network coding based retransmission schemes for 4g wireless
TRANSCRIPT
Network Coding Based Retransmission Schemes For 4G WirelessBroadcast Networks
Thesis submitted in partial fulfillmentof the requirements for the degree of
Master of Science (by Research)in
Computer Science Engineering
by
Mohammad Shaheer Zaman200707015
Communication Research CenterInternational Institute of Information Technology
Hyderabad - 500 032, INDIAJuly 2010
Copyright c© Mohammad Shaheer Zaman, 2010
All Rights Reserved
International Institute of Information TechnologyHyderabad, India
CERTIFICATE
It is certified that the work contained in this thesis, titled “Network Coding Based RetransmissionSchemes For 4G Wireless Broadcast Networks ” by Mohammad Shaheer Zaman, has been carried outunder my supervision and is not submitted elsewhere for a degree.
Date Adviser: Prof. G Rama Murthy
To my loving mother and father
Acknowledgments
Praise be to Allah, Lord of the worlds. I would like to thank Allah for blessing me with this oppor-tunity and making things easy for me. There is no help except from Allah.
I would like to thank my advisor Dr G Rama Murthy for his mentorship and guidance during thecourse of research. He has always given me the opportunity to explore different areas of research. Hisenthusiasm for research and open mindedness has been a great source of help for me.
I have had the fortune of working with a wonderful group of people at Motorola, Bangalore. NaveenArulselvan has been a great mentor and a warm friend. His insights into research problems are soclear. His eye for details and thorough professionalism will always be a source of motivation. SureshKalyansundaram was always very helpful. His critical comments and guidance made this thesis possible.
To my brothers in college Hussien and Faheem, I would always be grateful. Hussien has been likemy elder brother all along and a solid rock in my life. . Faheem had been very supportive all along. Mybrother Wajid for our discussions and debates during the late hours of the night.
And to my family, for being extremely supportive. Ammi has been so loving and caring that’s it’snot possible to thank her enough. Abba for his patience and his constant advice. Appi for her love forme. Zahid and Shoaib have been the apple of my eyes and delight of my life. And to my wife for beingloving and caring.
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Abstract
In the future, as communication networks become complicated new innovative techniques are re-quired to improve the performance of these networks. Error correction methods are being used to designtransmission schemes which deliver optimal data rate to the users. Retransmissions schemes in the pres-ence of feedback channel are another way to improve the performance of the network. In this thesis wedesign retransmission schemes for wireless broadcast networks which use simple network coding tech-niques. We analyze simple baseline schemes for retransmission using network coding principles. Twosimple feedback channels with different amounts of feedback available are considered and algorithmsare designed for them. In particular we consider a wireless system with a shared feedback channel,wherein the number of transmission errors can be estimated by polling. The feedback information isused to develop analytical and statistical models to characterize error correction and employ these resultsto design efficient retransmission algorithms. Optimization of algorithm is done to maximize the errorcorrection performance for a limited retransmission bandwidth. We compare the performance of thesealgorithms with the various baseline schemes. Substantial improvement in terms of bandwidth, powerrequired for transmission of data and data rate is achieved that is denoted by the number of retransmis-sions required. Various tradeoffs in the design and performance of these algorithms are discussed andtheir results are analyzed.
