since the broadcast transmission should be received by all the one-

33 Chapter 2. Related Work 17 hop neighbour nodes there is no acknowledgement (unlike the unicast mode). In addition there is no retransmission mechanism for broadcast transmissions. When a node has a packet to transmit, it sets a backoff timer with a uniform random number between 0 and W. After each idle time slot the backoff timer will be decremented. If the medium is busy the timer stops decrementing andif the medium remains idle for a DCF Inter Frame Space (DIFS) time, the decrementing starts over and the timer will be decremented after the next idle time slot. When the timer reaches zero the packet is transmitted and the node enters a post backoff regime to avoid back-to-back transmissions. The CSMA/CA substantially reduces the number of collisions and the main source of collision remains to be hidden terminals. As can be seen from the simple functionality of IEEE in the broadcast mode, several challenges should be addressed before considering the use of this MAC for safety applications and specifically for periodic safety messages. The unreliable vehicle to vehicle channel makes it more difficult to rely only on one transmission. On the other hand, each safety message has a lifetime after which it will not be useful. The random mechanism of IEEE makes it difficult to guarantee a stringent reception delay. In the presence of a dense traffic, the hidden terminals become the main source of collision. Furthermore when the backoff counter of two or more nodes reach zero at the same time there will be a collision. This situation is likely when the traffic density or the safety message generation is high. Unlike the unicast transmission in which the acknowledgement provides a feedback for a successful reception, in the broadcast mode no feedback is provided and lack of retransmissions as well as lossy dynamic channel makes the transmissions more susceptible to loss. Also since no feedback is provided to the sender, the Contention Window (CW) remains unchanged (CWmin) which would cause further collisions in a dense environment. The performance of IEEE broadcast mode for periodic safety messages has been recently analyzed in [65]. The probability that a specific message does not collide

34 Chapter 2. Related Work 18 with other messages (reliability) and the average number of transmitted messages in a random time slot (throughput) have been derived. The contention window size has been shown to be the key parameter in the reliability-throughput trade-off. Several improvements of IEEE broadcast mode to support a reliable broadcasting have been proposed in the literature and in the following we review some of the proposed schemes: Broadcast Support Multiple Access (BSMA) To resolve the hidden terminal problem in the IEEE broadcast mode, a RTS/CTS transmission mechanism for the broadcast messages is proposed in [59, 60]. When a node wants to broadcast a message to its neighbors, after a CSMA/CA phase it broadcasts a RTS message and sets its WAIT FOR CTS timer. The neighbor nodes who receive the RTS message and are not in the YIELD state (In the communication range of another broadcast session), set the WAIT FOR DATA timer and broadcast a CTS message in order to firstly inform the sender to broadcast its message and secondly inform the hidden terminals not to transmit. If the source receives a CTS message before the WAIT FOR CTS timer expires, it starts the data transmission and sets the WAIT FOR NACK timer. If no CTS is received another CSMA/CA phase starts. If WAIT FOR DATA timer expires and the potential receiver does not receive a data packet, it transmits a NACK message in order to inform the source of the reception loss. If before the WAIT FOR NACK timer expires the senders receives a NACK from one of the neighbors, it backs off and restart the data transmission after another CSMA/CA phase. If no NACK is received the broadcast is successful. The algorithm assumes data capture and in the presence of multiple CTS transmission the source can capture the strongest signal. The proposed algorithm has several disadvantages for vehicular networks. In the presence of a dense vehicular traffic (traffic jams), the capture assumption would not be

35 Chapter 2. Related Work 19 valid anymore as there could be several CTS messages with similar reception power. This also holds for the NACK packets and as mentioned in [29] the CTS and NACK collision is a critical drawback for this protocol. Assume a neighbor X is receiving a data session from another node and is in the YIELD state. Node X does not transmit a CTS message back to the sender but there is another neighbor of the sender that does transmit the CTS message. In this case the sender starts transmitting and will cause collision in node X. In lossy channels (the common case in vehicle-to-vehicle channel), the drop of control messages can further degrade the functionality of the algorithm. Since the safety message size is in the order of the control messages the reliability can be further deteriorated by the extra overhead. Broadcast Medium Window (BMW) To improve the reliability of the IEEE broadcast mode, the proposed scheme in [61] suggests a round robin RTS/CTS/DATA/ACK transmission to neighbours. Each node maintains a neighbour list, and a sender and receiver buffer. The sender buffer holds a copy of the packets that have been previously transmitted and may be needed later by other nodes. If a packet has been received by all the nodes, it will be removed from the sender buffer. After each successful reception, the sequence number of that packet will be recorded in the receiver buffer. When a node wants to broadcast, it first goes through a CSMA/CA phase. Then, it will send a RTS message indicating the lowest (oldest) sequence number in the sender buffer and the sequence number of the current packet. After successful reception of the RTS message, the receiver checks the receiver buffer and responds with a CTS message including the lowest sequence number that it needs and is also in the sequence number range of the RTS message. If the receiver asks for an older packet, the sender will transmit that packet and after receiving the acknowledgement it will transmit another RTS message to that node (without entering a CSMA/CA phase). Neighbour detection is done by sending periodic Hello messages. The reception of other

36 Chapter 2. Related Work 20 control messages can be also used to update the neighbour list and to minimize the Hello messaging overhead. The case of unreliable wireless channel has not been addressed in the BMW functionality. The protocol does not suggest any solution for the case of dropped RTS and CTS messages. The small periodic safety message size is comparable to RTS, CTS and ACK packet size. This noticeable extra overhead would not trade off with the benefit of hidden terminal removal. For just one transmission of a safety message to n neighbour nodes, n RTS, n CTS and n ACK messages should be transmitted. Although the protocol does not suggest multiple retransmission of a data packet, for a reliable transmission retransmission is inevitable. In that case the increased control message overhead degrades the performance. Batch Mode Multicast MAC (BMMM) To reduce the number of contention phases in the BMW protocol, the Batch Mode Multicast MAC Protocol (BMMM) has been proposed in [57]. When a sender wants to send a packet to n neighbour nodes and the wireless channel is lossless, in BMW, there are n contention phases to send the RTS packets to each neighbour. In BMMM there is only one contention phase after which RTS messages will be sent sequentially to all nodes. Similarly after the DATA transmission, RACK messages will be sent sequentially to all the nodes and the neighbours will respond by ACK packets. Although BMMM introduces new RACK messages to coordinate ACK transmissions, the reduced contention phase time still makes it a better choice than BMW in terms of broadcast delay. The periodic transmission of the RTS messages block any initiation for data transmission from the neighbours. Although BMMM minimizes the broadcast delay, it still suffers from excess number of control messages. Considering small size of the safety messages (comparable to the control messages), the overhead of the control messages is still high. The protocol ensures there is no collision for control messages, but message loss due to

37 Chapter 2. Related Work 21 channel unreliabilities is ignored. Location Aware Multicast MAC (LAMM) In order to minimize the overhead of control messages cover sets have been defined in [57]. A cover set for a receiver is a subset of receivers covering the same radio range as all other receivers. If only the nodes in the cover set transmit the CTS and ACK messages, it will push all other nodes into the yield state. The cover sets can be found by the position information provided by GPS Time Reservation Based MAC ADHOC MAC [13] is the proposed MAC protocol for the CarTALK2000 project in Europe. It is a dynamic Time Division Multiple Access (TDMA) protocol and each vehicle in the network knows about the two-hop neighbourhood in order to eliminate the hidden terminal problem. Similar to the conventional TDMA the time is divided into frames and each frame consists of several time slots. Each vehicle in the network acquires a time slot called Basic CHannel (BCH). All vehicles listen to the channel during the frame. If there is a successful transmission in each time slot they mark the corresponding field in their Frame Information (FI) with the ID of the transmitter. In each time frame vehicles transmit their FI in their assigned BCH. When a vehicle enters the network, it first listens to the channel and receives the FIs of neighbour vehicles during a frame time. If a free slot is found, the free time slot will be selected as BCH and the FI will be marked with the ID accordingly. After BCH allocation, the vehicle can start data transmission in the assigned time slot. In comparison to the CSMA/CA mechanism of the IEEE MAC, which can not guarantee any QoS requirements, the ADHOC MAC can satisfy the QoS requirements of real-time data traffic using the distributed TDMA protocol. However, CSMA/CA supports higher mobilities without a need for synchronization. In addition, the number of users in a two-hop neighbourhood in ADHOC MAC is limited

38 Chapter 2. Related Work 22 Figure 2.1: The D-MAC Protocol by the number of time slots in each frame MAC with Directional Antennas The use of directional antenna always increases the spatial reuse and lowers the collisions. Particularly in VANETs, where the mobility pattern of the vehicles is directional (along the roads, highways etc.), the directional antenna can be efficiently utilized to minimize the interference and number of collisions. In D-MAC[36], each vehicle knows the geographic location of itself and its neighbours. When a node receives a CTS or RTS, it gets blocked during the data transmission of its neighbour (indicated in the RTS and CTS). To describe the protocol mechanism let us consider the simple example in Fig The transmission space is divided into 3 transmission angles of 120 degrees. Node B initiates a data transmission to node A by sending a RTS packet. Since all the directional antennas of node B are unblocked, the RTS packet is sent by all of them. Node C receives the RTS by its left directional antenna and blocks it. So node C will not be able to send a RTS packet to B and initiates a data transmission. It can be seen that by using the directional antennas the other two directional antennas are unblocked and can be used by node C for other communications.

39 Chapter 2. Related Work Random Repetition-based MAC Repetitive broadcasting of safety messages has been proposed in [72, 69] as the first MAC protocol proposed for periodic safety messages. Each safety message is assumed to have a lifetime. The time is divided into time frames equal to the lifetime of a safety message consisting of L time slots. Several protocols with minor variations have been proposed. In Asynchronous Fixed Repetition (AFR), the protocol repeats the message w times in L time slots without checking the channel to be idle (No CSMA/CA). In Asynchronous p-persistent Repetition (APR) the message will be repeated during the message lifetime with w L probability in each timeslot without CSMA/CA. Synchronous Fixed Repetition (SFR) and Synchronous p-persistent Repetition (SPR) are the synchronous versions of AFR and APR in which the nodes are synchronized. APR-CS and AFR-CS are similar to APR and AFR but with CSMA/CA. At first glance carrier sensing may look beneficial as it can potentially reduce the number of collisions. But if nodes are able to capture then carrier sensing would not necessarily results in lower message loss probability. It has been shown in [72] that the number of repetitions, message transmission time (determined by modulation, coding and message size) and the number of interferers are the key parameters that determine the optimal performance. Increasing the number of repetitions results in higher reception probability up to a certain point after which excessive collisions increase the loss probability. Higher transmission power translates into higher rate (higher Signal-to-noise ratio (SNR) at the receiver for a given message range) and larger communication range. Higher rate results in smaller slot size and larger L which lowers the loss probability. On the other hand, there will be more interferers in larger transmission range which leads to higher loss rate. For higher transmission power, more messages are delivered to farther nodes which can be beneficial for some safety applications. The proposed algorithms in [72] do not result in low enough message loss probability and a need for better MAC protocols has been suggested. Nevertheless, their problem

40 Chapter 2. Related Work 24 formulation, the proposed performance metrics and characterizing the safety message generation have been a worthwhile contribution Using POCs in Repetition-based MAC To further minimize the message loss probability the use of Positive Orthogonal Codes (POCs) for deterministic transmission patterns has been proposed in [23]. A binary string is mapped into a time frame such that the repetitions represent the 1 s in the string. Each binary string with weight w corresponds to a repetition pattern with w repetitions. The POC property guarantees that each two codewords of length L have less than λ correlations. Based on the required QoS, the code weight can vary. In Constant Weight Codes (CWCs), all codes have the same weight with pair-wise limited correlation. Positive Orthogonal Codes (POCs) are a subset of CWCs such that every shifted version of two codes also have less than λ correlations. It has been shown in [23] that POCs can minimize the loss probability by limiting the number of collisions compared to random repetition patterns. Simulations using a more realistic vehicle-tovehicle channel, POC generation algorithm, POC assignment and system design issues later presented in [24, 25, 33].

