Network Coding for Real-time Wireless Communication for Automation
Vasuki Narasimha Swamy, Paul Rigge, Gireeja Ranade, Anant Sahai, Borivoje Nikolic
11 Network Coding for Real-timeWireless Communication for Automation
Vasuki Narasimha Swamy ˚ , Paul Rigge ˚ , Gireeja Ranade : ,Anant Sahai ˚ , Borivoje Nikoli´c ˚˚ University of California, Berkeley, CA, USA : Microsoft Research, Redmond, WA, USA
Abstract
Real-time applications require latencies on the order of a millisecond with very high reliabilities,paralleling the requirements for high-performance industrial control. Current wireless technologies likeWiFi, Bluetooth, LTE, etc. are unable to meet these stringent latency and reliability requirements, forcingthe use of wired systems. This paper introduces a wireless communication protocol based on networkcoding that in conjunction with cooperative communication techniques builds the necessary diversity toachieve the target reliability. The proposed protocol is analyzed using a communication theoretic delay-limited-capacity framework and compared to proposed protocols without network coding. The resultsshow that for larger network sizes or payloads employing network coding lowers the minimum SNRrequired to achieve the target reliability. For a scenario inspired by an industrial printing applicationwith nodes in the control loop, aggregate throughput of . Mb/s, MHz of bandwidth and cycletime under ms, the protocol can robustly achieve a system probability of error better than ´ witha nominal SNR less than dB under ideal channel conditions. Keywords
Cooperative communication, network coding, low-latency, high-reliability wireless, industrial con-trol, diversity, Internet of Things
I. I
NTRODUCTION
The Internet of Things (IoT) promises to enable many exciting new applications in health-care, robotics, transportation and entertainment. In particular, for real-time applications thatare interactive and immersive or involve machine control, reliable communication protocolswith latencies around ms are crucial [1]. Techniques used by existing wireless standards arefundamentally ill-suited for low-latency and high-reliability [2], [3].Wireless channels are inherently unreliable as movements of objects in the environment causethe channel to change over time. Diversity is the primary tool to overcome unreliable channels. a r X i v : . [ c s . I T ] M a r (a) Star message flow topology (b)
A generic message flow topology where one of thestreams originating at C1 has two subscribers: S1 and S2.
Fig. 1:
Information flow topologies
The availability of a large number of nodes in the network naturally creates opportunities toharvest spatial diversity. Cooperative communication techniques have been well studied in thewireless literature. Inspired by these cooperative communication techniques, our earlier works [4],[5] introduced a cooperative communication protocol (dubbed “Occupy CoW”) to meet thestringent QoS requirements. In this paper , we use network coding with our cooperative commu-nication protocol. Network coding is generally used to increase network throughput, sometimesat the cost of increased latency, but we show how to use network coding to use this improvedthroughput to decrease latency and reduce SNR. We show that integrating network coding withcooperative communication brings down the SNR required to meet the QoS requirements evenmore than the original cooperative-communication approach under ideal conditions.The protocol in this paper (“XOR-CoW”) targets a local wireless domain where nominally allnodes are in range, but fading might cause a pair of nodes to be unable to hear each other. Thetraffic patterns (referred to as “information topology”) considered are generic – any message‘stream’ might have several destination nodes (called subscribers) and there are several suchstreams in a network. Within a short period of time (referred to as “cycle time”), every streamneeds to deliver one packet reliably to each of its subscribers. The information topology can bearbitrary – something naturally centralized like a star topology as shown in Fig. 1a (e.g. witha central controller talking to many sensor/actuators collecting streams of measurements andsending streams of commands) or something more generic as in Fig. 1b. The XOR-CoW protocolcan support any arbitrary traffic but its performance peaks when the information topology is bi-directional (such as a star). This is due to the inherent two-way traffic that naturally facilitatesopportunities for network coding. This paper expands upon a conference version [6] that contained early forms of these results.
The main contributions of this paper are as follows: ‚ A new protocol framework, called XOR-CoW, for applications that require ultra-reliablelow-latency communication that combines cooperative communication and network coding. ‚ Analysis of the performance of XOR-CoW under a communication theoretic and delay-limited capacity framework. ‚ Comparison of various schemes’ (with and without network coding) performance bycomparing the minimum SNR required to meet the QoS requirements. ‚ Optimization of the parameters of XOR-CoW to show that XOR-CoW is relatively insen-sitive to parameter choices. Most of the benefit comes from cooperative communicationand network coding, so implementing more complicated schemes is not justified.The rest of the paper is organized as follows. In Section II we first briefly review some ofthe recent trends in wireless communications, the evolution of communication for industrialcontrol, cooperative communication, wireless diversity, and network coding techniques. Fora more detailed treatment of the related work, please refer to [5]. Section III describes theresource assumptions and high level overview of Occupy CoW and XOR-CoW. Section IV andSection V describe the design of XOR-CoW framework in detail for generic traffic and bi-directional traffic, respectively. Section VI analyzes the performance of XOR-CoW and presentshow it performs and compares it to hypothetical frequency-diversity-based scheme as wellas cooperative communication scheme without network coding. Additionally, it presents howXOR-CoW protocol’s internal parameters can be optimized and discusses the implications forimplementation. All the formulas used to generate the plots are derived in the Appendix.II. B
ACKGROUND
A. Recent development in G protocols
The current vision of G wireless standard not only focuses on increasing capacity and energyefficiency, but also on reducing latency. Tactile applications demanding latencies on the order of ms may be enabled by using mmWave frequencies [7], [8]. Recent works like [9] concentrateon the proposed 5GETLA radio interface and show that latencies below ms for payloads ofsize kb are achievable provided a bandwidth of 100MHz is available. Though the targetedlatency is on the same order as required by industrial control, they do not consider reliabilityguarantees or retransmissions. The feasibility, requirements, and design challenges of an OFDM based 5G radio interface that is suitable for mission-critical MTC (machine type communication)is discussed in [10] where various modulation schemes as well as different MIMO configurationswere considered. They concluded that for interference mitigation, multiple receive antennas werecrucial. In similar spirit, the coverage and capacity aspects based on evaluations consideringboth noise-limited and interference-limited operations for MTC were considered in [11]. Severalworks have studied the suitability of various signaling strategies for low-latency applications.Specifically alternatives for OFDM have been considered to relax synchronization requirementsand reduce out-of-band (OOB) transmissions such as Filter Bank Multi-carrier [12], UniversalFiltered Multi-carrier [13] and Generalized Frequency Division Multiplexing [14]. In this paper,we do not consider explicit signaling strategies and push it for future work. B. Industrial Control
Communication in industrial control is supported by wired fieldbus systems like HART,PROFIBUS, WorldFIP, Foundation Fieldbus, and SERCOS [15] meet these requirements. Severalwireless extensions of these fieldbus systems such as [16], [17] (as well as WirelessHART [18]and ISA100 [19]) which are based on wireless sensor network (WSN) techniques have beendeveloped. They have worst-case latencies on the order of hundreds on milliseconds [20] makingthem unsuitable for high-performance control applications. WISA [21] targeted wireless controlby employing frequency hopping techniques but it achieves latency on the order of ms witha reliability of ´ [22], which fails to meet the reliability achieved by wired fieldbuses. C. Cooperative communication and multi-user diversity
In our prior work [5], we discussed some of the relevant references on cooperative commu-nication and multi-user diversity in detail. Low-latency applications that we target cannot usetime diversity since the cycle time can be shorter than the coherence time. Additionally, TDMA-based schemes for industrial control considered in [23], [24] do not scale well with networksize. Commonly used frequency diversity techniques in WSN-inspired technologies [25] likechannel hopping and contention-based MACs aren’t sufficient to obtain the required diversity asthey cause unbounded delays. As there are multiple nodes in the system, harvesting cooperative,multi-user diversity is a viable option. Cooperation amongst distributed antennas can provide full C S123 C S123 + = + = Time 1 Time 2 Time 3Time 3
With Network Coding - need 3 time steps C S123 C S123
Time 1 Time 2 Time 4 Time 3
No Network Coding - need 4 time steps
Fig. 2:
Illustration of network coding along with simultaneous retransmissions where the C and S nodes haveinformation to convey to each other through 3 relays 1 - 3. The bold lines are active links and the dotted linesare inactive links. The blue packets are the downlink packets, the orange packets are the uplink packets and themaroon packets are the XORed packets. The XOR scheme can communicate the same amount of information in ashorter time because the uplink and downlink demands are satisfied simultaneously. diversity without physical arrays [26]. Even with noisy inter-user channels, multi-user cooperationincreases capacity and leads to achievable rates that are robust to channel variations [27].