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Contents
Chapter Page
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Network coding background and related work . . . . . . . . . . . . . . . . . . . . . . 21.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 General System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Simulation Network model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Random Retransmission, Partial Feedback and Perfect Feedback . . . . . . . . . . . . . . . 92.1 Random retransmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Random retransmission with coding . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Limited Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Perfect Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Shared Feedback Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Characterizing Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Properties of Expected number of Packets recovered . . . . . . . . . . . . . . . . . . . 253.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Design of Retransmission Scheme for shared feedback channel . . . . . . . . . . . . . . . . 274.1 t-errors for each packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 one errors for each packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 Algorithms for Retransmission for the Shared Feedback Channel . . . . . . . . . . . . . . . 335.1 Serpentine Packet Selection scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Serpentine Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Incremental Error Correction Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 Incremental Error Correction Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 385.5 Simulation results of shared feedback channel . . . . . . . . . . . . . . . . . . . . . . 395.6 Comparison across schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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viii CONTENTS
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
List of Figures
Figure Page
1.1 Multicasting over a communication network using network coding[9] . . . . . . . . . 21.2 Simple coding for broadcasting packets p1 and p2 . . . . . . . . . . . . . . . . . . . . 61.3 Figure for System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Example of one-interleave algorithm for Partial Feedback channel.10 users, 100 packets,5 retransmissions.Identity permutation for transmission . . . . . . . . . . . . . . . . . 14
2.2 Example of Disjoint Subset algorithm for Perfect Feedback channel. For 10 users, 100packets, 5 retransmissions.Identity permutation for transmission . . . . . . . . . . . . 16
2.3 Performance of Retransmission schemes . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 Venn Diagram for N = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1 Expected number of packets recovered Et(N) for different values of K and t . . . . . 314.2 Expected number of packets recovered E1(N) for different values of K . . . . . . . . 32
5.1 Performance of Algorithms for Shared feedback channel . . . . . . . . . . . . . . . . 405.2 Performance analysis for different Algorithms . . . . . . . . . . . . . . . . . . . . . . 42
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Chapter 1
Introduction
Introduction goes here... Broadcasting in Mobile Networks is expected to be very popular in the fu-ture. As the demand for broadcast services increases, 4G networks will have to incorporate mechanismsthat make sure that data is broadcast efficiently to users. Mobile broadcast services such as mobile TV,live sports/news casting, multi-party gaming, on-demand music etc which tend to be bursty are expectedto be popular. Financial services such as stock price information and the news of the stock market canbe sent as ticklers to the mobile users.
At present mobile TV is offered via streaming technology over point to point connections and inthe future it will be point to multipoint (broadcast/multicast). Telephony, messaging and on demandstreaming and download services are based on point to point (PTP) communication. The end pointsare either two telephones in a voice call or in the case of a download or streaming session, a clientserver connection. Broadcast and multicast on the other hand are synonyms for point to multipoint(PTM) communication where data packets are simultaneously transmitted from single source to multipledestinations. The term broadcast refers to the ability to deliver the content to all the users. Knownexamples are radio and TV services, which are broadcasted over the air (either terrestrial or via thesatellite) and over cable networks.
In reliable broadcast session, every receiver must correctly receive information that is sent by thesender. When the communication channels between a sender and receivers are lossy, some appropriateerror control schemes must be used to provide reliable retransmissions. Depending on applications,these schemes can be classified into two main approaches - retransmission based schemes like (ARQ)and forward error correction (FEC).
Using the retransmission approach, the sender may have to rebroadcast a lost packet to all the re-ceivers, although there may be only one receiver that did not correctly receive that packet. The ARQapproach assumes that a feedback channel is available so that receiver can communicate to the senderon whether it receives the correct data. On the other hand using the pure FEC approach, the sender gen-erates some redundancies then broadcasts both redundant and original information to the receivers. Thereceivers can then receive the data packets without any errors if the redundant information is sufficientenough to make up for the corruption of data [2],[10],[14].
1
UT
W
S
X
YZ
b1
b1
b1
b1
b2
b
2b
b
b
2
2
2
b1
1b 2
b
Figure 1.1 Multicasting over a communication network using network coding[9]
For all these services to be delivered in a scalable and reliable manner, standard bodies such as3Gpp and 802.16 have been shaping broadcast/multicast transmission guidelines. The main requirementduring the provision of such services is to efficiently use radio and network resources. However theindividual subscribers may be distributed throughout the cell and as a result experience varying channelconditions. Conservative broadcast coding schemes which cater to users with worst channel conditionwill invariably affect the overall system throughput.