41 Chapter 3 Network Coded Safety Message Broadcasting in VANETs An architecture for Wireless Access in Vehicular Environment (WAVE) is presented in the IEEE standard. As we mentioned before, IEEE p has been approved as an amendment to IEEE standard and specifies the MAC layer enhancements for vehicular environment. The significant changes are increased maximum transmission power and reduced channel bandwidth to 10Mhz. This provides a more reliable communication at lower rates. The IEEE standard also specifies some enhancements to the MAC layer to support multi-channel operation. Both IEEE p and are part of the WAVE architecture. Periodic broadcast and its related safety applications are one of the major driving forces for implementing VANETs [69]. In VANETs, a safety message is periodically generated (10Hz frequency) and transmitted to one-hop neighbour vehicles. These periodic heartbeat messages are the building blocks of many safety applications. By aggregating this local information, each vehicle can construct and maintain a local neighbourhood map that can be utilized by safety applications. The message contains the state of the vehicle, which consists of various sensor readings such as location, velocity, acceleration, 25

42 Chapter 3. Network Coded Safety Broadcasting 26 50ms 50ms CCH CCH CCH SCH SCH SCH time Figure 3.1: IEEE Periodic Channel Switching etc. The goal of the communication is to provide a reliable up-to-date neighbourhood information. Having an up-to-date local map could prevent accidents and collisions. Also this extra information assists the driver to choose alternative driving strategies such as taking over or turning. IEEE explains how channel coordination is performed for a WAVE device with a single radio [3]. The time is divided into 100ms sync intervals. Each sync interval consists of a Control CHannel (CCH) and a Service CHannel (SCH) (Fig. 3.1). Every 50ms the WAVE device switches to CCH (channel 178). The periodic broadcast of Wave Short Messages (WSM) takes place in the CCH intervals. SCH intervals are used for IP packets. Since WAVE devices should be in the same channel in order to be able to communicate, the channel switching for all nodes is synchronized. The synchronization mechanism is detailed in the IEEE standard [3]. The periodic broadcast is shown to perform well for low node densities (less than 10 nodes) [21]. In a dense network, however, congestion becomes a serious problem. It can produce excessive number of collisions and result in unacceptable reliability measures for safety applications. In [22] and [21], the authors have proposed a congestion control mechanism based on the channel occupancy. The message rate is adjusted in conjunction with the transmission range to alleviate the congestion problem. Other than the message rate control, adaptive control of the contention window size based on the node density is another way to avoid congestion [56]. The congestion problem is also discussed in IEEE where many nodes have a newly generated WSM for transmission and they all switch to CCH at the

43 Chapter 3. Network Coded Safety Broadcasting 27 m 1 m 2 No Coding U 3 U 1 U 2 U 4 a 1 m 1 +a 2 m 2 b 1 m 1 +b 2 m 2 Random Linear Network Coding U 3 U 1 U 2 U 4 Figure 3.2: Network Coding Example same time (synchronized collision). In this scenario, collision is expected if two nodes pick the same timeslot in the contention window. Naturally, for high node densities the collision probability increases. Aside from the congestion problem, unreliable vehicle-to-vehicle channel and channel errors are other concerns for safety applications. In the IEEE p broadcast mode there is no acknowledgement. Also, unlike the unicast transmissions, there is no retransmission for lost messages. In [70], several repetition based access schemes are proposed to guarantee a low message loss probability. Although message repetition improves the reliability, it can potentially aggravate the congestion problem. In this chapter we study the application of random linear network coding to provide reliability for safety message broadcasting. Nodes rebroadcast their safety message during a CCH interval in order to improve the reliability. However, when a node has a transmission opportunity, it can transmit a combination of the received messages instead of just broadcasting its own message. Let us consider the example in Fig In their first transmission opportunity, U 1 and U 2 broadcast their safety messages (m 1 and m 2 respectively). Suppose m 1 is received but m 2 is not received by U 3. If U 1 has a second transmission opportunity during the CCH interval, it rebroadcasts m 1 which is not helpful for U 3. If instead U 1 broadcast a random linear combination of m 1 and m 2 and U 3 receives the transmission, it can decode m 2. U 2 can act in the same way to deliver m 1 to U 4. However, if U 1 does not received the first U 1 s transmission but receives the second

44 Chapter 3. Network Coded Safety Broadcasting 28 transmission, if the second transmission is coded, U 3 can not decode any messages. This shows that network coding can be potentially beneficial for message delivery but a careful study is needed to provide insights into it s performance gain. Random linear network coding could asymptotically achieve the multi-cast capacity in a lossy wireless network [43]. However, the analysis in [43] only considers long term throughput, but our metric of interest is message loss probability which relates to short term throughput. In the context of gossip algorithms, the work in [18] considers the problem of disseminating k messages in a large network of n nodes. In gossip algorithms, at each timeslot, each node selects a communication partner in a random uniform fashion and only one message is transmitted, while in the broadcast scenario potentially more than one node can receive the message. For periodic safety message broadcasting, a fast local information broadcast is of interest. In gossip algorithms, at each timeslot, each node can contact at most one neighbour while a node can be contacted by multiple nodes. Our communication mechanism follows the opposite scenario: each node can potentially communicate with multiple nodes, but if a node is contacted by multiple nodes simultaneously, there will be collision. In the PULL mechanism [18], a node i contacting node j leads to a transmission from node j to node i. If all the nodes contact node j, it transmits a message to all nodes, which can be cast as a broadcast transmission. However, in a single cell scenario if more than two nodes are contacted, transmissions collide in the wireless channel while in the communication model in [18] transmissions are successful. The information dissemination considered in [18] intends to deliver global information through local communication. However, in our problem, fast local information dissemination is done through local communication and vehicles are not specifically interested in global information. Gossip algorithms for complete graphs have linear stopping time as compared to O(nlog(n)), the complexity of the sequential algorithms. Most recently it has been

45 Chapter 3. Network Coded Safety Broadcasting 29 shown that the stopping time of algebric gossip (for EXCHANGE gossip algorithm) is O( n) in which is the maximum degree of the network graph [14]. In contrast, in our scenario it is straightforward to show that for a complete graph, coding does not provide any gain. We propose a sub-layer in the application layer of the WAVE architecture (Fig. 3.3). This sub-layer handles the periodically generated messages in the application layer to improve the overall reliability of the updated local neighbourhood map and cope with the congestion problem and channel loss. Blue arrows in Fig. 3.3 represent the handshakings between the sublayer and the rest of the networking stack. The red arrow represents the timing feedback from the MAC layer. Since our sublayer is in the application layer it needs the timing information for synchronization. For example, the synchronization is necessary for the congestion control algorithm in order to transmit in predefined subframes. We study how and when the message repetition can be helpful to achieve more reliability. In our scheme, each node not only can send its own WSM, it can also transmit a random linear combination of previously received WSMs to cooperatively help delivering all WSMs. The low-level design and the exact description of the protocol is beyond the scope and space of our research, which aims to tackle the problem analytically. The main contribution of our work is the analytical study of random linear network coding in the periodic broadcast of heartbeat messages. To the best of our knowledge, this is the first analytical study of the application of random network coding for safety applications in vehicular networks. Our system model and performance metric definition are presented in Section 3.1. In Section 3.2, we present the proposed reliability sublayer design. Network coding overhead and reliability analysis can be found in Section 3.3. Numerical results and comparisons are presented in Section 3.4.

46 Chapter 3. Network Coded Safety Broadcasting 30 Application Layer Safety Applications Sensors, GPS SAE2735 Reliability sub-layer WAVE stack for Safety Applications Security & Management IEEE IEEE IEEE IEEE p IEEE p WSMP LLC MAC PHY Figure 3.3: Reliability Sub-layer: Interfaces with WAVE Short Message Protocol (WSMP), Sensors, and Safety Applications 3.1 System Model and Performance Metrics We consider a cluster of N nodes in the radio communication range of each other. At the start of each CCH interval (Fig. 3.1) each node has a safety message consisting of its state information such as the GPS location, velocity and braking status. Each user needs to receive the messages of the active nodes by the end of each CCH interval in order to update its neighbourhood map. Due to the congestion problem not all the nodes should transmit. Active nodes are the ones that transmit their generated messages and are selected according to a congestion control algorithm. This will be discussed further, in the next section. An erasure channel is assumed such that an uncollided transmission is received with probability 1 p e. The erasure channel has also been previously assumed in the context of safety communications in vehicular networks [73, 50]. To make our analysis tractable, we assume a symmetric erasure network in which every channel is a symmetric channel with erasure p e. This assumption does not consider higher reception probabilities for

47 Chapter 3. Network Coded Safety Broadcasting 31 closer neighbours (unlike the Nakagami channel model) and it essentially underestimates the successful message reception probability [50]. We define P s (n) as the probability of successful reception of all n messages if there are n active nodes. Because of the symmetry assumption, P s (n) is the same for all nodes. If a node receives all n messages, it can construct a full neighbourhood map based on the location and other sensory information included in the message. If R i,j is the event that node i receives a message from node j within a frame, P s (n) can be written as: P s (n) = Pr(R i,1 R i,2 R i,n ) Safety applications in VANETs require distinctively defined network performance metrics compared to MANETs. For safety critical applications, a vehicle needs to receive the messages of allitsneighbours bytheendofacch interval inorder tomakesure ofsafety. A message delivery is successful if it provides meaningful safety information about the neighbourhood. Even if n 1 messages are received successfully, the missing vehicle state can be hazardous and the received messages do not guarantee local safety. So a delivery is defined to be successful only if the messages of all other neighbour nodes are received successfully. Otherwise, even though we have received that message, the loss of other messages poses safety risk. Hence, for this particular application, throughput is naturally defined as np s(n) 100ms. We define the Map Throughput Metric (MTM) as the normalized system throughput: MTM = np s (n), which accounts for the average number of successful messages received in a sync interval. Since an active node does not need to receive its own message, the successful reception probability by an active node is lower bounded by P s (n). If there are no inactive nodes, we can consider a virtual inactive node and the resulted P s (n) will be a lower bound.

48 Chapter 3. Network Coded Safety Broadcasting 32 It has been shown that Nakagami distribution with properly estimated parameters would be a more realistic channel model for vehicle-to-vehicle communications ([64, 79, 58]). In our simulations we will study the Nakagami channel and will define an appropriate erasure probability by averaging the probability of reception in the Nakagami channel over a certain interval that corresponds to the coverage range in the WAVE standard. 3.2 Safety Message Reliability Control This section demonstrates different components of the proposed sublayer in Fig The message is generated based on the sensors output and according to the dictated format by SAE2735. The header for the new sublayer is essentially the network coding overhead which we will explain in Section 3.3. Instead of passing the message directly to the WAVE stack, our sublayer combines the received messages, attaches the network coding overhead (sublayer header) and sends it to the lower layer. The proposed reliability sublayer consists of three parts. The congestion control algorithm determines if a node should be active in a given CCH interval. If the node is notactive inacchinterval, thegeneratedmessage will bedropped. Ifthenodeisactive, it probabilistically transmits at each time slot during a CCH interval. We assume that the back off parameters of the IEEE p have been set to minimum and the physical carrier sensing is disabled. This can be done through the wireless card driver interface and ensures an instant transmission when a message is sent to the lower layer. The congestion control algorithm filters the generated messages in order to reduce the number of active nodes in each CCH interval. The Synchronized p-persistent Coded Repetition (SPCR) algorithm describes the main functionality of the sublayer for transmission. At each time slot a random linear combination of all the queued messages can be transmitted by an active node. In the following, first we introduce the congestion control algorithm that limits the

49 Chapter 3. Network Coded Safety Broadcasting 33 number of active nodes in a CCH. Then, the message repetition can be used to tackle the channel loss. Finally, message coding based on random linear network coding is used to further maximize the success probability Congestion Control When the number of vehicles in the cluster, N, is large the message reception probability drops considerably. We first derive the P s (n) for IEEE p to see how it behaves by increasing the number of vehicles in a cluster. Theorem 3.1. The success probability for IEEE p broadcast mode in a symmetric erasure channel with n active nodes at the beginning of the CCH, and the contention window size, CW, can be written as: P s (n) = n 1 ( ) i 1 min{x,cw 1}+1 i=1 ( ) n min{x,cw 1}+1 (1 p e ) n, CW where X is the number of idle timeslots in a subframe in which n packets have been successfully transmitted and is given by: X = T CCH T g n(t h + M +AIFS) R σ where T CCH is the CCH duration, T h is the PLCP (PHY Layer Convergence Procedure) duration, M is the Message size, R is the Channel rate, σ is the timeslot duration, T g is the Guard time, AIFS is the Arbitration inter-frame spacing, and p e is the Erasure probability. Proof. To find the success probability, we should find the probability that all nodes pick distinct backoff counters (C i ) and all transmissions are completed within the CCH period T CCH. Each successful transmission takes (T h + M +AIFS) seconds. So the maximum R

50 Chapter 3. Network Coded Safety Broadcasting 34 Table 3.1: Physical and MAC layer parameters: IEEE p R 3Mbps σ 16µs T h 40µs AIF S 32µs T g 0s CW 1023 backoff counter must be less than X = T CCH T g n(t h + M R +AIFS). The success probability σ can be written as follows: i,j: 1 i,j n and i j, P s (n) = Pr(C i C j max{c i } X) (1 p e ) n = Pr(C i C j max(c i ) X) Pr(max(C i ) X) (1 p e ) n. The first term is the complement of the Birthday Paradox Problem [47] when we have n people and min(x +1,CW) days in a year and is equal to: n 1 i=1 ( 1 ) i min(x,cw 1)+1 The second probability is the probability that all C i s are less than or equal to X. To see how the IEEE p broadcast mode performs when there is traffic congestion, we evaluated P s (N) with the parameters in Table 3.1, and for various erasure probabilities. The message size is assumed to be 200 Bytes. Fig. 3.4 shows how the probability of success declines by increasing the number of nodes N. Even when there is no channel loss (p e = 0) the success probability degrades to less than 0.3 for 50 nodes. For higher erasure probabilities the congestion problem is more pronounced. Ideally, we are interested in delivering all N messages with high probability, but for