D. Network Coding
The seminal work of Ahlswede et al., [28] showed that regarding information to be multicastas a “fluid” to be routed or replicated in general is not optimal and employing coding at nodescan lead to efficient use of bandwidth. This idea was further studied in [29], where a forwardingarchitecture for wireless mesh networks to improve throughput by introducing a coding layerin between the IP and MAC layers was proposed. They provide a practical implementation ofnetwork coding into the current network stack, addressing the common case of unicast traffic, anddynamic and potentially bursty flows. Recent results in [30] show that using randomized space-time block coding (RSTBC) in two-way relay networks improves throughput by exchangingdata through a bi-directional relay network. Like most works using network coding, we aimto increase throughput which translates to lower latency. Fig. 2 illustrates how we use networkcoding combined with simultaneous retransmissions in our work. Essentially, if there is a naturalviability for XORing then, only those nodes with the necessary packets help by broadcastingthe XORed packet.The proposed wireless communication system combines cooperative communication and net-work coding techniques to achieve the desired QoS requirements by exploiting multi-user di-versity and distributed space-time codes (such as those in [31]–[33], so that each receiver canharvest a large diversity gain) to achieve high-reliability and low latency. The key idea here is that relays simultaneously broadcast coded packets (as long as they are coding the same set ofpackets). III. P
ROTOCOL F RAMEWORK
The XOR-CoW protocol exploits multi-user diversity as well as side information at destinationnodes by using simultaneous relaying combined with network coding to enable ultra-reliablecommunication. The general setup considered is that the network consists of n nodes and eachmessage stream (size m bits) must reach its possibly many destinations within a cycle of time T . As we discussed in Section I, the information topology can be arbitrary. A. Resource assumptions
We make a few assumptions about the network, channel characteristics and hardware toabstract away some of the details and to support the exposition. These assumptions hold forall schemes discussed in this paper. The following assumptions are the same as the ones madein [5] for “Occupy CoW” protocol. ‚ We assume a local domain – that while normally, the nodes are within range of each other,bad fading events can cause transmissions to fail. Errors are caused only by bad fades. ‚ All nodes know the information topology. They share a universal addressing scheme andorder. Messages are of the same size and they contain their destination addresses. ‚ Channels are assumed to be reciprocal. All nodes are half-duplex, but can switch instantlybetween transmit mode and receive mode. ‚ Channels are assumed to be quasi-static and remain the same during a cycle. ‚ Channel sounding to aid channel estimation is assumed to take a constant fraction ofthe cycle time T . All nodes are assumed to estimate channels that are being sounded.When multiple nodes simultaneously broadcast a message during the relaying phase, theywould not need to spend time again sounding each channel and can do a short combinedsounding as the intended receivers only need to identify which of the nodes that it canhear are transmitting. ‚ Clocks on each of the nodes are perfectly synchronized in both time and frequency. Onecould achieve adequate synchronization with low overhead by adapting techniques suchas [34]. Thus we can schedule time slots for specific nodes without significant overhead. ‚ The protocol relies on time/frequency synchronization to achieve simultaneous retrans-mission of messages by multiple relays. We assume that if k relays simultaneously (withconsciously introduced timing jitter ) transmit the exact same information, then all receiverscan realize signal diversity k . B. Overview of Occupy CoW Protocol Framework
We briefly summarize the Occupy CoW protocol which would be the benchmark protocol forcomparison purposes. For a detailed description, refer to [5]. The XOR-CoW protocol shares thesame network setup and aims to meet the same requirements as Occupy CoW. In the OccupyCoW protocol, the source of different message streams transmit the message in a round-robinfashion. After all messages have been transmitted once, each message is then re-transmitted simultaneously by all the nodes which have the message (either the source or nodes that decodedthe first transmission) using some appropriate distributed space-time code (DSTC) again in around-robin fashion. Consider how this would play out for a star information topology as shownin Fig. 1a. There is a downlink and uplink phase (corresponding to the first time the messages aretransmitted) of length T D and T U respectively. This is followed by a scheduling phase whereall “strong” nodes get to know the state of each message (whether it has reached the intendeddestination or not). This is followed by the relaying phases – first the downlink phases II (length T D ) and III (length T D ), where the controller and strong nodes alter the broadcast messageto remove already-successful messages for the strong nodes and simultaneously broadcast theadapted packet. The unsuccessful nodes are listening. At the end of this phase, the nodes whoreceived their messages from the controller have also received the global ACK information.which allows these nodes to participate as relays in the uplink phases. The uplink phases II(length T U ) and III (length T U ) are similar to their downlink counterparts. The protocol caneither have two hops – such that there are only two downlink and uplink phases or three hops –where there are a total of three downlink phases and three uplink phases (as described above). C. Overview of XOR-CoW Protocol Framework
Consider two nodes (say A and B) that have messages to each other i.e., node A has amessage for B and node B has a message for A. If the direct channel exists (link AB), then To transform spatial diversity into frequency-diversity [33].