1.1 Network coding background and related work
Network coding generalizes network operation beyond traditional routing, or store-and-forward, ap-proaches. Traditionally, coding is employed at source nodes for compression of redundant informationor to provide protection against losses in the network; coding is also employed at the link level to pro-tect against random errors or erasures on individual links. The network’s usual task is to transport,unmodified, information supplied by source nodes.
Network coding, in contrast, treats information as mathematical entities that can be operated upon,rather than as unmodifiable objects to be transported. It allows interior network codes to perform ar-bitrary operations on information from different incoming links. Its interest is in network-wide effectsarising from coding across multiple links.
Assume that we multicast two bits b1 and b2 from the source node S to both the nodes Y and Z asshown in Figure 1.1. Each of the links in the above network can carry only 1 bit per unit time. Thechannel from W to X transmits b1
⊕b2 , which is replicated at X before passing it on to Y and Z. Then
node y can decode b2 from b1 and b1⊕b2. Similarly Z can decode b2 .In this way all 9 channels in the
network is used only once. The computation of the exclusive-OR bit is a simple form of network coding.If the same communication objective is to be achieved simply by bit replication at the intermediatenodes without coding, at least one channel in the network must be used twice so that the total number of
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channel usage would be at least 10. Thus, network coding offers the potential advantage of minimizingboth latency and energy consumption and at the same time maximizing bit rate.
In [1] the authors first suggested the idea of network coding to increase efficiency for multicast.They show that coding within a network allows a source to multicast information at a rate approachingthe smallest minimum cut between the source and any receiver, as the coding symbol size approachesinfinity. Li et al. [8] show that linear coding with finite symbol size is sufficient for multicast connec-tions, where linear coding techniques are used to combine packets at intermediate nodes of a multicastnetwork and shown to be optimal.
In [11], Luby introduced a class of codes which are rateless and near optimal with respect to anyerasure channel. He showed that if the original data consists of K input symbols, all the input symbolscan be recovered at the receiver from any K + O(
√K log2Kδ ) of the encoded symbols for some
arbitrary δ. The decoding cost KlogK is not prohibitive either. The performance analysis is predictedon a large number of receptions. As a result it is less effective for smaller file sizes and playback buffers.The encoding and decoding costs can be further reduced for Raptor codes [17]. In [17], Shokrohalliimproved upon Luby’s scheme with an outer LT code and inner LDPC code. The encoding and decodingcomplexities are linear. Again this works very well for large files. While Luby and Raptor codes areindeed universally-optimal, they work more effectively for larger sized files. This is imperative forthe relevant packet-combining distribution to be realized while encoding. In [16], the authors studya network coding scheme when full information of all erroneous packets of all users in the broadcastsystem is available to the base station. Assuming no constraint on the feedback, the authors in [16] havestudied a heuristic to solve the resulting NP-hard retransmission design problem. In [18] a network-coding scheme has been studied for media streaming with coding constraints. In this thesis we studyretransmission schemes for a generic class of bursty applications when there is limited uplink feedback.In [6], the authors study a distributed randomized coding approach to multi-source multicasting. Theauthors do not consider quantifying the improvement in communication reliability.
This work is rooted in the recent developments of network coding for wireless ad hoc networks[19], [3], [7], [5]. Wu et al [19] proposed the basic scheme that uses XOR of packets to increase thebandwidth efficiency of a wireless mesh network. Katti et al. [7] implemented a XOR-based scheme ina wireless mesh network and showed a substantial bandwidth improvement over the current approach.Eryilmaz et al. [4] also have recently proposed a model single hop broadcast wireless network. In thatpaper, Eryilmaz et al. employed a random network coding scheme for multiple users that are down-loading a single file or multiple files from a wireless base station. Rather than using XOR operations,their scheme encodes every packet using coefficients that are randomly taken from a sufficiently largefinite field. This scheme guarantees that the receivers can decode the original data with a high prob-ability. In [13], the authors consider the problem of broadcast for wireless networks using Networkcoding. They employ Network coding at the transmitter to reduce the number of transmissions requiredto efficiently broadcast the packets. In [13], the authors assume perfect feedback information at thebase station. They then use this information to develop analytical formula for the number transmissions
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required to successfully broadcast packets to the users. They focus more on the bandwidth calculationsfor a memoryless receiver with traditional ARQ schemes. They also discuss an algorithm which usesnetwork coding to increase bandwidth efficiency. Analysis of bandwidth requirements for this algorithmis carried out and the improvements in performance are compared.