51 Chapter 3. Network Coded Safety Broadcasting pe=0 pe=0.1 pe=0.2 pe= Ps(n) n Figure 3.4: Success probability vs. Number of nodes: IEEE p Broadcast mode L f Subframes CCH CCH CCH CCH CCH CCH CCH CCH CCH CCH w subframes are randomly chosen Figure 3.5: Congestion Control higher traffic densities, it is unattainable. The main solution to the congestion problem is message rate control. Rate control algorithms drop some of the messages to reduce congestion. Nonetheless, this is the only way to manage excessive collisions and provide an overall acceptable reliability. Our goal is to optimize the number of active nodes in a CCH interval and the overall message reception reliability, which is done through maximizing the MTM. Here, we consider a simple rate control algorithm, which randomly filters the load. We define a frame to consist of L f CCH intervals. We call a CCH interval a subframe. We assume nodes are synchronized with synchronization information inferred from the MAC layer (Fig. 3.3). At the beginning of every frame each node randomly picks w subframes and only transmits in those subframes (Fig. 3.5). Hence, the number of active nodes in a subframe is a random number n N. Unlike the congestion control

52 Chapter 3. Network Coded Safety Broadcasting 36 algorithm in[22], in which the rate is controlled based on the channel occupancy feedback, the feedback parameter in our method is the number of neighbours N, which is derived based on the most up-to-date neighbourhood map. Since MTM is a random variable, we use E(MT M) for performance comparisons. The w that maximizes the E(MTM) is set based on N. There are ( L f w) ways to choose w subframes for each of N users. In order to have n active users in a subframe, N n users must choose w and the n active nodes must choose w 1 subframes out of the remaining L f 1 subframes. Therefore, the expected value of MTM is: E(MTM) = ( N N Lf ) 1 (N n) ( Lf ) 1 n n)( w w 1 ) N np s (n). (3.1) n= Message Retransmission ( Lf w The IEEE p broadcast mode does not have a retransmission mechanism. The presented rebroadcasting algorithms in[72] do not take into account the recently emerged congestion problem in VANETs. In this section, we study how a rebroadcasting scheme performs when there is both congestion and channel loss. At first glance, the use of message retransmission, in the presence of congestion, seems counter intuitive. On the other hand, since we intentionally drop some messages to avoid congestion, further message losses might not be tolerated. Based on the presented congestion control algorithm in the previous section, the number of active nodes in each subframe can be controlled and therefore retransmission can still be a viable solution to alleviate message loss. We assume each subframe is slotted and the size of each timeslot is equal to the message transmission time M. We consider the Synchronized R Persistent Repetition (SPR) scheme [72]. At each timeslot, each active node retransmits the generated message with probability p (Fig. 3.6). If there are n active nodes in a subframe it is easy to show that the optimal p is 1. We assume each node attaches L n f bits to its message to represent its activity pattern in each subframe inside the frame.

53 Chapter 3. Network Coded Safety Broadcasting 37 L f Subframes CCH CCH CCH CCH CCH CCH CCH CCH CCH CCH w subframes are randomly chosen transmission at each timeslot with probability " LTimeslots Figure 3.6: SPR in an active subframe The 1 bits represent the subframes in which the node is active. Also we assume nodes do not change their transmission pattern. Since each node knows its neighbours and theirs transmission pattern based on the received messages, it knows the number of active nodes in each subframe and can adjust its retransmission probability accordingly. Next, we find the P s (n) for this retransmission scheme. All users want to collect the messages of all n active nodes by the end of the subframe. At each timeslot, and independent of other nodes, the transmitted packet of an active node does not collide with other transmissions with probability p s = p(1 p) (n 1). If we consider the messages as coupons, our problem can be seen as a modified version of the coupon collection problem [47]. The completion time D (reception delay of all n messages) can be divided into n rounds such that by the end of the j th round the j th distinct message is received. The duration of the j th round, D j, has a Geometric distribution with the success probability p j = (n (j 1))p s (1 p e ). The completion time D is the sum of n Geometric random variables with distinct success probabilities and can be modelled as a discrete phase type distribution PH d (τ,t) with the following

54 Chapter 3. Network Coded Safety Broadcasting 38 transition matrix: T = 1 p 1 p 1 1 p 2 p p n 1 p n 1 1 p n. Using the CDF of D, the probability of failure (loss), P l (n), can be computed as: P l (n) = Pr(D > L) = 1 F D (L) (3.2) = τt L 1. where τ = [1,0,,0] and L = T CCH R is the number of timeslots in a subframe. M 3.3 Message Coding In this section, we propose an algorithm that uses random linear network coding in combination with message rebroadcasting. Based on the introduced repetition-based scheme in the previous section, each node potentially has multiple transmission opportunities in a subframe. The same copy of the message is retransmitted to account for the channel loss. However, a node can linearly combine the already heard messages and transmit the coded message. In this section, we consider the application of random linear coding together with SPR (Fig. 3.7). We call this algorithm Synchronized p-persistent Coded Repetition (SPCR). The random linear coding algorithm is simple: each node enqueues all the received messages and when it has a transmission opportunity based on its retransmission pattern

55 Chapter 3. Network Coded Safety Broadcasting 39 CC m 1 m 1 m 1 U 1 $ m 2 m 2 m 2 U 2 m 1 a 1 m 1 +a 2 m 2 U 1 $ m 2 U 2 b 1 m 1 +b 2 m 2 U 3 U U 1 U 2 4 Figure 3.7: SPCR vs. SPR it broadcasts a random linear combination of all the already received messages in its queue with the coefficients in GF(q). At the end of the subframe, if the node has n linearly independent coded vectors it can decode all the original packets. At the end of each subframe, all nodes empty their queues and start a new transmission for the next subframe. Algorithm 1 outlines the transmission algorithm of the sublayer. All nodes actively listen to the channel, when they are not transmitting, and queue all the received packets. The queue resets at the end of each CCH interval. Let us consider a single cell of n active nodes (n 1, n 2,, n n ). The time is slotted and the nodes are synchronized. Each subframe consists of L timeslots. At the beginning of each subframe each active node generates a message that should be received by all other nodes within the subframe. The generated message is retransmitted several times during a subframe. When a node has a transmission opportunity, it transmits a random linear combination of all already received packets in its queue. If m k i represents the k th retransmission of the n i s message, then the m k i can be expressed as a linear combination

56 Chapter 3. Network Coded Safety Broadcasting 40 Algorithm 1 Transmission Algorithm for every subframe do message generated (congestion control:) if node is not active then drop the message end if (SPCR:) for every time slot in a subframe do if transmitting (with probability p) then create a random linear combination of the messages in the queue broadcast the coded message end if end for reset the queue end for of all the original messages: m i k = n c j m j (3.3) j=1 in which c j is a random coefficient in GF(q). The coefficient vector c k i = (c 1,c 2,,c n ) is a vector in q n. Note that some of c i s can be zero. If a node receives n linearly independent equations, it can decode all the original messages. The coefficient vectors for the original messages at the beginning of a subframe are the standard basis for q n Coding Overhead In random linear network coding, the random coefficients should be attached to the coded message. In a congested network with large n the coding overhead can be comparable to the size of the message. For example, for GF(2 8 ), if there are 200 vehicles in a cluster, the maximum coding overhead will be 200 Bytes, which is comparable to the message size of Bytes. To reduce the overhead, in [42], the authors have suggested to attach the seed of the random number generator. The seed specifies the sequence of random coefficients in a coded message. New coded messages can be generated from received coded messages.

57 Chapter 3. Network Coded Safety Broadcasting 41 In this case, the seeds of all the included coded messages should be attached to the new coded message. This can potentially result in excessive overhead. As a result, the introduced algorithm in [42] can only encodes the uncoded messages. However, in our algorithm, we need to encode all the received coded messages. In the following, we show how we can indeed find the corresponding seed to the new coded message which considerably reduces the overhead. Linear Feedback Shift Registers(LFSRs) are an efficient way for implementing Pseudo Random Number Generators (PRNGs) in a Galois field. A LFSR implementation based on a primitive polynomial of GF(2 m ) has m shift registers and a period of 2 m 1. The state of the shift register represents the binary coefficients of the corresponding polynomial to a member of the Galois field. Based on an initial seed (LFSR state), the sequence of the states of the LFSR is equivalent to the sequence of pseudo random numbers from a Galois field. The following lemma shows how the seed of the random coefficients of the random linear combination of coded messages relates to the seeds of the coded message coefficients. Lemma 3.1. If s 1,s 2,,s k GF(2 m ) are the seeds corresponding to k received coded messages m 1,m 2,,m k, the seed for the corresponding random number sequence to k i=1 c im i is s c = k i=1 c is i, where c i s are randomly chosen from GF(2 m ). It is easy to prove this lemma. Assume C i (x) and Si 0 (x) are the polynomial representation of c i and s i. Subsequent states are related through S n+1 i (x) = XSi n (x) for a Galois LFSR. If the first state of the Galois LFSR is a linear combination of the received seeds with coefficients c i s, we have S 0 c(x) = k i=1 C i(x)s 0 i(x). Multiplying both sides by X n, we get Sc n(x) = k i=1 C i(x)si n (x), which demonstrates that every subsequent shift is also a linear combination of the shifted version of the received seeds with the same coefficients. Using this lemma, the coding overhead is limited to m bits. If we use a large enough field, the random sequences are distinct with high probability. For example, if we use

58 Chapter 3. Network Coded Safety Broadcasting 42 GF(2 32 ) and a primitive polynomial, there are distinct random sequences and the coding overhead is only 4 Bytes. Since each coded message does not necessarily contain all the original messages, we should attach another n bits together with the PRNG seed to specify the original messages included in each coded message. So the total overhead of SPCR is M h = m+n 8 Bytes. To fairly compare SPCR with other algorithms and to account for the coding overhead, we assume the message size for SPCR is M = M +M h, in which M is the original message size. For a fixed channel rate, this increased message size results in smaller number of timeslots in a subframe Analysis Assume S l t is the spanned subspace by all the received equations of n l up to time t. We can partition the set of all other nodes into two disjoint subsets: the innovative nodes, I l t, and non-innovative nodes, I l t, defined as: i I l t : S i t S l t i I l t : Si t S l t At the beginning of the subframe (t=0), for an inactive node all the active nodes are innovative and Ii 0 = n. For an active node, all other active nodes are innovative and I i 0 = n 1. The success probability for node n l is defined as the probability that it decodes all the original messages by the end of the subframe. The success probability for node n l can be written as follows: P s (n) = Pr(dim(S L l ) = n).

59 Chapter 3. Network Coded Safety Broadcasting i-1 i Starting State (The first time that the dimension of S becomes i and there exist at most i non-innovative nodes) A Absorbing State (Dimension of S becomes i+1) Figure 3.8: Markov chain model The set of original packets creates an n-dimensional subspace. Network coding creates a linear combination of the data vectors, which belongs to the same subspace since the subspace is closed under linear operation. If dim(s t l ) = i, then we should have at least n i innovative nodes since the packets of the innovative nodes and S t l should span the whole space. It is impossible to span a space with dimension n i with less than n i vectors. We summarize this fact in the following proposition: Lemma 3.2. If dim(s t l ) = i, then the minimum number of innovative nodes for n l at time t is n i, or the maximum number of non-innovative nodes is i. This is an important result that shows why SPCR should perform better than SPR. In SPR, the number of innovative nodes is equal to n i, which is the minimum value of that achieved by SPCR. The following theorem gives an upper bound on the loss probability of SPCR in a network of n nodes with symmetric erasure channel. Theorem 3.2. Using SPCR in a symmetric erasure channel with erasure probability p e, an upper bound for the loss probability can be obtained using the following inequality: P l τt L 1

60 Chapter 3. Network Coded Safety Broadcasting 44 where : τ = [1,0,,0] t i = 1 T i 1 T 0 [t 0 0] T 1 [t 1 0] T = T n 2 [t n 2 0] T n 1 and T i s are defined as: ( ) i j [T i ] jk = (1 p e ) k j pe i k+1 p j, j k k j [T i ] jj = 1 p j (1 p i j+1 e ) p j = (n (i j))p(1 p) n 1 0 j k i n 1 in which L is the number of timeslots in a subframe, n is the number of active nodes in the cluster, and p is the transmission probability at each timeslot in the SPCR scheme. Proof. For an inactive user, we define the random variable D i as: D i = {min t S t = i+1} {min t S t = i}. The first time that the dimension of S t becomes i, Proposition 1 guarantees that there are at most i non-innovative nodes. To find D i, we assume the maximum number of non-innovative nodes exists. It is straightforward to show that the computed D i with this worst case assumption stochastically dominates D i. Di can be modelled as the