A’s message to B as well as B’s message to A succeeds in reaching the destination. If thereis no direct channel, then A’s message to B may succeed if there is at least one node (say C)that has connection to both A and B. If there exists such a node, then both A’s message to B and
B’s message to A succeeds via the same node (or set of nodes). Essentially, when thereis a bi-directional traffic, the paths of ‘success’ in both direction are the same. When we havesuch bi-directional traffic patterns, then relay nodes can ‘XOR’ the packets and broadcast theresulting packet simultaneously using a DSTC as shown in Fig. 2. This is what we leverage inXOR-CoW – opportunistically network code packets. Network coding also provides throughputbenefits (and as a result reduction in latency or reduction in SNR needed) when the trafficpatterns are multicast (messages need to reach multiple destinations). We consider this scenarioin detail in Section. IV.IV. XOR-C O W FOR G ENERIC I NFORMATION T OPOLOGY
The XOR-CoW scheme for a generic information topology can be summarized as follows. Allnodes know the information topology – the origin and destinations of the messages. Therefore,all nodes know which messages can be XORed. The schedule of messages G are determined andall nodes know the schedule. For the first phase, the schedule is simple: each message streamis allocated one slot. However, in the second phase (XOR phase), the schedule G X is different:whenever bi-directional traffic exists in the information topology, allocate one slot for those twomessages in G X , else allocate one slot for that single message in G X (as shown in Fig. 3a). In thefirst phase, nodes take turn according to the schedule to transmit the messages. All nodes listenwhen they are not transmitting. In the XOR phase, all nodes that can transmit a message (or anXORed message) transmit according to the XOR phase schedule simultaneously using a DSTC.In the following section, we focus on the star topology as network coding yields maximumbenefits when the traffic is bi-directional [35], [36].V. XOR-C O W FOR B I - DIRECTIONAL I NFORMATION T OPOLOGY
In this section, we consider bi-directional topologies wherein if a node A has information fornode B, then node B also has information for node A. A simple case of bi-directional traffic is thestar topology which we will consider here for exposition purposes. A centralized control systemcan be modeled as a star topology where the network consists of a central controller C and n (a) Schedule during first phase and the XOR phase fora generic topology. Pairs of message streams that areinherently bi-directional i.e, p C ´ C , C ´ C q and p C ´ S , S ´ C q are the only ones that are XORed(shown in teal colored boxes). (b) Fixed and flexible scheduling for the star topologyexample considered in Fig. 4. The green boxes corre-spond to the downlink packets from the controller tothe client nodes (the destinations are labelled: S i ). Thepink boxes correspond to the uplink packets from theclient nodes to the controller (the origins are labelled).The purple boxes correspond to the XOR packets wherethe label corresponds to client node whose DL and ULpackets are XORed. Fig. 3:
Scheduling for generic and star topology client nodes. In each ‘cycle’ of time T , the controller has m distinct bits of message for each client node (downlink messages - DL) and each client node has m distinct bits of message for thecontroller (uplink messages - UL). As in [5], we assume that while normally, the controller and allthe nodes are in-range of each other, bad fading events can cause transmissions to fail. Successfulnodes, namely those that have received both the downlink message from the controller and theuplink message for a client node in need, XOR the uplink and downlink messages together toform a single packet. They then broadcast the XORed packet simultaneously. The controller usesthe XORed packet as well as the downlink information that it already has to decode the uplinkpacket. The destination node uses the XORed packet as well as the uplink information that italready has to decode its downlink packet.This scheme has three phases: downlink phase, followed by uplink phase and then the XORphase. Let the time allocated for the downlink phase be T D , the uplink phase be T U and theXOR phase be T X such that T D ` T U ` T X “ T . We will describe the protocol with the aid ofFig. 4 where the network consists of one controller and nodes (S1 - S4). To the left of thefigure are the downlink buffers at each node (controller and clients) and to the right of the figure Relaying Phases C S1 S2S3 S4
Uplink State C S1S2S3S4
Downlink State
Downlink Phase C S1 S2S3 S4
Uplink StateDownlink State
XOR for S3 C S1 S2S3 S4
Uplink StateDownlink State C S1 S2S3 S4
Uplink StateDownlink State
Uplink Phase XOR for S4 C S1 S2S3 S4
Uplink State C S1S2S3S4
Downlink State C S1S2S3S4 C S1S2S3S4 C S1S2S3S4 C S1S2S3S4 C S1S2S3S4 C S1S2S3S4 C S1S2S3S4 C S1S2S3S4 C S1S2S3S4 (1) (2) (3) (4) (5) =+ =+ Fig. 4:
Simple example of XOR-CoW with one controller and 4 nodes. The graph illustrates which links are activeduring that phase. The downlink and uplink tables at each stage represent the information each node has at the endof that phase. Striped cells indicate message origins and starred cells indicate message destinations. are the uplink buffers, also at each node. They get populated as messages are decoded. Initially,the controller’s downlink buffer is full as it is the origin of all downlink messages (shown bythe striped buffers) and its uplink buffer is empty. S1 - S4 start with their corresponding uplinkbuffer being full (shown by the striped buffers) and their downlink buffers are empty. The starredmessages are those that each user is interested in receiving. The controller is interested in theuplink messages of nodes and the nodes are interested in receiving the specific downlink messageintended for them.
Schedules:
There are two versions of the XOR-CoW protocol that can be employed: a) fixed scheduleprotocol and b) flexible schedule protocol. The difference between these two mainly lies in therelaying phase – do all nodes get another shot at getting their message across or only those inneed? This is illustrated in Fig. ?? .
1) Fixed schedule:
In this scheme, time is allocated equallyfor all nodes in the XOR phase – such that they get another shot at sending their messages.Since the schedule is predetermined, the time at which the message of a particular node is tobe transmitted is also known to all users and there is no real need for a scheduling phase todetermine the schedule for the XOR phase.