It is well known that Fountain codes such as LT codes and Raptor codes ideal for when the filebeing transmitted is very large. The same codes are not optimal for downloading small files. In stream-ing applications, the files are usually small in size. We will require engineering solutions to designingretransmission schemes for such applications. As we will see in later chapters of this work the perfor-mance of our algorithms does not depend on the size of the file. These algorithms are general and canbe used for files of different sizes.Our work differs significantly from that of [13]. In this thesis we pro-pose different feedback methods, carry out analysis and design retransmission algorithms. In particularwe focus on a feedback model, that we call shared feedback model, that is similar to blind statistics, todevelop statistical and analytical results. These results are used to design retransmission algorithms. Forthese algorithms our criterion is to maximize successful retransmissions(error correction). We discussour results in terms of number of user packets recovered and savings in retransmission bandwidth.
1.2 Contributions
In this work we present some efficient retransmission-based broadcast schemes in single-hop wire-less networks to be used for WiFi or WiMAX networks to improve reliability in broadcast transmission.Specifically we propose broadcast schemes that combine network coding and retransmission to effi-ciently utilize the bandwidth. This in turn leads to the following design issues
1. Provisioning an uplink feedback channel in broadcast settings which enable users to send infor-mation about the packet delivery status.
2. Devising a strategy to minimize feedback overhead while collecting information representative ofall users in the cell. This is necessary as the available feedback bandwidth tends to be limited.
The design of our algorithms are similar to systematic codes, in the sense that repair packets are sentafter transmitted packets are reported to be in error. These schemes are based on preliminary networkcoding principles and exploit the fact that error packets tend to be distributed among the various cellusers.
We are interested in designing a retransmission scheme that works effectively when a feedbackchannel is provisioned but may be constrained in terms of bandwidth. Early contributions for E-MBS802.16m and 3GPP MBMS have indicated the possibility of a feedback channel. In this work we suggestthree different types of feedback channels, each of which has different amount of bandwidth and assum-ing that these types of feedback information are available at the transmitter, we propose retransmissionalgorithms which efficiently utilize this information.
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We focus on three different types of bandwidth for the feedback channel that is perfect feedback, par-tial (limited feedback) and shared feedback channel. For these different bandwidth-constraint feedbackchannels we design network coding based retransmission algorithms and compare their performanceswith each other and with other simple retransmission schemes.
In particular we consider retransmission schemes where there is no feedback information availableto the base station and the best the base station can do is to use random retransmission of packets. Alsoconsidered is the simple ARQ algorithm which does not employ network coding. These transmissionschemes serve as our baseline scheme to benchmark other schemes.
In the presence of an uplink feedback we consider two cases where different amount of feedbackinformation is sent to the transmitter by the users. We analyze limited feedback scenario and the fullfeedback scenario and propose algorithms for these cases. Finally a shared feedback channel, whosebandwidth is independent of the number of users present in the cell, is proposed and analyzed. Trans-mission algorithms to maximize the expected number of packets recovered are designed and analysis ofthese transmission schemes are carried out.
The performance of these algorithms are analyzed in terms of the number of packets corrected andthe bandwidth used. Retransmission slots indicate the amount of bandwidth used. And for a particularamount of bandwidth available, the number of packets recovered by successful retransmission is theperformance indicator which we use for all the algorithms.Maximizing the number of packets recoveredfor a given number of retransmission slots is design goal of these schemes.