61 Chapter 3. Network Coded Safety Broadcasting 45 absorption time of the Markov chain depicted in Fig This Markov chain model is inspired by [18]. The state of the Markov chain relates to the number of non-innovative nodes. At state j (1 j i) there are (i j) non-innovative nodes. The first state represents the first time that the dimension of the node becomes i. The time transition from this state till the time that the dimension of S t l becomes i+1 (absorbing state) has a discrete time phase type distribution PH d (τ,t i ). T i is an upper triangular matrix and τ = [1,0,,0]. The absorbing state is the first time that the dimension of S t becomes i+1. Starting from the initial state (State 0), after each transmission, if the node receives an innovative packet, it moves to the absorbing state otherwise depending on how many of non-innovative nodes receive this message, the number of non-innovative nodes may decrease. Now, we find the transition probabilities of the Markov chain [T i ] jk (0 j k i). We define the following events: A = {One of n (i j) innovative nodes transmits successfully} B = {(k j) non-innovative nodes become innovative} C = {dim(s t ) increases to i+1} When j k, we can compute the transition probabilities as follows: [T i ] jk = P(B C A)P(A) = P(B A)p e (n (i j))p(1 p) n 1 The B A has a binomial distribution with parameters (i j) and p e : [T i ] jk = ( ) i j (1 p e ) (k j) pe i k+1 P(A) (3.4) k j

62 Chapter 3. Network Coded Safety Broadcasting 46 for j = k, we have [T i ] jj = p i j+1 e P(A)+1 P(A) (3.5) In this Markov chain, D i is the number of timeslots starting from state i to get to the absorbing state. Since D i dominates D i the loss probability P l can be upper bounded as follows: n 1 n 1 P l = Pr( D i L) Pr( Di L). i=0 The random variabled = n 1 i=0 D i i=0 is thesum of nindependent phase type distributions. Theorem in [38] shows that D has a discrete phase type distribution PH d (τ,t) with the following parameters : τ = [1,0,,0] t i = 1 T i 1 T 0 [t 0 0] T 1 [t 1 0] T = T n 2 [t n 2 0] T n 1 Evaluating the CDF of D at L completes the proof. To evaluate how tight the derived bound is, simulations are performed for a binary symmetric channel with p e = 0.1. Two network topologies with low (n = 10) and high

63 Chapter 3. Network Coded Safety Broadcasting N=10 Loss Probability 10 1 SPCR Sim.: pe=0.1 SPCR Upper bound: pe= β Figure 3.9: Simulation results vs. theoretical upper bound: n = 10 node density (n = 50) are considered. The number of timeslots in a subframe is assumed to be a function of node density: L = β nlog(n). As can be seen in Fig. 3.9 and Fig. 3.10, for both cases the derived loss probability bound is tight for a low erasure probability. For higher erasure probabilities (p e = 0.2 and p e = 0.4) and for low and high densities, the comparison is shown in Fig and Fig As can be seen for higher erasures the bound is not as tight but still good for p e = 0.2. Also the tightness does not change drastically by increasing the number of nodes. 3.4 Numerical Results The erasure probability is a key factor for determining the reliability. To see how the erasure probability changes in a typical vehicular environment, first we consider a realistic vehicle-to-vehicle channel and find the erasure probability as a function of distance. We use the Nakagami channel model for the vehicle-to-vehicle communication link. The

64 Chapter 3. Network Coded Safety Broadcasting N= Loss Probability 10 2 SPCR Sim.: pe=0.1 SPCR Upper bound: pe= β Figure 3.10: Simulation results vs. theoretical upper bound: n = N=10 Loss Probability 10 1 SPCR Sim.: pe=0.2 SPCR Upper bound: pe=0.2 SPCR Sim.: pe=0.4 SPCR Upper bound: pe= β Figure 3.11: Simulation results vs. theoretical upper bound: n = 10

65 Chapter 3. Network Coded Safety Broadcasting N= Loss Probability 10 2 SPCR Sim.: pe=0.2 SPCR Upper bound: pe=0.2 SPCR Sim.: pe=0.4 SPCR Upper bound: pe= β Figure 3.12: Simulation results vs. theoretical upper bound: n = 50 probability density function of the signal amplitude Y based on this channel model is: f Y (y) = 2mm 1 1 y 2m 1 1 Γ(m 1 )Ω m 1 e m 1 y2 Ω m 1 1 2, Ω 0 in which Ω is the average received power and m 1 is the fading figure. A path loss component of is reported for highways [46]. Here, we assume the path loss component of 2 for our evaluations. In [16], the parameter m 1 has been estimated based on empirical measurements for a vehicle-to-vehicle link in a highway: 1.5 d < 80 m 1 = 0.75 d > 80 (3.6) This estimate of m 1 with the physical and MAC layer parameters listed in Table 3.1 has been used in our evaluations. We assume the transmission power is 20 dbm (Class C)

66 Chapter 3. Network Coded Safety Broadcasting Reception Probability Mbps Mbps 6Mbps 9Mbps Mbps 18Mbps Mbps 27Mbps Distance(m) Figure 3.13: Reception Probability vs. Distance [2] and the reception and transmission antenna gain is 2. The minimum reception threshold for a 10Mhz channel is specified in IEEE standard and can be found in Table 3.2. As we expect, for higher rates the threshold is higher. Fig shows how the successful reception probability drops as a function of distance for allowed rates in IEEE p. The erasure probability is a function of distance. However, our analysis in previous section was for symmetric erasure channels. For the numerical evaluation, we assume the p e for a communication range of R d is the average of erasure probability over that range: p e = 1 Rd p e (x)dx. (3.7) R d 0 This seems to be a valid choice as it reasonably relates the erasure probability to the rate. Fig is the comparison of loss probability for SPCR, SPR and IEEE802.11p as a function of the number of active nodes in a subframe and for R d = 500m. We have used the derived loss probability upper bound in the previous section for numerical

67 Chapter 3. Network Coded Safety Broadcasting Ps(n) p: 3Mbps SPCR: 9Mbps SPR: 9Mbps 0.3 SPCR: 12Mbps SPR: 12Mbps SPCR: 18Mbps 0.2 SPR: 18Mbps SPCR: 24Mbps 0.1 SPR: 24Mbps SPCR: 27Mbps SPR: 27Mbps Number of active nodes in a subframe Figure 3.14: Loss probability comparison for different rates evaluations of SPCR. For SPR and SPCR, increasing the rate results in more timeslots in a subframe and more retransmission opportunities. On the other hand, by increasing the rate, the erasure probability increases. For IEEE p, channel rate is set to 3Mbps, which corresponds to the lowest channel error. As can be seen, the optimal rate for SPR is 12 Mbps and for SPCR is 18Mbps. By increasing the number of nodes, the loss probability for repetition based schemes increases with a higher slope compared to IEEE p. This is due to excessive number of collisions when there is congestion. For safety applications, we are generally interested in lower loss probabilities. For lower loss probabilities, SPR performs much better than IEEE p. The SPCR performs significantly better than SPR over the entire range. It is also interesting to see that for the similar rates, increasing the rate widens the gap between the SPCR and SPR. Lower rates correspond to smaller number of timeslots in a subframe. The decoding probability for SPCR drops for smaller subframes. For 30 nodes and L f = 10, Fig shows the E(MTM) versus w, the number of active subframes in L f subframes. As can be seen, there is an optimal w that maximizes E(MTM). The optimal w depends on the total number of nodes, which for N = 30, is

68 Chapter 3. Network Coded Safety Broadcasting 52 Table 3.2: Minimum reception threshold for IEEE p Rate (Mbp) Minimum RX threshold (dbm) , 5 and 7, for IEEE p, SPR and SPCR, respectively. The SPCR has the highest optimal w and allows more nodes to be active in each subframe with a higher E(MTM). Note that the IEEE p curve indeed represents the congestion control algorithm in [22], in which the rate is adjusted according to the channel occupancy time feedback. The number of neighbours is strongly correlated with the channel occupancy time, and different w s correspond to different rates. For a more congested cluster with 60 nodes, a similar trend can be seen in Fig Although the optimal w is smaller, the optimal E(MTM) is not much lower than the less congested scenario E(MTM) IEEE802.11p SPR SPCR w Figure 3.15: E(MTM) vs. w (number of active subframes in 10 subframes): n = 30

69 Chapter 3. Network Coded Safety Broadcasting IEEE802.11p SPR SPCR E(MTM) w Figure 3.16: E(MTM) vs. w (number of active subframes in 10 subframes): n = 60

70 Chapter 3. Network Coded Safety Broadcasting 54 Table 3.3: Table of Notations V ANET : Vehicular Ad-hoc NETwork W AV E : Wireless Access in Vehicular Environment CCH : Control CHannel SCH : Service CHannel W SM : Wave Short Message SF R : Synchronized Fixed Repetition SP R : Synchronized Persistent Repetition SP CR : Synchronized Persistent Coded Repetition P OC : Positive Orthogonal Code GP S : Global Positioning System P s (n) : The probability of receiving a full map of size n T CCH : The length of CCH interval MT M : Map Throughput Metric P LCP : PHY Layer Convergence Procedure R : Data rate T h : PLCP duration : Guard time duration T g p e : Erasure probability AIF S : Arbitration inter-frame spacing M : Safety Message length CW : Contention Window size σ : Time slot duration in IEEE p L f : Number of subframes in a frame N : Number of nodes in a cluster n : Number of active nodes w : Number of subframes in which a node is active p : Transmission probability in a time slot D : Completion delay L : Number of time slots in a subframe LF SR : Linear Feedback Shift Register P RNG : Pseudo Random Number Generator q : Finite field size M h m 1 : Network coding overhead : Fading figure Ω : Average received power R d : Communication range

71 Chapter 4 Network Coded Safety Broadcasting: Delay Analysis In this chapter, we use the proposed Phase-type distribution in Chapter 3 to prove fundamental delay bounds for periodic safety broadcasting application. First, we prove a general asymptotic upper bound for completion delay. Then, we find a tighter upper bound for a range of p e. These bounds provide useful analytical insights into delay performance of random network coding. In the following section, we review our proposed phase-type distribution model. 4.1 Phase-type Model In previous chapter, we showed thecompletion delay of SPCR (D SPCR ) isupper bounded by the absorption time of a discrete phase-type distribution PH(T,τ) with the following parameters: τ = [1,0,,0] t i = 1 T i 1 55

72 Chapter 4. Asymptotic Delay Bound i-1 i Starting State (The first time that the dimension of S becomes i and there exist at most i non-innovative nodes) A Absorbing State (Dimension of S becomes i+1) Figure 4.1: Markov chain model T = T 0 [t 0 0] T 1 [t 1 0] T n 2 [t n 2 0] T n 1 and T i s are defined as: ( ) i j [T i ] jk = (1 p e ) k j pe i k+1 p j, j k k j [T i ] jj = 1 p j (1 p i j+1 e ) p j = (n (i j))p(1 p) n 1 0 j k i n 1 in which p e is the channel erasure probability, L is the number of timeslots in a subframe, n is the number of active nodes in the cluster, and p is the transmission probability at each timeslot in the SPCR scheme. In the next sections, we use this model to prove asymptotic bounds for completion delay.

73 Chapter 4. Asymptotic Delay Bound Delay Analysis If D i is the absorption delay for the Markov chain defined in the previous section, then D SPCR is the sum of all absorption delays: D SPCR = n 1 i=0 D i. Since D i has a discrete phase-typedistributionph d (τ,t i ),E(D i )canbefoundusingthefollowingformula([38]): E(D i ) = τ(i T i ) 1 1, therefore: n 1 E(D SPCR ) = τ(i T i ) 1 1. (4.1) i=0 The following theorem provides an asymptotic upper bound for the expected SPCR delay: Theorem 4.1. For erasure probability p e and number of nodes n, an asymptotic upper bound for E(D SPCR ) is: E(D SPCR ) = O( en(log(n(1 p e))+λ) 1 p e ) in which λ is the Euler-Mascheroni constant. Proof. We consider the modified discrete phase type distribution in Fig. 4.2, PH(T,τ). We assume the transition probabilities are defined as follows: [T i ] j,j+1 = i [T i ] jk = k=j+1 i k=j+1 = p e p j (1 p i j e ) 0 j i 1 [T i ] jj = [T i ] jj = 1 p j (1 p i j+1 e ) ( ) i j (1 p e ) k j pe i k+1 p j k j

74 Chapter 4. Asymptotic Delay Bound i-1 i Starting State (The first time that the dimension of S becomes i and there exist at most i non-innovative nodes) A Absorbing State (Dimension of S becomes i+1) Figure 4.2: Modified Markov chain model p j = (n (i j))p(1 p) n 1 ItiseasytoseetheabsorptiontimeofthisMarkovchainstochasticallydominatesD SPCR, providing an upper bound for the expected completion delay: E(D SPCR ) n 1 i=0 i l=0 [I T i ] 1 0l. Since T i is a triangular matrix, for l = 0 we have: [I T i ] 1 00 = 1 1 [T i ]. 00 For l 1: [I T i] 1 0l = C l0 I T i = ( 1)l M l0 I T i In which C is the matrix of cofactors and M l0 is the minor for [I T i ] l0. M l0 is the

75 Chapter 4. Asymptotic Delay Bound 59 determinant of a block diagonal matrix: M l0 = G l 0 0 G u, in which G l and G u are lower and upper triangular matrices: G l = G u = [T i ] [T i ] 11 [T i ] [T i] l 1,l 1 [T i] l 1,l 1 [T i ] l+1,l+1 [T i ] l+1,l+2 1 [T i ] l+2,l+2 [T i ] l+2,l [T i] i 1,i 1 [T i ] ii. For the block diagonal matrix: M l0 = G l G u, so we have: [I T i] 1 0l = = ( 1) l G l G u i j=0 (1 [T i ] jj) l 1 j=0 [T i ] j,j+1 i j=l+1 (1 [T i j=0 (1 [T i ] jj) i ] jj), by substituting and simplifications: [I T i ] 1 0l = = l 1 e ) j=0 p ep j (1 p i j l j=0 p j(1 p i j+1 e ) p l e p l (1 p i+1 e ).