2) Flexible schedule:
In this scheme, time is allocated equally only for the nodes which need help in the XOR phase (and no time is given for the messages that have already reached thedestination). This scheme requires a scheduling phase since the relays need to be told about thenodes that need help.Keeping these schemes in mind, we describe the protocol under these schemes. A. Downlink and Uplink Phases
During these phases, all the nodes are listening whenever they are not transmitting. Thedownlink phase is common in both the fixed and flexible scheduling schemes. The cycle startswith a downlink phase in which the controller broadcasts a single packet consisting of all m -bitmessages to all n nodes at rate R D “ m ¨ nT D . In Fig 4 column 2, S1 and S2 successfully decodethe entire downlink message. Their starred buffers are filled along with the downlink bufferscorresponding to other nodes. Fixed Schedule Scheme:
This is followed by the uplink phase, in which the individual nodestransmit their messages to the controller one by one according to a predetermined schedule atrate R U “ mT U { n “ m ¨ nT U by evenly dividing the time slots among all nodes. In Fig 4 column 3, thecontroller successfully decodes the uplink messages of S1 and S2 and the starred uplink buffersof the controller corresponding to these nodes are filled. Since all nodes are listening wheneverthey are not transmitting, S1 receives the uplink messages of S3 and S4 while S2 receives theuplink message of S4. The nodes which have successfully received the downlink message aswell as successfully transmitted their uplink message to the controller are referred to as strongnodes . In Fig. 4, S1 and S2 are the strong nodes. Flexible Schedule Scheme:
In the uplink phase of the flexible scheduling scheme, the nodes alsotransmit a one bit ACK to the controller (indicating whether they’ve successfully received thedownlink packet or not). Therefore, the individual nodes transmit their messages (including onebit for an ACK) to the controller one by one according to a predetermined schedule at rate R U “ m ` T U { n “ p m ` q¨ nT U by evenly dividing the time slots among all nodes. B. Scheduling Phase
This phase is crucial when the flexible scheduling scheme is employed. In this phase thecontroller transmits acknowledgments to the strong nodes (at the same rate as the downlink phase). This is just bits of information per node for downlink and uplink. The common-information about the system’s state enables the strong nodes to share a common schedule forrelaying messages for the remaining nodes. Note that the schedule only reaches the strong nodesbut the nodes which need help do not know the schedule. How will they know which messageis intended for them without the knowledge of the schedule? This can be addressed by buildingin identification of the destination node in the packet such that the nodes can figure out whichpacket was addressed to them while keeping the transmission rate the same. This approach hasbeen discussed in detail in [37]. Thus, for the remainder of the paper we’ll assume that the nodesknow which packet was meant for them. C. XOR phase:
Depending on the scheduling scheme, the time allocated for this phase can either be equallydivided among all nodes – corresponding to the rate of transmission is R X “ m ¨ nT X , or only thosethat need help – corresponding to the rate of transmission is R X “ m ¨ n T X where n are the numberof unsuccessful nodes. In either case, the strong nodes XOR the downlink and uplink messagesof each of the unsuccessful nodes they’ve heard. During the slot of an unsuccessful node (saynode Y ), all the strong nodes that have successfully heard node Y act as simultaneous broadcastrelays and transmit the XORed packet using a DSTC.In Fig. 4, S3 and S4 are the unsuccessful nodes. In the XOR slot allocated for S3 (Fig. 4column 3), S1 XORs the downlink and uplink packet of S3 (represented by the purple packet)and broadcasts it. Using the downlink packet of S3, the controller can now recover the uplinkpacket. Using its own uplink packet, S3 can now recover the downlink packet. The processfor S4 is similar and the difference lies in the fact that S1 and S2 simultaneously transmit theXORed packet for S4. VI. A NALYSIS OF
XOR-C O WIn this section, we analyze the performance of XOR-CoW. The performance of XOR-CoW’sperformance for a generic information topology is the same as the performance of Occupy CoWfor a generic topology. Therefore, we refer the readers to [5] for the analysis and performanceof XOR-CoW when the traffic is not strictly bi-directional. In this paper, we focus on theperformance of XOR-CoW for star topology only as we reap maximum benefits in this case. A. Behavioral assumptions for analysis
Our analysis depends on the following behavioral assumptions in addition to the resourceassumptions in Sec. III-A. We assume a fixed nominal SNR on all links with independentRayleigh fading on each link. We also assume channel reciprocity. Our model assumes a single-tap channel (hence flat-fading). Because the cycle-time is so short, we use the delay-limited-capacity framework [38], [39].A link with channel coefficient h and bandwidth W is deemed good (thus no errors or erasures)if the rate of transmission R is less than or equal to the link’s capacity C “ W log p `| h | SNR q . Consequently, the probability of link failure is defined as p link “ P p R ą C q “ ´ exp ´ ´ R { W ´ SNR ¯ .As in [4], if there are k simultaneous transmissions, then each receiver harvests perfect senderdiversity of k . For analysis, this is treated as k independent tries that only fail in communicatingthe message if all the tries fail. We do not consider any dispersion-style finite-block-length effectson decoding. This can be justified in spirit by [40]. We assume that transmission related errorsare always detected [41]. B. XOR-CoW probability of failure
The complete analysis of the performance of the XOR-CoW protocol is described in theA. In this section we mainly present the results and state two theorems which are useful inunderstanding the results.
Theorem 1:
If an instance of fixed schedule two-hop Occupy CoW protocol (i.e., no rateadaptation in the relaying phases) with equal downlink and uplink phases ( T D “ T U “ T D “ T U “ T M ) succeeds, then there is a common downlink and uplink success path for each nodein the network. Proof:
If a node successfully decoded the downlink message in one hop, its uplink messagealso gets through successfully to the controller in one hop (due to channel reciprocity). If a nodesuccessfully decoded the downlink message in two hops via a relay Z , then the same relay helpsuplink as well – again due to channel reciprocity. Performance would improve if we reliably had more taps/diversity. Fig. 5:
The performance of XOR-CoW for a star information topology compared with reference schemes for varyingnetwork size, and a ms cycle time, aiming at ´ probability of failure for a MHz channel. The numbers nextto the frequency-hopping scheme show the frequency diversity needed and those next to the non-simultaneousretransmission scheme show the optimal number of relays per message stream.
Theorem 2:
If an instance of fixed schedule two-hop Occupy CoW protocol with equaldownlink and uplink phase 1 ( T D “ T U “ T D “ T U “ T M ) and a given SNR succeeds,then the fixed scheduling version of XOR-CoW with downlink and uplink phase lengths bothequal to T M and XOR phase length also equals to T M succeeds at the same SNR. Proof:
From Theorem 1 we know that the paths for downlink and uplink success when T D “ T U “ T D “ T U “ T M are the same – i.e., either they directly succeed to the controlleror they have the same relay helping in both downlink and uplink. These relays essentially havethe capability the XOR the packets as they have both the packets as well as good links fortransmission. Hence, as long as the rate in the XOR phase stays the same (this is ensured by T D “ T U “ T X “ T M ), the XOR-CoW protocol also succeeds at the same SNR. A corollary of Theorem 2 is that while two-hop Occupy CoW would require time ˆ T M tosucceed, XOR-CoW succeeds in time ˆ T M – i.e., a throughput improvement of .C. Results and comparison We explore the performance of XOR-CoW with parameters taken from a contemporarypractical application, the industrial printer case described in [2]. The SERCOS III protocol [42]supports the printer’s cycle time of ms with system error probability of ´ . We target thefollowing system requirements for the application: moving printing heads that move at speedsup to m/s over distances of up to m. Every cycle lasts ms and in each cycle the controller transmits bytes of actuation data to each head and each of the sensors transmit bytesof sensory data to the controller. Assuming access to a single MHz wireless channel, this . Mbit/sec throughput corresponds to an overall spectral efficiency of approximately . bits/sec/Hz. SERCOS supports a reliability of ´ and for our protocol we target a reliabilityof ´ .We define the cycle failure probability as the probability that any packet transmitted duringthe cycle did not reach at least one of its destinations. Following [4], [5] and the communication-theoretic convention, we use the minimum SNR required to achieve ´ reliability as our metricto compare XOR-CoW to other schemes. Fig. 5 compares the performance of the followingprotocols a) XOR-CoW, b) Occupy-CoW (the cooperative-communication-based protocol notemploying network coding), and c) Frequency hopping based protocols. We see that optimizedversion of Occupy CoW (the best performance that can be obtained without using network cod-ing) and XOR-CoW with a simple equal-time allocation to different phases perform comparablyfor m “ bits (the dot-dashed lines). The advantage of XOR-CoW is clear for high aggregaterates and large networks as shown by the solid in Fig. 5. We see that XOR-CoW beats theperformance of Occupy CoW for m “ bits and network size ą while also being asimpler scheme. The dotted purple curves represent a hypothetical (non-adaptive) frequency-hopping scheme that divides the bandwidth W “ MHz into k sub-channels that are assumedto be independently faded, for m “ bits and m “ bits. The curves are annotated withthe optimal k . As k (and thus frequency hops) increases, the available diversity increases, butthe added message repetitions force each link’s instantaneous data rate to be higher. For low n the scheme prefers more frequency hops to exploit diversity benefits. The SNR cost of doingthis is marginal because the throughput is low enough that we are still in the linear-regime ofchannel capacity. For networks with fewer than nodes, this says that using frequency-hoppingis great — as long as we can reliably count on about independently faded sub-channels torepeat across, which is not always practical. D. Optimization1) Network Coding Optimization:
XOR-CoW scheme only allows for the opportunity to XORtwo packets and not more. Are we making sub-optimal decisions by restricting to XORingonly two packets? We are not and the reason is as follows. In undirected network (wireless networks considered here can be modeled as undirected networks) the throughput improvementthat network coding provides when compared to routing only schemes is upper bounded at [43].We showed in Sec. VI that the throughput improvement for the best case i.e., the star-topologyis actually ă .Furthermore, we can model the generic information topology as a multicast session. It hasbeen shown that asymptotically network coding provides no benefits when compared to a purerouting schemes [44]. Additionally, even if we end up with a network realization which canprovide significant network coding benefits (a rare event in itself), the coding points (whichperform network coding operation) need to know the state of each packet and the networkrealization to compute the optimium code. The overhead of acquiring this network informationstate is significant (similar in spirit to why backpressure routing isn’t implemented as-is in currentnetworks).