1.3 Motivation
In the Figure 1.2, the base station (BS) wants to broadcast packets p1 and p2 to the two users A andB. After the initial transmissions by the BS, user A informs the BS by means of NACK that it did notreceive packet p2. Similarly user B informs the BS by means of NACK that it did not receive packet p1.The BS XOR’s packet p1 and p2 and forms a single packet. This coded packet is now broadcast to boththe users. Both the users receive the packet p1
⊕p2 Since user A already has packet p1, it decodes p2
by XORing the received packet p1⊕
p2 with p1 and recovers p2. Similarly user B already has packetp2, it decodes p1 by XORing the received packet p1
⊕p2 with p2 to recover p1.
The number of retransmissions required to broadcast these packets using the simple coding is one.Using traditional schemes like ARQ would require two retransmissions to broadcast these packets. Inthe first retransmission the Base station would broadcast packet p1 and in the second retransmissionthe Base station would broadcast packet p2 to the users. We observe that coding scheme saves 50%bandwidth as compared to ARQ.
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Figure 1.2 Simple coding for broadcasting packets p1 and p2
1.4 General System model
The general broadcast network model we consider in this thesis in described here. For this model,different types of feedback information which is available at the base station and the algorithms forthose feedback channels will be considered in detail and discussed in later chapters in the thesis.
Perfect
Partial
Shared
Feedback channel
NC-based
retransmission strategy
1 2 ……. N 1 2 ... RData slots Retransmission
slots
MS1
MS2
MSi
MSn
Basestation
Pi:iid packet loss probability
No packet loss for transmissionFigure of Merit
Average number of
erroneous packets
recovered by
retransmissions
Figure 1.3 Figure for System model
The system model is described in Figure 1.3. There are K users in the system, scattered randomlyin the cell. The base station wants to broadcast N packets to all these users. After transmission of thesepackets, various users inform the base station about the packets which they have lost. This informationcan be sent to the base station in the form of NACK for each packet, voice vote mechanism or someother feedback means. After this initial transmission, the base station is given R retransmission slots toretransmit some of the packets which have not been received by the users. The packets retransmitted
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during each of the retransmission slot is either a single original packet or a coded packet (multipleoriginal packets are XORed together to form a coded packet).
1.5 Assumptions
All the analysis we carry out in this thesis use the following assumptions mentioned below.
1. The information is sent in packets. All the packets are of the same size and use the same modula-tion and demodulation techniques. Even the XORed packets sent during each retransmission areof the same size as the original packets.
2. For all the algorithms, we assume that the packets are not lost during the retransmissions. Thismeans that every retransmitted packet, transmitted during the retransmission slot, is received by allthe users in the cell without any error. Since we use this assumption even for the baseline schemesand traditional ARQ scheme, we can compare performances across the schemes.In 4G systems,an outer error control loop is commonly provisioned. Modulation and coding schemes (MCS) areadapted so that frame error rate for the first transmission is controlled at Xpercent (X 20-30).There will be type-II HARQ combining benefits for the retransmitted packets. Moreover thesepackets can also be allocated conservative resources to reduce error rate to a negligible range, forexample using lower MCS or larger resource blocks.
3. The decoding used at the mobile receiver is memory less. That means that the mobile user doesnot store the received coded packet if it cannot decode it in that retransmission slot. It just discardsthe packet. It does not store packets that cannot be decoded at that moment for future decodingwhen there is more information available.
4. The probability that a user i does not receive a transmitted packet follows a Bernoulli’s distributionwith parameter pi. This model is very simple and not sufficient for many real cases. However thismodel is intended to capture the essence of wireless broadcast network.