76 Chapter 4. Asymptotic Delay Bound 60 So E(D SPCR ) can be bounded as: E(D SPCR ) n 1 i=0 i l=0 p l e (n i+l)p(1 p) n 1 (1 p i+1 e ), (4.2) p s = p(1 p) n 1 and since we assume p = 1 n in order ro maximize p s, we have p s = Θ( 1 ne ). E(D SPCR ) then can be bounded as: n 1 E(D SPCR 1 ) = O(ne (1 p i+1 i=0 n 1 = O(ne = O(ne The second term is: i=0 q=0 n 1 p q e q=0 i=0 e ) i l=0 p q e n i+q pi+1 e p l e (n i+l) ) (1 p i+1 e ) q=0 p q e n+1+q n 1 1 (n i+q)(1 p i+1 e ) ne i=0 ) p i+1 e 1 p i+1 e q=0 p q e n+1+q ). n 1 p i+1 e ne 1 p i+1 e i=0 q=0 p q e n+1+q n 1 p i+1 e 1 = O(ne 1 p i+1 e n i=0 e = O( 1 p e n 1 i=0 p q e ) q=0 p i+1 e (1 p i+1 e ) ). Since the sum is convergent: E(D SPCR ) = O(ne = O(ne = O(ne = O(ne n 1 p q e q=0 i=0 1 (1 p i+1 e )(n i+q) ) n 1 p q e p k e q=0 k=0 i=0 p q e q=0 k=0 p k(1+n) e n 1 z +ne z=1 (p k e) i ) (geometric expansion) n i+q n z=1 p q e q=1 z=1 p kz e ) (change of variables: n i = z) z +q n 1 z +q +ne p q e q=0 k=1 p k(1+n) e n z=1 p kz e z +q ),

77 Chapter 4. Asymptotic Delay Bound 61 the last term can be bounded as: en p q e q=0 k=1 p k(n+1) e n z=1 p kz e z +q = O(en n z=1 k=1 en = O( 1 p e en = O( 1 p e e = O( 1 p e = O(1) n z=1 n z=1 n i=1 p k(n+1 z) e 1 z k=1 q=0 p k(n+1 z) e ) pe n+1 z z(1 pe n+1 z ) ) p i e (1 p i e )) p q e z +q ) The n th harmonic number H n = n 1 k=1 can be asymptotically upper bounded as H k n = O(log(n)+λ) ([47]) in which λ is the Euler-Mascheroni constant. Finally E(D SPCR ) can be bounded as: E(D SPCR ) = O(en(log(n)+λ)+en = O(en(log(n)+λ)+en p q e (H n+q H q )) q=1 q=1 p q eh n+q en log( 1 1 p e ) 1 p e ) p e = O(en(log(n)+λ)+en(log(n)+λ) en log( 1 ) 1 p e 1 p e = O( en(log(n(1 p e))+λ) 1 p e ). 1 p e ) In order to evaluate the absorption time of the modified Markov chain in Fig. 4.2, we have numerically compared the expected completion time for SPR, SPCR and the SPCR upper bound in (4.2). The comparisons for n = 20 and n = 50 can be found in Fig. 4.3 and Fig For smaller n, the SPCR bound is tighter. This is due to the fact that for denser networks after one innovative transmission, the probability that more than one non-innovative node receive the transmission becomes larger.

78 Chapter 4. Asymptotic Delay Bound SPR SPCR SPCR: Bound I n= E(D) (timeslots) pe Figure 4.3: Delay Bound based on Modified Markov Chain: n = 20 To see how tight the asymptotic bound is for a small n, we have compared the SPCR upper bound in (4.2) and the corresponding asymptotic bound in Theorem 4.1. As can be seen in Fig. 4.5 and 4.6 for relatively small n = 50 the asymptotic bound is almost exact. From Section 3.2.2, the completion delay of SPR can be written as: E(D SPR ) = n i=1 1 ip(1 p) n 1 (1 p e ) = θ( en(log(n)+λ) 1 p e ). Using this and the bound in Theorem 4.1 the following corollary is easily followed: Corollary 4.2. E(D SPR ) E(D SPCR ) = Ω(en log( 1 1 pe ) 1 p e ). This proves that asymptotically SP CR provides a linear factor improvement over SP R expected competition delay.

79 Chapter 4. Asymptotic Delay Bound SPR SPCR SPCR: Bound I n= E(D) (timeslots) pe Figure 4.4: Delay Bound based on Modified Markov Chain: n = n= E(D) (timeslots) SPCR: Bound I SPCR: Bound I Asymptotic pe Figure 4.5: Delay Bound Performance for Small n: n = 20

80 Chapter 4. Asymptotic Delay Bound n= E(D) (timeslots) SPCR: Bound I SPCR: Bound I Asymptotic pe Figure 4.6: Delay Bound Performance for Small n: n = Tighter Delay Upper Bound Here we prove a tighter delay upper bound for SPCR for a range of erasure probabilities. First, we prove Lemma 4.1 and Theorem 4.2, then we use these results to prove a tighter upper bound for completion delay in Theorem 4.3. The following Lemma relates the expected SPCR delay to T n 1 : Lemma 4.1. n 1 n 1 E(D SPCR ) = [I T n 1 ] 1 ij i=0 j=0 Proof. From the definition of T i s in Section 4.1 we have: T i = [T i] 00 [T i ] 0i 0 T i 1

81 Chapter 4. Asymptotic Delay Bound 65 Since T i is a triangular matrix, it is easy to show: [I T i ] 1 = [I T i] 1 00 [I T i ] 1 0i 0 [I T i 1 ] 1 Using this recursive equation, the following formula for [I T n 1 ] 1 can be derived: [I T n 1 ] 1 = [I T n 1 ] 1 00 [I T n 1 ] 1 01 [I T n 1 ] 1 0,n 1 [I T n 2 ] 1 00 [I T n 2 ] 1 01 [I T n 2 ] 1 0,n [I T 0 ] 1 00 From (4.1): E(D SPCR ) = = n 1 i i=0 l=0 n 1 n 1 i=0 j=0 [I T i ] 1 0l [I T n 1 ] 1 ij Theorem 4.3. For m,n,k Z +, 0 p 1, k = αm for some 0 < α < 1 and m : m n=k ( ) n (1 p) n = k (1 p) k p k+1 1 (1 p) k 2 p k+1 if α < p if α = p 0 otherwise Proof. f(k,m) = = m ( n k m n=k n=k ) (1 p) n ( n 1 k 1 ) (1 p) n + m ( ) n 1 (1 p) n k n=k

82 Chapter 4. Asymptotic Delay Bound 66 = (1 p)(f(k 1,m 1)+f(k,m 1)) ( ) m = (1 p)(f(k 1,m)+f(k,m) (1 p) m ( + k 1 = (1 p)f(k 1,m) ( ) m+1 k (1 p) m+1 p ( = 1 p ( ) m+1 f(k 1,m) )(1 p) m p k ( ) m )) k (4.3) For k=0 we have: f(0,m) = 1 (1 p)m+1 p Using this and the recursion formula in (4.3) we can find f(k,m): f(k,m) = (1 p)k p k+1 (1 p) m j=0 = (1 p)k p k+1 (1 (1 p) m+1 = (1 p)k p k+1 ( 1 k j=0 k ( 1 p ( ) m+1 p )k j+1 j k ( ) p m+1 ( 1 p )j ) j j=0 ) ) (1 p) m+1 j p j ( m+1 j The summation is the CDF of a binomial distribution with parameters (m+1,p). By using the Chernoff bound for k < (m+1)p we have: k j=0 (1 p) m+1 j p j ( m+1 j ) 2 e ((m+1)p k) 2p(m+1) = 0 (for m ), for k > (m+1)p in the neighbourhood of (m+1)p using de Moivre-Laplace theorem: ( ) m+1 (1 p) m+1 j p j = j 1 e (k (m+1)p) 2 2(m+1)p(1 p) = 0, 2π(m+1)p(1 p) for k = (m + 1)p, following the central limit theorem the binomial distribution converges

83 Chapter 4. Asymptotic Delay Bound 67 to a gaussian distribution with µ = (m+1)p and σ 2 = (m+1)p(1 p): (m+1)p j=0 (1 p) m+1 j p j ( m+1 j ) = Q(0) = 1 2 Finally in the following theorem we prove a tighter asymptotic upper bound for expected SPCR delay when p e 1 2 ( 5 1): Theorem 4.4. For channel erasure probability p e 1 2 ( 5 1) and n nodes, the expected SPCR delay can be asymptotically upper bounded as: E(D SPCR ) = O(enlog(n)+ef(p e )n) 1 f(p e ) = log( )+ Li 2(p e ) 1 p e log( 1 p e ) +λ in which Li 2 is Spence s function and λ is Euler-Mascheroni constant. Proof. For notational simplicity we assume T = T n 1 and Y = [I T] 1. Y ij is the expected delay in state j before absorption, given the initial state i (E(D ij )). Y jj = 1 1 T jj is the expected delay at state j before absorption. For j > i using the total probability theorem we can write: E(D ij ) = k E(D ij next state is k)pr(next state is k). When the next state is the absorbing state or any other state with k > j, the conditional expectation is zero: E(D ij ) = = j E(D ij next state is k)pr(next state is k) k=i j E(D kj )T ik +T ii E(D ij ), k=i+1

84 Chapter 4. Asymptotic Delay Bound 68 which results in the following recursive formula for Y ij : Y ij = j k=i+1 T ik 1 T ii Y kj, (4.4) Y 00 = 1 1 T 00 and for j = 1,2 we have: j = 1 : Y 11 = Y 01 = 1 Y i1 = i=0 j = 2 : Y 22 = = Y 12 = Y 02 = 2 Y i2 = i=0 1 1 T 11 T 01 Y 11 1 T 00 1 T T 11 (1 T 00 )(1 T 11 ) 1 1 T 22 T 12 Y 22 1 T 11 1 T 01 T 12 (T 01 Y 12 +T 02 Y 22 ) = 1 T 00 (1 T 00 )(1 T 11 ) Y 22 + T 02 Y 22 1 T 00 1 T T 22 (1 T 11 )(1 T 22 ) + T 02 (1 T 00 )(1 T 22 ) T 01 T 12 + (1 T 00 )(1 T 11 )(1 T 22 ) Following the same procedure and evaluating the recursive formula in (4.4), the following formula for E(D SPCR ) can be derived: E(D SPCR ) = = n 1 n 1 Y ij i=0 j=0 n 1 n 1 j T j=0 jj j=1 i=0 }{{} S 0 T ij (1 T ii )(1 T jj ) } {{ } S 1

85 Chapter 4. Asymptotic Delay Bound 69 n 1 j 1 k 1 + j=2 k=1 i=0 T ik T kj (1 T ii )(1 T kk )(1 T jj ) } {{ } S T 01 T 12 T (n 2)(n 1) (1 T 00 )(1 T 11 ) (1 T (n 1)(n 1) ) } {{ } S n 1. T jk can be written, according to Section 4.1, as: T jj = 1 (j +1)p s (1 pe n j ) ( ) n 1 j T jk = (1 p e ) k j pe n k (1+j)p s. n 1 k S 0 can be upper bounded as follows: S 0 = = = n 1 j=0 n 1 j=0 n 1 j=0 1 1 T jj 1 (j +1)p s (1 p n j e ) 1 (j +1)p s + k=1 p kn e n 1 j=0 (p k e )j (j +1)p s (geometric expansion) p s = p(1 p) n 1 and since we assume p = 1 n in order ro maximize p s, we have p s = Θ( 1 en ). S 0 then can be upper bounded as: S 0 = O(en(log(n)+λ)+en k=1 p (n+1)k e log( 1 1 p k e the sum is convergent so S 0 = O(en(log(n) + λ)). λ is the Euler-Mascheroni constant. For S 1 we have: )), S 1 = n 1 j 1 T ij (1 T ii )(1 T jj ) j=1 i=0