2) Phase Length Optimization:
We consider the XOR-CoW protocol and look at the optimalallocation of time which minimizes the SNR required to meet the performance specifications.Although the phase length allocations are uneven (as seen in the figure 6b), we find that theSNR saving that we achieve by having different lengths is minimal (as seen in the figure 6a).The complexity of building a system which can operate at variable rates is extremely difficultand ultimately negates out the small SNR savings achieved by optimization. The strength of theprotocol lies in the fact that a simple scheme with equal time allocations with fixed scheduleperforms almost as good as the optimal scheme – thus paving the way for a practical system.VII. C
ONCLUSIONS & F
UTURE W ORK
In this work, we designed a network coding based wireless communication protocol frameworkfor high-performance control-like systems. We have additionally shown that simple phase lengthallocations are sufficient and optimizations only provide marginal benefits. In the future, weaim to address the impact of modeling assumptions such as spatial independence, quasi-staticbehavior, etc., on cooperative communication protocols. Understanding the impact of imperfectsychronization as well as imperfect channel estimation would also be important in making theseschemes practical. A
PPENDIX
We analyze the XOR-CoW protocol by looking at all the ways at least one of the messagesdid not reach the destination within the cycle as in [4], [5]. We achieve this by partitioning the Number of nodes -2-1012345 M i n i m u m S NR ( d B ) Equal phase length allocation with fi xed schedulingOptimal phase length allocation with fl exible scheduling (a) SNR comparison of optimized flexile-scheduleXOR-CoW and fixed-schedule XOR-CoW for m “ bit and varying network size with MHz bandwidthand a ms cycle time, aiming at ´ . (b) The phase allocation for optimized XOR-CoW withflexible scheduling for m “ bit messages andvarying network size with MHz bandwidth and a mscycle time, aiming at ´ is shown. Fig. 6:
Optimization of XOR-CoW protocol nodes into various sets which depend on various aspects like downlink/uplink success and thestate of node-node as well as node-controller links in different phases. Before continuing withthe analysis itself, we define some notation.
Notation:
To effectively present the derived expressions, we provide a guide to the notation thatwill be used in the following sections. Let a transmission over a single link be an “experiment.”A binomial distribution with n independent experiments, probability of success ´ p , and numberof success m will be referred to as B p n, m, p q “ ˆ nm ˙ p ´ p q m p n ´ m . (1)Note that the probability p is the probability of failure, not the probability of success. Theprobability of at least one out of n independent experiments failing will be denoted as F p n, p q “ ´ p ´ p q n . (2) A link with fading coefficient h and bandwidth W is considered “good” (thus decodable) if therate of transmission R i is less than or equal to the link’s capacity, C “ W log p ` | h | SNR q . Weassume that the nominal operating SNR is held consistent across the entire system. Consequently,for a rate R , the assumption of Rayleigh fading tells us that the probability of an unsuccessfultransmission is defined as p “ P p R ą C q “ ´ exp ˆ ´ R { W ´ SNR ˙ . (3)We assume that if R exceeds capacity, the transmission will surely fail (with probability 1). If R is less than capacity, the transmission will surely succeed and decode to the right codeword. Set Notation:
We describe the various sets used in the analysis. Following general convention,the set itself will be represented in script font. The random variable representing the number ofnodes in that set will be presented in uppercase letters. Finally, the instantiation of that randomvariable (the cardinality of the set), will be in lowercase letters. The sets being considered are: ‚ A : the set of nodes successful in the downlink phase. Further divided into disjoint sets r A and q A such that A “ r A Ť q A . ˝ r A : the set of nodes which succeed in downlink as well as uplink phases. This isfurther partitioned into r A X (the set which connects to the controller in the XORphase) and r A U (the set which cannot connect to the controller in the XOR phase). ˝ q A : the set of nodes which do not succeed in uplink. This set is further partitionedinto q A X (which can connect to the controller in the XOR phase) and q A D (whichcannot connect to the controller in the XOR phase). ‚ B : the set of nodes that weren’t successful in downlink phase but were successful in uplinkphase. Further partitioned into disjoint sets r B (has link to the controller in the XOR phase)and q B (doesn’t have link to controller in the XOR phase) such that B “ r B Ť q B . ‚ C : the set of nodes that succeed only in the XOR phase – both uplink and downlinksuccesses happen in this phase. They can only succeed through relays. Analysis of XOR-CoW:
Let the time allocated for the downlink phase be T D , the uplink phase be T U and the XORphase be T X such that T D ` T U ` T X “ T where T is the given cycle time. If we choseto do fixed scheduling then the transmission rates for downlink, uplink and XOR phases are fixed at R D “ m ¨ nT D , R U “ m ¨ nT U and R X “ m ¨ nT X respectively. If adaptive scheduling schemeis employed, then the transmission rates for downlink, uplink and XOR phases are given by R D “ m ¨ nT D , R U “ p m ` q¨ nT U and R X “ m ¨p n ´ r a q T X where r a is the number of nodes that succeededin both uplink and downlink phases. These ˜ A are called “strong nodes” and all the others needhelp. Without loss of generality we consider the flexible schedule scheme and proceed with theanalysis. Depending on the time allocations for different phases and the number of strong nodes r a , we get the following theorem. Theorem 3:
Let the time allocated for downlink, uplink and XOR phases be T D , T U and T X respectively, the number of non-controller nodes be n , and message size be m bits. Thedownlink and uplink transmission rates are given by R D “ m ¨ nT D and R U “ p m ` q¨ nT U respectively.The corresponding probability of a single link failure, p D & p U , is given by Eq. (3). The XORphase transmission rate is given by R r aX “ m ¨p n ´ r a q T X where r a is the number of “strong nodes” inboth downlink and uplink phases and the corresponding probability of a single failure p X , isgiven by Eq. (3). The probability XOR-CoW failure is then P p fail q “ n ÿ a “ « n ´ a ÿ b “ P p fail q p R D ě R U ą R X q ` n ´ a ÿ b “ b ÿ r b “ P p fail q p R D ą R X ě R U q` a ÿ r a “ P p fail q p R U ě R D ą R X q ` a ÿ r a “ a ´ r a ÿ q a X “ P p fail q p R U ą R X ě R D q` a ÿ r a “ r a ÿ r a X “ P p fail q p R X ě R U ą R D q ` a ÿ r a X “ n ´ a ÿ b “ P p fail q p R X ą R D ě R U q ff where, P p fail q “ B p n, a, p D q ˆ B p n ´ a, b, q UD q ˆ F p n ´ a ´ b, p aU q is the probability of failure if the relationship between the rates is R D ě R U ą R X , P p fail q “ B p n, a, p D q ˆ B p n ´ a, b, q UD q ˆ B p b, r b, r UX,UD q ˆ F p n ´ a ´ r b, p aX q is the probability of failure if the relationship between the rates is R D ą R X ě R U , P p fail q “ B p n, a, p D q ˆ B p a, r a, s UD q ˆ F p n ´ a, p aU q is the probability of failure if the relationship between the rates is R U ě R D ą R X , P p fail q “ B p n, a, p D q ˆ B p a, r a, s UD q ˆ B p q a, q a X , r DX,DU q ˆ F p n ´ r a ´ q a X , p r a ` q a X U q is the probability of failure if the relationship between the rates is R U ą R X ě R D , P p fail q “ B p n, a, p D q ˆ B p a, r a, s UD q ˆ B p r a, r a X , s XU q ˆ p ´ P p success qq P p success q “ p ´ p r a X X q q a ˆ ˜ r a X ÿ k “ B p r a X , k, p U q ´ ´ s kXU ` s kXU p ´ p r a U X q ¯¸ n ´ a are the probabilities of failure and success if the relationship between the rates is R X ě R U ą R D , P p fail q “ B p n, a, p D q ˆ B p r a, r a X , s XD q ˆ B p n ´ a, b, q UD q ˆ p ´ P p success qq P p success q “ p ´ p aX q b ˆ ˜ r a X ÿ k “ B p r a X , k, p U q ´ ´ s kXU ` s kXU p ´ p r a U X q ¯¸ n ´ a ´ b are the probabilities of failure and success if the relationship between the rates is R X ą R D ě R U , where: ‚ q UD “ P p C ă R U | C ă R D q “ p U p D ‚ s UD “ P p C ă R U | C ą R D q “ p U ´ p D ´ p D ‚ s XU “ P p C ă R X | C ą R U q “ p X ´ p U ´ p U ‚ s XD “ P p C ă R X | C ą R D q “ p X ´ p D ´ p D ‚ r UX,UD “ P p R U ă C ă R X | R U ă C ă R D q “ p X ´ p U p D ´ p U ‚ r DX,DU “ P p R D ă C ă R X | R D ă C ă R U q “ p X ´ p D p U ´ p D Proof:
All potential relays get the schedules in the scheduling phase where the rate oftransmission is the same as downlink rate as stated earlier in Sec. III. This ensures that allpotential relays (those that have the downlink information) know when to transmit. Additionally,all nodes that need help also know which packet is intended for them as their identity is builtinto the packet. We look at each case to understand the subtle effects that may arise.
Case 1: R D ě R U ą R X The rates of transmission are as described earlier and the probabilities of a link succeeding indownlink, uplink and XOR phases are given by p D , p U and p X respectively. Fig. 7a shows theexhaustive list of ways to succeed in the first case of the XOR-CoW protocol. ‚ A node can succeed directly to the controller in downlink – these nodes are in set A . Asthe rate in downlink phase R D is greater than the rate in uplink phase R U , these nodesalso succeed in uplink directly to the controller (so they are an overall success). In thiscase, r A “ A as all of A retain links in the uplink phase and they are potential relays. ‚ A node can gain a link to the controller at the lower uplink rate of R U – these nodes arein set B . They get their downlink message directly from the controller in the XOR phaseas all of them retain the link to the controller in the XOR phase. (a) Case 1: R D ě R U ą R X . (b) Case 2: R U ě R D ą R X . Fig. 7:
Different ways to succeed in XOR-CoW protocol. The links between the controller and nodes are annotatedwith the rates in which they are present. The links to C are only denoted for the rate at which the links are important. ‚ A node can have both downlink and uplink successes during the XOR phase, if theyconnected to A in the uplink phase and as the rate R X in the XOR phase is less than R U ,the links do not disappear.To calculate the probability of error of the XOR-CoW protocol, we will unroll the statespace and sum over all possible instantiations of the sets of interest that result in failure. Theprobability of A “ a depends on the point to point link to the controller which has a failureprobability of p D (we use Eq. (3)). Thus we have P p A “ a q “ B p n, a, p D q . The probability thata node does not gain a link to the controller in the uplink phase given it did not have a linkin the downlink phase is given by q UD “ P p C ă R U | C ă R D q “ p U { p D . Conditioned on therealization that A “ a , the probability that B “ b nodes gain links to the controller is given by P p B “ b | A “ a q “ B p n ´ a, b, q UD q .Given A “ a and B “ b , the probability of a node in S z p A Ť B q , failing in the XOR phaseis the probability that it doesn’t connect to A in the uplink phase. The probability of a singlenode failing is given by p aU . Thus the overall probability of failure given A “ a and B “ b is F p n ´ a ´ b, p aU q . Thus we get that the probability of failure of the XOR-CoW protocol whenthe relationship between the rates is R D ě R U ą R X is given by n ÿ a “ n ´ a ÿ b “ P p fail q p R D ě R U ą R X q where, P p fail q “ B p n, a, p D q ˆ B p n ´ a, b, q UD q ˆ F p n ´ a ´ b, p aU q . Case 2: R U ě R D ą R X The rates of transmission are as described earlier and the probabilities of a link succeeding indownlink, uplink and XOR phases are given by p D , p U and p X respectively. Fig. 7b shows theexhaustive list of ways to succeed in the third case of the XOR-CoW protocol. ‚ A node can succeed directly to the controller in downlink – these nodes are in set A . Asthe rate R D in the downlink phase is lower than the rate R U in the uplink phase, this setis further divided into two disjoint sets r A (which retains the connection to the controllerin the uplink phase) and q A (which loses the connection to the controller in the uplinkphase). The nodes in r A are the potential uplink message helpers in the XOR phase. ‚ The nodes in q A succeed directly to the controller in the XOR phase as they have thedownlink as well as uplink packets to XOR. ‚ A node can have both downlink and uplink successes during the XOR phase, if theyconnected to A in the uplink phase and as the rate in XOR phase R X is less than R U , thelinks do not disappear.To calculate the probability of error of the XOR-CoW protocol, we will unroll the statespace and sum over all possible instantiations of the sets of interest that result in failure. Theprobability of A “ a depends on the point to point link to the controller which has a failureprobability of p D (we use Eq. (3)). Thus we have P p A “ a q “ B p n, a, p D q . Given A “ a ,the probability that a node in A loses its link to the controller in the uplink phase is givenby s UD “ P p C ă R U | C ą R D q “ p p U ´ p D q{p ´ p D q . Thus we get the probability that r A “ r a (these do not lose the links) given A “ a is B p a, r a, s UD q . Given A “ a and r A “ r a ,the probability of a node in S z A , failing in the XOR phase is the probability that it doesn’tconnect to A in the uplink phase. The probability of a single node failing is given by p aU . Thus,the overall probability of failure given A “ a and r A “ r a is F p n ´ a, p aU q . Thus, we get thatthe probability of failure of the XOR-CoW protocol when the relationship between the rates is R U ě R D ą R X is given by n ÿ a “ a ÿ r a “ P p fail q p R U ě R D ą R X q where, P p fail q “ B p n, a, p D q ˆ B p a, r a, s UD q ˆ F p n ´ a, p aU q . Case 3: R D ą R X ě R U The rates of transmission are as described earlier and the probabilities of a link succeeding in (a) Case 3: R D ą R X ě R U . (b) Case 4: R U ą R X ě R D . Fig. 8:
Different ways to succeed in XOR-CoW protocol. The links between the controller and nodes are annotatedwith the rates in which they are present. The links to C are only denoted for the rate at which the links are important. downlink, uplink and XOR phases are given by p D , p U and p X respectively. Fig. 8a shows theexhaustive list of ways to succeed in the second case of the XOR-CoW protocol. ‚ A node can succeed directly to the controller in downlink – these nodes are in set A . Asthe rate in downlink phase R D is greater than the rate in uplink phase R U , these nodesalso succeed in uplink directly to the controller (they are an overall success). In this case r A “ A as all nodes in A retain links in uplink phase. All of these will be potential relays. ‚ A node can gain a link to the controller at the lower uplink rate of R U – these nodes are inthe set B . Some of these nodes lose the link during the XOR phase as (since R X ě R U ).The nodes that retain the links constitute the set r B and the ones which lose the linkconstitute the set q B . The set r B get their downlink message directly from the controller inthe XOR phase but the set q B doesn’t. They need to connect to at least one node in A inthe uplink as well as XOR phase to successfully receive their downlink message. ‚ A node can have both downlink and uplink successes during the XOR phase, if theyconnected to A in the uplink phase as well as in the XOR phase (similar to q B ).To calculate the probability of error of the XOR-CoW protocol, we will unroll the state space andsum over all possible instantiations of the sets of interest that result in failure. The probability of A “ a depends on the point to point link to the controller which has a failure probability of p D (we use Eq. (3)). Thus we have P p A “ a q “ B p n, a, p D q . The probability that a node does notgain a link to the controller in the uplink phase given it did not have a link in the downlink phaseis given by q UD “ P p C ă R U | C ă R D q “ p U { p D . Conditioned on the realization that A “ a , the probability that B “ b nodes gain link to the controller is given by P p B “ b | A “ a q “ B p n ´ a, b, q UD q . Given B “ b , the probability that a node in B , loses the connection to the controller inthe XOR phase is given by r UX,UD “ p p R U ă C ă R X | R C ă C ă R D q “ p p X ´ p U q{p p D ´ p U q .Thus the probability that r B “ r b given B “ b is given by B p b, r b, r UX,UD q . Given A “ a , B “ b and r B “ r b the probability of a node in S z ´ A Ť r B ¯ , failing in the XOR phase is the probabilitythat it doesn’t connect to A in the uplink and XOR phases. The probability of a single nodefailing is given by p aX . Thus, the overall probability of failure given A “ a , B “ b and r B “ r b is F p n ´ a ´ r b, p aX q . Thus, we get that the probability of failure of the XOR-CoW protocol whenthe relationship between the rates is R D ą R X ě R U is given by n ÿ a “ n ´ a ÿ b “ b ÿ r b “ P p fail q p R D ą R X ě R U q where, P p fail q “ B p n, a, p D q ˆ B p n ´ a, b, q UD q ˆ B p b, r b, r UX,UD q ˆ F p n ´ a ´ r b, p aX q . Case 4: R U ą R X ě R D The rates of transmission are as described earlier and the probabilities of a link succeeding indownlink, uplink and XOR phases are given by p D , p U and p X respectively. Fig. 8b shows theexhaustive list of ways to succeed in the fourth case of the XOR-CoW protocol. ‚ A node can succeed directly to the controller in downlink – these nodes are in set A .As the rate in downlink phase R D is lower than the rate in uplink phase R U , this set isfurther divided into two disjoint sets r A (which retains the connection to the controller inthe uplink phase) and q A (which loses the connection to the controller in the uplink phase). ‚ The nodes in q A are further divided to q A X (those that regain the link to the controller inthe XOR phase) and q A D (those that do not regain the link to the controller). The nodesin q A X successfully transmit their own uplink message to the controller in the XOR phaseas they have the downlink messages to XOR and the link to transmit. ‚ The nodes in q A D succeed only by connecting to r A Ť q A in the uplink phase (the link backto them will automatically exist in the XOR phase since R X ă R U ). ‚ Any other node can have both downlink and uplink successes during the XOR phase, ifthey connected to r A Ť q A X in the uplink phase and as the rate in XOR phase R X is lessthan R U , the links do not disappear.To calculate the probability of error of the XOR-CoW protocol, we will unroll the statespace and sum over all possible instantiations of the sets of interest that result in failure. The (a) Case 5: R X ě R U ą R D . (b) Case 6: R X ą R D ě R U . Fig. 9:
Different ways to succeed in XOR-CoW protocol. The links between the controller and nodes are annotatedwith the rates in which they are present. The links to C are only denoted for the rate at which the links are important. probability of A “ a depends on the point to point link to the controller which has a failureprobability of p D (we use Eq. (3)). Thus we have P p A “ a q “ B p n, a, p D q . Given A “ a ,the probability that a node in A loses link to the controller in the uplink phase is given by s UD “ P p C ă R U | C ą R D q “ p p U ´ p D q{p ´ p D q . Thus we get the probability that r A “ r a (these do not lose the links) given A “ a is B p a, r a, s UD q . Given A “ a and r A “ r a , theprobability of a node in q A gaining a link to the controller in the XOR phase is given by ´ P p R D ă C ă R X | R D ă C ă R U q “ ´ r DX,DU . Thus, we get that q A X “ q a X nodes gainlinks to the controller in the XOR phase with probability B p q a, q a X , r DX,DU q .Given A “ a , r A “ r a and r A X “ r a X , the probability of a node in S z ´ r A Ť q A X ¯ failing inthe XOR phase is the probability that it doesn’t connect to r A Ť q A X in the uplink phase. Theprobability of a single node failing is given by p r a ` q a X U . Thus the overall probability of failuregiven A “ a and r A “ r a is F p n ´ r a ´ q a X , p r a ` q a X U q . Thus we get that the probability of failure ofthe XOR-CoW protocol when the relationship between the rates is R U ą R X ě R D is given by n ÿ a “ a ÿ r a “ a ´ r a ÿ q a X “ P p fail q p R U ą R X ě R D q where, P p fail q “ B p n, a, p D q ˆ B p a, r a, s UD q ˆ B p q a, q a X , r DX,DU q ˆ F p n ´ r a ´ q a X , p r a ` q a X U q . Case 5: R X ě R U ą R D The rates of transmission are as described earlier and the probabilities of a link succeeding indownlink, uplink and XOR phases are given by p D , p U and p X respectively. Fig. 9a shows theexhaustive list of ways to succeed in the fifth case of the XOR-CoW protocol. ‚ A node can succeed directly to the controller in downlink – these nodes are in set A .As the rate in downlink phase R D is lower than the rate in uplink phase R U , this set isfurther divided into two disjoint sets r A (which retains the connection to the controller inthe uplink phase) and q A (which loses the connection to the controller in the uplink phase). ‚ The nodes in r A are further divided to r A X (those that retain the link to the controller inthe XOR phase – thus can act as uplink message relays) and r A U (those that lose the linkto the controller). The set r A U can still act a relays for downlink messages. ‚ The nodes in q A succeed only if they connect to r A X in the uplink phase. ‚ The nodes in S z A succeed only in the following way: they must connect to r A X in theuplink phase (to convey their uplink message). They can receive their downlink messageeither by connecting to r A X in the XOR phase (this is not guaranteed as the rate in theXOR phase is higher) or by connecting to r A U in the uplink and XOR phase.To calculate the probability of error of the XOR-CoW protocol, we will unroll the state spaceand sum over all possible instantiations of the sets of interest that result in failure. The probabilityof A “ a depends on the point to point link to the controller which has a failure probability of p D (we use Eq. (3)). Thus we have P p A “ a q “ B p n, a, p D q . Given A “ a , the probability thata node in A loses link to the controller in the uplink phase is given by s UD “ P p C ă R U | C ą R D q “ p p U ´ p D q{p ´ p D q . Thus we get the probability that r A “ r a (these do not lose the links)given A “ a is B p a, r a, s UD q . Given A “ a and r A “ r a , the probability that a node in r A loses linkto the controller in the XOR phase is given by s XU “ P p C ă R X | C ą R U q “ p p X ´ p U q{p ´ p U q .Thus, the probability that r A X “ r A X is given by B p r a, r a X , s XU q .The probability that nodes in q A succeed is the probability that they connect to r A X in the uplinkphase which is given by ´ p r a X U . Thus the probability that all nodes in q A succeed is p ´ p r a X U q q a .For the rest of the nodes, let us calculate the probability of success. To succeed, a node must connect to r A X in the uplink phase. Let us consider that the node is connected to k nodes in r A X .The probability of this event is B p r a x , k, p U q . This is essential for uplink success. Downlink cansucceed either by connecting to one of these k nodes in r A X in the XOR phase or by having aconnection to r A U in the uplink as well as XOR phases. Thus we have the probability of downlinksuccess is ´` ´ s kXU ˘ ` s kXU p ´ p r a ´ r a X X q ¯ . Combining the uplink and downlink success we getthat a node in S z A succeeds with a probability B p r a x , k, p U q ˆ ´` ´ s kXU ˘ ` s kXU p ´ p r a ´ r a X X q ¯ . Thus, probability of success in Case 5 is given by P p success q “ p ´ p r a X U q q a ˆ ˜ r a X ÿ k “ B p r a X , k, p U q ´` ´ s kXU ˘ ` s kXU p ´ p r a ´ r a X X q ¯¸ n ´ a . (4)Thus we get that the probability of failure of the XOR-CoW protocol when the relationshipbetween the rates is R X ě R U ą R D is given by n ÿ a “ a ÿ r a “ r a ÿ r a X “ P p fail q p R X ě R U ą R D q where, P p fail q “ B p n, a, p D q ˆ B p a, r a, s UD q ˆ B p r a, r a X , s XU q ˆ p ´ P p success qq . Case 6: R X ą R D ě R U The rates of transmission are as described earlier and the probabilities of a link succeeding indownlink, uplink and XOR phases are given by p D , p U and p X respectively. Fig. 9b shows theexhaustive list of ways to succeed in the second case of the XOR-CoW protocol. ‚ A node can succeed directly to the controller in downlink – these nodes are in set A . Allthe nodes in set A succeed in uplink as the rate R U is less than R D . Thus, A “ r A . ‚ A node can gain a link to the controller at the lower uplink rate of R U – these nodesare in set B . Note that these succeeded only at R U and not at R D and hence these nodescannot help to get to the controller in the higher XOR phase rate of R X . ‚ The nodes in r A are further divided to r A X (those that retain the link to the controller inthe XOR phase) and r A U (those that lose the link to the controller in the XOR phase).Only r A X can effectively relay the uplink messages of the nodes in need. ‚ The nodes in S z A succeed only in the following way: they must connect to r A X in theuplink phase (to convey their uplink message). They can receive their downlink messageeither by connecting to r A X in the XOR phase as well (this is not guaranteed as the ratein the XOR phase is higher) or by connecting to r A U in the uplink as well as XOR phase.To calculate the probability of error of the XOR-CoW protocol, we will unroll the statespace and sum over all possible instantiations of the sets of interest that result in failure. Theprobability of A “ a depends on the point to point link to the controller which has a failureprobability of p D (we use Eq. (3)). Thus we have P p A “ a q “ B p n, a, p D q . The probabilitythat a node does not gain a link to the controller in the uplink phase given it did not have alink in the downlink phase is given by q UD “ P p C ă R U | C ă R D q “ p U { p D . Conditioned on the realization that A “ a , the probability that B “ b nodes gain link to the controller isgiven by P p B “ b | A “ a q “ B p n ´ a, b, q UD q . The probability that nodes in B succeed isthe probability that they connect to r A X in the uplink phase which is given by ´ p r a X U . Thusthe probability that all nodes in A succeed is p ´ p r a X U q b . Given A “ a , r A “ r a and B “ b ,the probability that a node in r A loses its link to the controller in the XOR phase is given by s XD “ P p C ă R X | C ą R D q “ p p X ´ p D q{p ´ p D q . Thus, the probability that r A X “ r a X isgiven by B p r a, r a X , s XD q .For the rest of the nodes, let us calculate the probability of success. In order to succeed, anode must connect to r A X in the uplink phase. Let us consider that the node is connected to k nodes in r A X . The probability of this event is B p r a x , k, p U q . This is essential for uplink success.Downlink can succeed either by connecting to one of these k nodes in r A X in the XOR phaseor by having a connection to r A D in the XOR phase. Thus we have the probability of downlinksuccess is ´` ´ s kXU ˘ ` s kXU p ´ p r a ´ r a X X q ¯ . Combining the uplink and downlink success we getthat a node in S z A succeeds with a probability B p r a x , k, p U q ˆ ´` ´ s kXU ˘ ` s kXU p ´ p r a ´ r a X X q ¯ .Thus, probability of success in case 6 is given by P p success q “ p ´ p aX q b ˆ ˜ r a X ÿ k “ B p r a X , k, p U q ´` ´ s kXU ˘ ` s kXU p ´ p r a ´ r a X X q ¯¸ n ´ a ´ b . 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