86 Chapter 4. Asymptotic Delay Bound 70 = = = = n 1 j 1 ( n 1 i j i ) (1 pe ) j i p n j e (1+i)p s (i+1)(j +1)p j=1 i=0 2 s(1 pe n i )(1 pe n j ) ( n 1 j 1 n 1 i ) (1 pe ) j i p j i e n j (j +1)p j=1 i=0 s (1 pe n i )(1 pe n j ) ( n 1 pe n j (1 p e ) j j 1 n 1 i ) (1 pe ) i n 1 j (j +1)p j=1 s (1 pe n j ) 1 p n i i=0 e n 1 j 1 j=1 pe n j (1 p e ) j (j +1)p s (1 pe n j ) k=0 p nk e i=0 ( n 1 i n 1 j ) ((1 p e )p k e ) i. (4.5) We first prove the following lemma which we refer to in the rest of the proof: Lemma 4.2. j 1 ( ) n 1 i p i = n 1 j i=a p j O( ) if j (n 1)p+a(1 p) (1 p) n j 0 otherwise Proof. By change of variables l = n 1 i: j 1 i=a ( n 1 i n 1 j n 1 a )p i = p 1 n l=n j ( ) l p l n 1 j If we consider the l = n j 1 in the summation, we get the following upper bound: j 1 i=a ( n 1 a n 1 i ( )p i p 1 n n 1 j l=n j 1 l n 1 j ) p l. Applying Theorem 4.3 to the summation concludes the proof. By applying Lemma 4.2 to (4.5), S 1 can be asymptotically upper bounded as: S 1 = O(en = O(en n 1 j=(n 1)(1 p e)p k e n 1 pe n j (1 p e ) j (j +1)(1 pe n j ) j=(n 1)(1 p e)p k e k=0 k=0 1 (j +1)(1 p n j p nk e e ) ( p k+1 e ((1 p e )p k e ) j (1 (1 p e )p k e) n j) ) n j ) 1 (1 p e )p k e

87 Chapter 4. Asymptotic Delay Bound 71 Let p k = p(k+1) e 1 (1 p e)p. p k 0 = 1 and for any k 1, p k < 1, we have: e S 1 = O(en = O(en n 1 k=0 j=(n 1)(1 p e)p k e n 1 p n j k (j +1)(1 p n j e ) ) j=(n 1)(1 p e) (j +1)(1 pe n j ) }{{} H 1 1 +en n 1 p n j k (j +1)(1 p k=1 j=(n 1)(1 p e)p k e n j ) e }{{} H 2 ). H 2 can be proved to be asymptotically constant. By change of variables n j = q: H 2 = O(en n k=1 q=1 p q k (n q +1)(1 p q e) ), For q = βnfor some 0 < β < 1 the nominator is exponentially dominant andsummations converge to zero. For finite vales of q, the denominator (n q+1) is asymptotically equal to n so we have: n p q k H 2 = O(e 1 p q ) e k=1 q=1 e = O( p k ) 1 p e k=1 e p k = O( ). 1 p e 1 p k Using the ratio test, it can be shown that the summation is convergent so H 2 = O(1). H 1 can be bounded similar to S 0. Finally, we can bound S 1 as follows: k=1 S 1 = O(en(log(n) log(n(1 p e )))) 1 = O(enlog( )). 1 p e

88 Chapter 4. Asymptotic Delay Bound 72 For S j : S j = = n 1 i j+1 =j n 1 i j+1 =j i 3 1 i 2 1 T i1 i 2 T i2 i 3 T ij i j+1 j+1 l=1 (1 T i l i l ) i 2 =1 i 1 =0 i 3 1 i 2 1 j ( n 1 il l=1 i 2 =1 i 1 =0 ) (1 pe n 1 i l+1 ) i l+1 i l p n i l+1 e (1+i j+1 )p j+1. s l=1 (1 pn i l e ) For notational simplicity, we define the following variables: ( ) n 1 il W l = (1 p e ) i l+1 i l p n i l+1 e n 1 i l+1 Z l = 1 p n i l e S j can be written as: S j = = = n 1 i j+1 =j n 1 i j+1 =j n 1 i j+1 =j i 3 1 i 2 1 i 2 =1 i 1 =0 i 3 1 i 2 =1 i 3 1 i 2 =1 (1 p e ) i 2 p n i 2 e j l=1 W l (1+i j+1 )p j+1 s l=1 Z l j l=2 W i 2 1 l (1+i j+1 )p j+1 s l=2 Z l i 1 =0 j l=2 W l (1+i j+1 )p j+1 s l=2 Z l i 2 1 k 1 =0 p k 1n e i 1 =0 ( n 1 i1 n 1 i 2 ( n 1 i1 ) n 1 i 2 (1 pe) i 2 i 1 p n i 2 e 1 p n i 1 e ) ((1 p e )p k 1 e ) i 1 (geometric expansion) Last summation can be found using Lemma 4.2: = O(en n 1 i j+1 =j i 4 1 i 3 =2 k 1 =0 i 3 1 i 2 =(n 1)((1 p e)p k 1 e ) j l=2 W l (1+i j+1 ) j l=2 Z l ( p k 1+1 e 1 (1 p e )p k 1 e ) n i 2 ).

89 Chapter 4. Asymptotic Delay Bound 73 Let P 1 = pk 1 e +1 S j = O(en 1 (1 p e)p k 1 e n 1 i j+1 =j (1 p e ) i 3 p n i 3 e = O(en n 1 i j+1 =j (1 p e ) i 3 p n i 3 e : i 4 1 j l=3 W l (1+i j+1 ) j+1 i 3 =2 k 1 =0 i 4 1 i 3 =2 i 3 1 i 2 =(n 1)(1 p e)p k 1 e l=3 Z l ( n 1 i2 j l=3 W l (1+i j+1 ) j+1 l=3 Z l k 1 =0 k 2 =0 (p k 2 e P 1 ) n ) n 1 i 3 (1 pe) i 2 P n i p n i ) 2 e i 3 1 i 2 =(n 1)((1 p e)p k 1 e ) By applying Lemma 4.2 to the last summation: S j = O(en n 1 i j+1 =j i 5 1 i 4 1 i 4 =3 k 1 =0 k 2 =0 i 3 =(n 1)(1 p e)p k 1 e (1+p k 2 +1 e ) By repeatedly applying Lemma 4.2, we have: j l=3 W l (1+i j+1 ) j+1 l=3 Z l ( n 1 i2 ( n 1 i 3 p k 2+1 e P 1 1 (1 p e )P 1 p k 2 e ) ((1 pe)p 1 p k 2 e ) i 2 ). ) n i 3 ). S j = O(en k 1 =0 k 2 =0 n 1 k j =0 i j+1 =(n 1)F j n i P j+1 j (1+i j+1 )(1 p n i j+1 e ) ), in which F j is defined as follows: F j = (1 p e )p k 1 e (1+pk 2+1 e +p k 2+1 e p k 3+1 e + +p k 2+1 e p k 3+1 e p k j+1 e ), and P j = p k j +1 e P j 1 1 (1 p e)p j 1 p k j e, P 0 = 1. Using this recursive formula an upper bound for P j

90 Chapter 4. Asymptotic Delay Bound 74 can be obtained: P j = j l=1 pk l+1 e 1 F j j l=1 pk l+1 e 1 (1 p e )p e j l=1 p2(l 1) e = j l=1 pk l+1 e 1+pe 2j+1. (4.6) When k 1 = 0,k 2 = 0,,k j = 0, it is easy to see P j = 1 and F j = 1 p j e: S j = O(en +en n 1 i j+1 =(n 1)(1 p j e) 1 (1+i j+1 )(1 p n i j+1 e ) n i P j+1 j k 1 =1 k 2 =1 k j =1 i j+1 (1+i j+1 )(1 p n i j+1 e ) ). }{{} C j (4.7) Similar to bounding S 1 we have: e C j = O( 1 p e k 1 =1k 2 =1 k j =1 P j 1 P j ). Using the upper bound in (4.6): e C j = O( 1 p e k 1 =1 k 2 =1 k j =1 e = O( (1 p e )(1 (1 p e )p 2j e ) j l=1 pk l +1 e 1+p 2j+1 e 1 p2j e 1+p 2j+1 e k 1 =1k 2 =1 ) e = O( (1 p e )(1 (1 p e )p 2j e ) ( p 2 e ) j ). 1 p e j k j =1 l=1 p k l+1 e ) The first term in (4.7) can be computed similar to S 0. So S j can be asymptotically

91 Chapter 4. Asymptotic Delay Bound 75 bounded as: 1 S j = O(enlog( 1 p j )+C j ). e Finally the upper bound for E(D SPCR ) can be computed as follows: n 1 E(D SPCR 1 n 1 ) = O(en(log(n)+λ)+en log( 1 p j )+ C j ), e j=1 j=1 n 1 j=1 C j is convergent if p 2 e 1 p e or p e 1 2 ( 5 1)). For p e 1 2 ( 5 1)) we have: E(D SPCR 1 1 ) = O(en(log(n)+λ)+enlog( )+en log( )dx) (integral bound) 1 p e 1 1 p x e 1 = O(en(log(n)+λ)+en(log( )+ Li 2(p x e) 1 p e log(p e ) 1 ) 1 = O(en(log(n)+λ)+en(log( )+ Li 2(p e ) 1 p e log( 1 p e ) )). in which Li 2 is Spence s function. According to this theorem the dominant term in the completion delay is not a function of erasure probability which gives us a fundamental insight into the network coding delay performance. This shows that for this range of p e, asymptotic delay performance is not dependent on the erasure probability. As we mentioned earlier, unlike the application of random network coding in gossip algorithm in which for an error free channel network coding provides considerable gain, for p e = 0 it does not provide any gain in our application. This finding demonstrates the achieved gains of network coding for p e > 0. In order to see how tight the asymptotic upper bound is for smaller n, in Fig. 4.7 and 4.8 we have compared the asymptotic upper bound in Theorem 4.3 and the SPCR completion delay in (4.1). By Asymptotic Bound I and II we mean the asymptotic upper bounds in Theorems 4.1 and 4.3, respectively. As can be observed in Fig. 4.8, even for a

92 Chapter 4. Asymptotic Delay Bound SPR SPCR SPCR: Bound I Asymptotic SPCR: Bound II Asymptotic n=20 E(D) (timeslots) pe Figure 4.7: Delay Bound II Performance for Small n: n = 20 relatively small n = 50, the Asymptotic Upper Bound II is a comparatively tight bound and for larger n the gap becomes smaller.

93 Chapter 4. Asymptotic Delay Bound SPR SPCR SPCR: Bound I Asymptotic SPCR: Bound II Asymptotic n= E(D) (timeslots) pe Figure 4.8: Delay Bound II Performance for Small n: n = 50

94 Chapter 5 Opportunistic Network Coding for Broadcasting in VANETs In Chapter 3 we introduced the MTM and discussed its significance in safety critical applications in VANETs. We also showed how random linear network coding can be efficiently utilized to optimize MTM. The intrinsic nature of SPCR requires each node to collect enough coded messages for decoding. It is certain if a node can collect n linearly independent messages from n original messages, it can decode and extract all original messages. Each node can do partial decoding based on the coded messages in the queue, however partial decoding it is not necessarily guaranteed. Numerous safety applications with distinct safety requirements have been introduced by United States Department of Transportation(USDOT) and related agencies([17, 52]). There are certain safety applications in which the average vehicle to vehicle reception delay and reliability are important. Since random linear network coding tries to optimize the maximum delay it might not perform well for average delay. This indeed motivates the work in this chapter. In Section 5.1 we propose XOR-based instantly decodable network coding algorithm, inspired by solutions to index coding problem, which intends to optimize the average reliability and delay in a greedy manner. Furthermore we introduce 78

95 Chapter 5. Opportunistic Network Coding 79 a distributed feedback mechanism which provides side information for the network coding algorithm. In Section 5.2, we compare the performance of the proposed coding algorithm with random linear network coding through realistic simulations. 5.1 Opportunistic Network Coded Broadcasting Various MAC protocols have been devised for VANET. Among these are Time Division Multiple Access (TDMA) slot-reservation schemes such as RR-ALOHA [12] and contention-based techniques such as Fast Collision Resolution [37]. Recently, repetitionbased techniques have been proposed in [71, 72] and [24] to address the requirements on delay and transmission reliability, as well as the short useful lifetime of the messages in vehicular safety applications. The use of Positive Orthogonal Codes (POC) in the repetition patterns proposed in [24] have produced promising results in reducing the interference between nodes. This work aims to improve upon this POC-based repetition broadcasting. The index coding problem has attracted much interest for its application to transmissions over broadcast channels [10, 9]. When information is available to the server about the packets received and those demanded by its clients, the server can combine its packets so that a maximum number of clients demands are satisfied in each transmission. We propose an enhancement of repetition-based broadcasting by applying index coding on repetition-based MACs. The required side information about the received packets are transmitted in the header of forward packets. In maximizing the number of informed destination nodes (clients) per transmission, we aim to achieve a higher probability of reception for a fixed number of transmissions on a wireless channel.

96 Chapter 5. Opportunistic Network Coding System Model and Performance Metrics A cluster of N nodes U = {u 1,...,u N } is considered. Each node has a safety message that should be delivered to all neighbour nodes within the message lifetime. The message lifetime can be determined based on several factors such as the GPS updating period, velocity, etc. Each node uses a repetition-based MAC and a high reception success probability should be achieved. A timeslotted system is assumed and the message transmission duration is one timeslot. The message lifetime is L timeslots and each user s message will be sent repeatedly in w timeslots out of L. The communication is frame-synchronous, meaning that all nodes start their frames simultaneously. Frame-asynchronous cases can also be handled as shown in [33]. We consider an erasure channel in which all the nodes in the transmission range of a sender receive the message with probability 1 p e. A similar channel model has also been assumed in the context of safety communications in vehicular networks [73, 50]. Since the optimization of the channel busy time mostly favors non-safety applications, we shall focus instead on the message loss probability, which is safety critical. However, as has been shown in [72] and [70], with a CSMA/CA mechanism the channel busy time could nevertheless be minimized. The message loss probability is defined as the probability that a vehicle fails to receive the safety message of another vehicle in its communication range within the frame. The delay of a received message is measured from beginning of the transmission frame to the time of successful reception and is in terms of timeslots. The maximum delay is the delay of the last message to be successfully received Index Coded MAC Towards a highly reliable, fully automatic safety message delivery MAC in a lossy vehicular environment, our proposed scheme tries to minimize the safety message loss probability using a distributed feedback and message combining algorithm. We allow packet

97 Chapter 5. Opportunistic Network Coding 81 combining for each retransmission: when a node has a transmission opportunity it can XOR some of its already-received messages and send the result. In contrast to the previous repetition-based MAC algorithms, a node does not only send repetitions of its own message, but rather can also transmit combinations of its message with other messages that it has overheard. A distributed feedback mechanism is designed to provide reception information for the packet coding algorithm. In the following two sections, we explain the distributed feedback mechanism and packet coding algorithms in detail. Distributed Feedback During a frame, once a node successfully receives a message from another node, it takes note of this successful reception and passes this information to other nodes. Furthermore, each node should also disseminate its knowledge of successful transmissions between other node pairs. Each user u k (1 k N) in the network maintains a binary square feedback matrix F u k such that: F u k ij = 1 if node j has received the message of node i 0 otherwise Since every user has its own message, the diagonal elements of the feedback matrix are equal to one. At the beginning of the frame, each feedback matrix has only a single element, which is set to 1 (corresponding to each user s own message). Each node sends the most recent version of its feedback matrix in the packet header along with the known users indices. When a node receives a new message from another node, it takes note of new reception information in the feedback matrix contained in the received message. This is done by changing 0s to 1s for the corresponding entries in its own feedback matrix. As the set of users known to a node increases, the size of the node s stored feedback matrix should also be increased. For example, assume that at a specific timeslot the feedback

98 Chapter 5. Opportunistic Network Coding 82 matrix of the nodes A and B are as the following: F A = A C {}} { and F B = B D { }} { Now if node A receives the message of node B in the next timeslot, it will update its feedback matrix to the following: A B C D { }} { F A = and the set of users known to user A will be {B,C,D}. In multi-hop scenarios, each vehicle maintains a feedback matrix which only contains information about its neighbours. Vehicles can identify their neighbours using the GPS information contained in the safety messages. The presented feedback mechanism increases the message size. For a matrix F u k of size N, the amount of overhead is N 2 N + N logn bits: the N 2 N term is the size of the feedback bit matrix included in the message, with the well-known diagonal elements omitted, and the N logn term is the number of bits required to transmit the identifiers for all known users. Users can obtain unique identifiers through a process similar to the POC code distribution protocol mentioned in [24] and described in [33]. To account for this overhead and have a fair comparison, we compare our proposed MAC with L timeslots in a frame with a repetition-based MAC without feedback and L = L( S M+S F S M ) timeslots in a frame. S M is the safety message size and S F is the maximum feedback overhead length.

99 Chapter 5. Opportunistic Network Coding 83 Index Coding Algorithm Based on the updated feedback matrix, at each timeslot, each node has the knowledge (possibly incomplete) of all the receptions. Returning to our previous example, assume that node A, after updating its feedback matrix, has scheduled a retransmission. In the previous repetition-based MAC schemes node A will send its own message (P A ) and nodes B and D will have the chance to receive a new message from node A. Could node A instead send some combination of its received messages to increase the number of newlyinformed nodes? Node A has the messages of nodes B and C as well. If node A sends P A P B (bitwise XOR), then in addition to nodes B and D, which will have the chance to receive P A, node C also will have the chance to receive P B. This example shows that by intelligently combining messages together, one can increase the number of message receptions. Here, a sender has some packets and also some side information about which packets have been received and which ones are still needed by its neighbours. The sender finds the best packet combining strategy to maximize the number of received messages in one transmission. It is possible to show that this problem is NP-hard. In the sequel, we provide a heuristic for this problem. In contrast to the index coding problem in [9, 19], here the sender has only one broadcast opportunity and wants to maximize the number of received messages with its transmission. We modify a heuristic solution for index coding [15] to solve our problem. Let us assume R(u s ) and N(u s ) are the set of messages received by the sender and the set of messages that are still needed (not yet received), respectively. U us is the set of users that are known to the sender (users already present in the feedback matrix of the sender). N(u i ) and R(u i ) (u i U us ) are respectively the set of needed and received messages of each known user to the sender. We represent a known user with m needed messages as m virtual users, each with one needed message. Each virtual user has the same set of received messages as the original user. We call the new set of users Ú. Now a graph

100 Chapter 5. Opportunistic Network Coding 84 G(V,E) can be constructed such that the vertices represent the users of Ú. We connect two vertices corresponding to distinct users ú i,ú j Ú if one of the following rules holds: N(ú i ) R(ú j ) and N(ú j ) R(ú i ) and N(ú i ),N(ú j ) R(u s ) (1) N(ú i ) = N(ú j ) and N(ú i ),N(ú j ) R(u s ) To illustrate, let us continue with the example from the previous section. Node A s expanded feedback matrix with virtual users is: B1 B2 C1 D1 D2 { }} { 0 X 1 0 X F = X 0 1 X 0 where Don t cares from virtual user decomposition are expressed with Xs, and A s own column is omitted. The virtual users are denoted above the matrix. For example, node B has been decomposed into virtual users B1 and B2, etc. An inspection of the two rules presented in (1) shows that only those entries in R(u s ) are important, since the sender can only perform coding with messages in its memory. Here, since u s = A, we may safely ignore the fourth row in the matrix F. Moreover, we can also ignore any virtual users that need a message which is not in R(u s ). This is why C2 does not appear in F. Figure 5.1: Graph G(V,E) for node A, with two cliques of size three.

101 Chapter 5. Opportunistic Network Coding 85 Using the two rules, the corresponding graph G(V,E) is generated and is shown in Fig The users corresponding to the vertices of each clique of this graph can receive their needed messages with a single broadcast. The transmitted message is the XOR of all the needed messages of the clique users. Contrary to the index coding problem where we solve the clique partition problem, here we should find the maximum clique of the graph which shows the maximum number of users that can receive their needed message with a single broadcast. The maximum clique problem is NP-hard and many algorithms have been proposed during the last decades. [48] and [63] are among the most recent efficient algorithms, using the branch and bound algorithm together with vertex coloring. The coloring scheme helps to find a better upper bound of the maximum clique size and accelerate the pruning step. For practical values of message size and channel rate, the timeslot duration is on the order of a few milliseconds or less. In the worst case, the sender should find the best message combination within a timeslot. For a fast algorithm and a nominal computer system, finding the maximum clique for a sparse random graph with 100 vertices takes more than 10ms on average [48]. Therefore, instead of finding the maximum clique, we use a simple, fast, sub-optimal algorithm. We simply assume the maximum clique contains the highest degree vertex in the graph. Then, we use a greedy algorithm that, at each step, chooses the maximum degree vertex in the potential set of clique vertices to join the current clique. Simulation results show that this simple algorithm performs well enough (specially in lower loads) in our application. Even for the index coding problem, it has been shown that if we partition the graph to 3-vertex and 2-vertex cliques, it still performs well in terms of the minimum number of transmissions [15]. When a node has a transmission opportunity, if the clique size is larger than the number of nodes that have not received its message, it will transmit a combination of the needed messages in the clique. Otherwise (coding has no benefit), it will transmit its own message.

102 Chapter 5. Opportunistic Network Coding Simulation Results In this section, we evaluate the proposed index coded scheme on top of the POC MAC protocol in comparing it with two other repetition-based MAC protocols: SFR MAC([72, 70]) and POC-based MAC ([23, 24]). A binary erasure channel is assumed for modeling the wireless link. The communication range is limited to a nominal value of 300m [72]. Each pair of vehicles within the maximum range can successfully receive each other s messages with probability 1 p e. Based on a 3-lane highway and minimum distance of 30m between vehicles in each lane (including the length of the vehicle), a maximum of 30 vehicles can fit within the maximum communication range. We considered a cluster of 15 vehicles. The simulations have been done for 128 timeslot frames and high (p e = 0.5) and low (p e = 0.2) error probabilities. Fig. 5.2 shows the loss probability averaged over all vehicles versus the number of repetitions each vehicle makes within a transmission frame, where p e = 0.2, for the three schemes. The POC codes have been generated based on the presented algorithm in [24]. For higher values of w, more repetitions create more opportunities for updating the feedback matrices. This allows each vehicle to gain a more complete picture of the transmissions and receptions of their neighbours, which leads to more effective packet coding. As we can see in Fig. 5.2, for w = 12, the loss probability of our scheme is almost one order of magnitude smaller than the loss probability of the POC-based MAC. For higher channel error p e = 0.5, as shown in Fig. 5.3, POC s performance degrades and its loss probability approaches that of SFR. The channel loss decreases the number of successfully received repetitions within a transmission frame. For lower numbers of repetitions, the random transmission patterns used in SFR become similar to POC transmission patterns. However, even for such a lossy channel, our scheme still provides a loss probability of 10 3 for w = 12 which indicates the enhanced reliability of the proposed MAC for safety applications. Fig. 5.4 and 5.5 show the average delays of received messages, averaged over all

103 Chapter 5. Opportunistic Network Coding Loss Probability Index Coded POC POC SFR W Figure 5.2: Message loss probability versus number of retransmissions. L = 128, p e = 0.2 and N = Loss Probability Index Coded POC POC SFR W Figure 5.3: Message loss probability versus number of retransmissions. L = 128, p e = 0.5 and N = 15

104 Chapter 5. Opportunistic Network Coding Average Delay(time slots) Index Coded POC POC SFR W Figure 5.4: Average delay versus number of retransmissions. L = 128, p e = 0.2 and N = 15 vehicles in the cluster, for the three MAC schemes with low and high channel error rates, respectively. We see that the proposed index coded scheme suffers a slight penalty in terms of average delay. For the low channel error rate, this difference is negligible. Even when the channel error rate is high as in Fig. 5.5, the difference in the average delay is at most 5 timeslots. The maximum delays of received messages, averaged over all vehicles, for the three MAC schemes are compared in Fig. 5.6 and 5.7, for p e = 0.2 and p e = 0.5 respectively. For smaller values of w, the incomplete feedback information leads to a less efficient packet coding which results in slightly higher maximum delay for our MAC. For higher values of w, more successfully received repetitions allow feedback information to be effectively disseminated throughout the network. Thus, each vehicle is able to perform message coding efficiently, which allows for more messages to be delivered in each timeslot. This effectively lowers the maximum message delivery delay. Fig. 5.8 and 5.9 show the average size of the message header used for feedback versus the number of retransmissions performed by each vehicle in a frame when p e = 0.2 and

105 Chapter 5. Opportunistic Network Coding Average Delay(time slots) Index Coded POC POC SFR W Figure 5.5: Average delay versus number of retransmissions. L = 128, p e = 0.5 and N = Index Coded POC POC SFR Maximum Delay(time slots) W Figure 5.6: Maximum delay versus number of retransmissions. L = 128, p e = 0.2 and N = 15

106 Chapter 5. Opportunistic Network Coding Maximum Delay(time slots) Index Coded POC POC SFR W Figure 5.7: Maximum delay versus number of retransmissions. L = 128, p e = 0.5 and N = 15 p e = 0.5, respectively. We can see that a greater number of retransmissions made by each vehicle in a frame allows each vehicle to gather more feedback information about the network. Consequently, this increases the average amount of feedback information included in the transmitted messages. 5.2 Comparison to Random Linear Network Coding A repetition-based MAC can be visualized as a two-layer design. First layer is the design of the repetition pattern in a time frame. The number of repetitions could be deterministic with w repetitions, or probabilistic such that the expected number of repetitions is w. The second layer is the content of each repetition. In the previously proposed MAC schemes, the content of each retransmission is fixed over time and each node only transmits its own message based on the repetition pattern. It has been shown in [43] that random linear network coding could asymptotically achieve the unicast and multi-cast capacity in a wireless network with lossy links. The presented results in[43] only consider

107 Chapter 5. Opportunistic Network Coding Average Feedback Header Size (Bytes) Index Coded POC W Figure 5.8: Average feedback header size versus number of retransmissions. L = 128, p e = 0.2 and N = Average Feedback Header Size (Bytes) Index Coded POC W Figure 5.9: Average feedback header size versus number of retransmissions. L = 128, p e = 0.5 and N = 15

108 Chapter 5. Opportunistic Network Coding 92 long term throughput, but unlike a data traffic session, here we are interested in safety critical metrics such as message success probability and delay. In this section we consider a more realistic vehicle-to-vehicle channel model and assume the perfect feedback is provided. Furthermore, we study the random linear network coding performance in our application. We show through simulations that given perfect feedback, the proposed opportunistic network coding has superior performance in terms of average delay and message loss probability. We should emphasize that the goal of our work is not to study the network coding algorithm, but rather to investigate the reliability gain that we can achieve by using network coding in repetition-based MAC schemes through simulations System Model and Performance Metrics We assume the same topology and MAC protocol, and load as in Section Our metrics of interest are the message success probability and average delay. Message success probability is the average probability that a node receives the safety message of another node within the time frame. Average delay is the average delay that it takes that a node receives another node s safety message. The average delay is over successful message receptions within the time frame. An erasure channel is assumed such that an uncollided transmission is received from node i with probability 1 p i e where p i e depends on the distance from node i as well as the stochastic channel model. The erasure channel has also been previously assumed in the context of safety communications in vehicular networks [73, 50]. It has been shown that Nakagami distribution with properly estimated parameters would be a more realistic channel model for vehicle-to-vehicle communications [64]). Our simulation follows a realistic Nakagami channel model with the parameters chosen from the IEEE p standard.

109 Chapter 5. Opportunistic Network Coding Index Coded MAC using perfect feedback Compared to the feedback matrix in the previous section, here we assume that there is a unique feedback matrix which contains the perfect knowledge of the reception information ofallthenodes. Thereceptioninformationofthenodesisrepresented intermsofabinary square matrix as follows: 1 if node j has received the message of node i F ij = 0 otherwise After each retransmission, depending on which nodes have received the transmitted message successfully, the feedback matrix will be updated accordingly in all nodes. These updates can be done through a dedicated perfect feedback channel. Based on the updated feedback matrix, at each timeslot, each node has the knowledge of all the receptions and this information can be used to devise an efficient message combining strategy. Then when a node has a transmission opportunity it uses the same algorithm as in Section to find the combining strategy. Random Linear Network Coding The random linear network coding is similar to the presented algorithm in Chapter 3. Each node enqueues all the received messages and when it has a transmission opportunity based on its retransmission pattern, it broadcasts a random linear combination of all the already received messages in its queue with the coefficients in GF(q). After each message reception, a node tries to do the Gaussian elimination on all the received coded messages in the queue to find the newly decodable messages. Utilizing random linear network coding scheme introduces overhead to each message. In random linear network coding, higher number of neighbours and larger Galois field size results in higher message overhead. Considering a fixed message lifetime, frame size and data rate, higher message overhead results in a smaller number of timeslots in a frame, which for a fixed number of

110 Chapter 5. Opportunistic Network Coding 94 repetitions results in more collisions. On the other hand, by taking advantage of network coding schemes higher information entropy can be achieved and the probability of an innovative retransmission (a transmission that potentially satisfies more users need) will be higher than blindly repeating the same message. To have a fair comparison and to consider the coding overhead the frame size for random linear network coding has been scaled down to L M = L ; in which M is the message size and n is the number M+nlog(q) of active users in that time frame Simulation Results Here, we evaluate the performance of the proposed opportunistic network coding algorithm through simulations. We have used the Nakagami channel model for the vehicle-to-vehicle communication link. The distribution of the signal amplitude X based on this channel model is: f X (x) = 2mm x 2m 1 mx 2 Γ(m)Ω m e Ω m 1 2, Ω 0 in which Ω is the average received power and m is the fading figure. In [64] the fading figure parameter m has been estimated based on empirical measurements for a vehicle-to-vehicle link in a highway: m = 3 d < 50m m < d < 150m 1 150m < d (5.1) in which d is the distance between the sender and the receiver. This estimation of m with the physical and MAC layer parameters listed in Table 5.1 have been used in our simulations. Considering the antenna gains, the Effective Isotropic

111 Chapter 5. Opportunistic Network Coding Reception Probability Distance(m) Figure 5.10: Reception Probability vs. Distance Table 5.1: Physical and MAC layer parameters Data Rate 1Mbps Message Lifetime 100ms Slot Duration 1ms Message Size 100B Transmit Power 10dBm Reception Threshold -90dBm Frequency 5.9GHz Gt 4dB Gr 4dB Radiated Power (EIRP) will be less than 20 dbm, which is the maximum allowed EIRP by the IEEE p standard. The reception probability versus distance is shown in Figure As can be seen, compared to the traditional two-ray ground channel model, the packet reception probability changes over the radio range. We consider a road segment of length 1000m (Figure 5.11). The distance between each two vehicle is 30m. There are a total of 34 vehicles placed in the road segment. At each time frame each vehicle transmits a safety message with probability µ p. The POCs have been used to specify the repetition pattern. The POCs have been

112 Chapter 5. Opportunistic Network Coding 96 generated by using the proposed algorithm in [24]. The simulations have been done for three different MAC schemes: 1) POC MAC without network coding 2) POC MAC with random linear network coding and 3) POC MAC with the proposed opportunistic network coding. We have used GF(2 8 ) in our simulations. Two sets of simulations for low load (µ p = 0.2) and higher loads (µ p = 0.4 and µ p = 0.6) have been performed. As we can see in Figure 5.12, in low load both coding algorithms have much lower loss probability compared to the uncoded scheme. The opportunistic coding performs slightly better than the random linear network coding. For µ p = 0.2, opportunistic network coding has the best average delay performance, which is closely followed by the uncoded scheme (Figure 5.13). Although the average delay of the uncoded scheme is close to the opportunistic coding case, we should take into account that the average delay has been measured based on the number of successfully received messages which is much less for the uncoded scheme. The poor delay performance of random linear network coding is related to the fact that it does not guarantee instant decodability and a node should collect enough coded messages to decode new messages. For higher loads (µ p = 0.4 and µ p = 0.6), since the coded packets consist of a larger number of original messages, the random linear network coding loss probability gets worse (Figure 5.14). In this case, a node would need more coded messages in order to decode the original messages. In high load, the opportunistic coding still has the best performance in terms of message loss probability and again has a better average delay performance compared to random linear network coding (Figures 5.15 and 5.16). Figure 5.11: Simulation Topology

113 Chapter 5. Opportunistic Network Coding Loss Probability Uncoded Random Linear Coding Opportunistic Coding W Figure 5.12: Loss Probability vs. Number of repetition (µ p = 0.2) Average Delay (Time Slots) Uncoded Random Linear Coding Opportunistic Coding W Figure 5.13: Average Delay vs. Number of repetition (µ p = 0.2)

114 Chapter 5. Opportunistic Network Coding Uncoded Random Linear Coding Opportunistic Coding µ p =0.6 Loss Probability 10 1 µ p = W Figure 5.14: Loss Probability vs. Number of repetition (µ p = 0.4,0.6) Uncoded Random Linear Coding Opportunistic Coding Average Delay (Time Slots) W Figure 5.15: Average Delay vs. Number of repetition (µ p = 0.4)

115 Chapter 5. Opportunistic Network Coding Average Delay (Time Slots) Uncoded Random Linear Coding Opportunistic Coding W Figure 5.16: Average Delay vs. Number of repetition (µ p = 0.6) 5.3 Heuristics for XOR-based Broadcasting In this section, first we formulate the problem as a l 0 minimization. We assume each node in the network maintains a feedback matrix F. If a user has m neighbors and n uncoded packets then the feedback matrix is a m by n matrix which is defined as follows: 0 user i has received the j th packet F ij = 1 otherwise If our coding strategy is to XOR a subset of n original messages, the problem is to find the optimal coding pattern such that after one broadcast the number of nodes that can instantly decode an original message is maximized. We consider a binary decision vector x of size n in which each element is 1 if the corresponding original message exists in the coded message. For example, if we have the following feedback matrix about the three

116 Chapter 5. Opportunistic Network Coding 100 neighbor nodes: F = the optimum coding vector is x = [ ] T. This corresponds to broadcasting of m 1 m3 m6 and the first, third and the sixth packet will be decoded instantly by the second, third and first neighbors, respectively. In previous sections, we showed a graph based algorithm for finding the optimal coding combination. It relied on constructing a graph based on the feedback matrix. The maximum clique in the graph represents the optimum coding strategy (the clique size is the number of packets that we should XOR). However, here we formulate the problem in the form of an l 0 norm minimization problem. If f i is the i th row of the feedback matrix, the i th user can decode a message if and only if f i x = 1. So in order to minimize the number of nodes which can not decode, we should solve the following optimization problem to find the optimum coding vector: min Fx 1 l0 (P0 ) s.t. x i {0,1} In the rest of this section, we provide heuristics for the coding problem. First we describe the greedy algorithm. We then show how the feedback matrix can be represented as a Tanner graph and Belief propagation algorithm then can be utilized as an efficient algorithm Greedy Algorithm The greedy algorithmbegins with the vector x = 0 or the empty set asan initial solution. In each iteration, it finds the unselected packet whose addition to the solution set yields the lowest objective function value in (P0 ). The algorithm continues adding packets to

117 Chapter 5. Opportunistic Network Coding 101 Algorithm 2 Greedy Algorithm x, v m, v m loop for all [ do Find x with minimizing i] i / x t i x if Ft 1 l0 < v then x t, v Ft 1 l0 end if end for if v < v then x x, v v elsereturn x end if end loop the solution in this greedy manner until the objective function value cannot be further minimized by the addition of any of the unselected packets Tanner Graph Representation and Belief Propagation Algorithm Low-density parity check (LDPC) codes [27], first proposed by Gallager, are specified by a parity-check matrix which is binary and sparse. Belief propagation (BP) decoding algorithms [44] have been shown to perform very well. Since the renewed interest in LDPC codes in the 1990 s, many variations on the basic BP algorithm have been proposed to improve the performance and reduce the computational complexity of the decoding process. In particular, different schedules for iterative message passing have been proposed. In the Tanner graph [62] representation, messages are exchanged between the check and bit nodes. In the original flooding schedule, the messages are updated in two separate steps in each iteration. In the horizontal step, all of the check-to-bit messages are updated in parallel; in the vertical step, all of the bit-to-check messages are updated in parallel. In sequential belief propagation schemes such as Layered Belief Propagation (LBP)

118 Chapter 5. Opportunistic Network Coding 102 f 1 f 2 f 3 x x 1 2 x 3 x 4 x 5 x 6 Figure 5.17: Tanner Graph representation for the feedback matrix [45, 51] and Shuffled Belief Propagation (SBP) [81] the updating schedule is serialized. In LBP, the horizontal and vertical steps are applied to each check node and its adjacent bit nodes sequentially. Updated messages are used in update steps for subsequent check nodes within the same iteration. SBP applies the same technique except the update sequence is over the bit nodes. Both these techniques have been demonstrated to have improved convergence and also allow for simplified implementations. Recently, dynamic scheduling belief propagation techniques have been proposed [35, 20]. These residual BP schemes adapt the sequence of updates by prioritizing those nodes with the highest residual, that is, those whose messages have varied the most. These techniques can further reduce the convergence time of the iterative decoding algorithm. The feedback matrix can be represented as a Tanner graph [62]. The check nodes represent the users and the variable nodes correspond to the packets. A check node is connected to a variable node if the associated user needs that corresponding packet. The graph model for the F in Section 5.3 is depicted in Fig The belief propagation algorithm can be used to find the optimal x. We assume C j is the set of neighbour check nodes for variable node x j and V i is the set of neighbour variable nodes for check node f i. We also assume µ j = Pr(x j = 1), and ν i,j is the probability of x j being 1 from the perspective of f i. The update rule for the

119 Chapter 5. Opportunistic Network Coding 103 f i x 1 x 2 x j x n f 1 f 2 f i f m x j Figure 5.18: Update Rules belief propagation algorithm can be written as (Figure 5.18): ν t i,j = k V i,k j (1 µ t 1 k ) µ t j = k C j ν t 1 k,j k C j ν t 1 k,j + k C j (1 ν t 1 k,j ) To improve the numerical stability of the algorithm, we use the log-likelihood ratios (LLR) as messages instead of the actual probabilities. The LLR of a Bernoulli random ( ) Pr(xj =1) variable is defined as λ(x j ) = ln Pr(x j. We use the identity tanh(λ/2) = eλ 1 to find =0) e λ +1 the following relationships between the raw probabilities and the LLR in terms of the hyperbolic tangent function: Pr(x = 0) = 1 e λ +1