1.6 Simulation Network model
Our network model consists of a single base station and K(>1) users scattered randomly in the cell.Instead of using Rayleigh fading parameters, we use packet loss rates to characterize the wireless chan-nel. The average packet error rate is around 5% for each of the users, which is similar to that of IEEE802.11 standard [12]. Bursts of 100 packets are transmitted periodically followed by a retransmissioninterval of γ slots. This buffer size is usually sufficient for multimedia applications and other applica-tions where short files are to be broadcasts. The packet errors are considered to be independent amongthe users and identically distributed across different transmissions. These simulation runs were carried
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using the above model for 500 different user error rate scenarios and performance is averaged out. Theseresults are then plotted.
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Chapter 2
Random Retransmission, Partial Feedback and Perfect Feedback
In this chapter, we consider baseline random retransmission schemes without feedback channels andschemes with feedback channels, with different amounts of feedback information available to the trans-mitter. The analysis of these schemes for our application is carried out and some results are developed.Limited and partial feedback models are suggested, and retransmission algorithms for these feedbackmodels are suggested. Performance of these algorithms are shown by carrying out simulation.
Some of these schemes serve as baseline schemes which are then compared with other intelligent re-transmission schemes in later chapters. For this chapter we consider the following network setup. Thereare K users in the system, to whom N packets are to be broadcast. This is followed by a retransmissioninterval of R slots.
2.1 Random retransmission
In random retransmission strategy, the base station randomly picks R packets among the N originalpackets and retransmits them. Suppose a given user i, does not receive ’x’number of packets of theoriginal N transmitted packets.
Definition 1 Let X be the number of colored balls that are chosen when m balls are randomly selected
from a bin containing n balls of which k balls are known to be colored. Then the random variable X
follows hypergeometric distribution with parameters (m, k, n). The distribution is defined as [15]
fHGX (x;m, k, n) =
(n−km−x
)(kx
)(nm
) ∀ x = 0, 1, 2, . . . .,min (m, k)
Lemma 1 The probability that j of the R transmitted packets were among the x packets that were not
received, follows hypergeometric distribution with parameters N, x, R.
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Proof : The probability that j of the of the R transmitted packets were among the x packets notreceived, is given by
pj (N, x,R) =
(N−xR−j
)(xj
)(NR
) ∀ j = 0, 1, 2, . . . .,min (x,R) (2.1)
This is nothing but a hypergeometric distribution with parameters N, x,R.
The mean of hypergeometric distributed random variable is known to be xRN . This means that if we
had R = N, retransmission slots we would be able to receive all x packets on the average if we appliedrandom retransmissions.
But the probability the user had not received x packets follows binomial distribution with parameterpi. In other words,
pe(x) =
(N
x
)pix(1− pi)N−x ∀ x = 0, 1, 2, . . . ., N. (2.2)
Therefore the average number of packets that can be recovered the random retransmission strategyfor a particular user is given by
n =N∑x=0
pe(x)
min(x,R)∑j=0
jpj(N, x,R),
=
N∑x=0
(N
x
)pxi (1− pi)N−x
min(x,R)∑j=0
j
(N−xR−j
)(xj
)(NR
) ,
=N∑x=0
(N
x
)pix(1− pi)N−x
xR
N,
= piR.
The average total number of packets that can be recovered for all the users using this strategy will be∑Ki=1 piR.
2.2 Random retransmission with coding
In this scheme, in the first retransmission, the base station randomly selects NR packets out of the
N original packets, which were transmitted earlier and XORs them to form a coded packet, which isthen transmitted. Each coded packet is a XOR of exactly N
R packets, selected from the set of N packetstranmitted earlier. During the second retransmission, a set of N
R packets, different from those selectedduring the first retransmission, are selected and coded to form a new packet and then retransmitted.Therefore during each retransmission, a new set consisting of N
R packets not selected in any of the
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previous retransmission, is encoded and transmitted. This procedure continues till all the retransmissionslots are finished. So after R retransmissions, all the N original packets would have been constituent ofsome coded packet and sent to the users. The packets that are XORed to form a coded retransmissionpacket in each retransmission can be generated by a scrambling sequence known both to the users andthe base station of the network. This will minimize the overhead that needs to be sent over-the-air. Ifa user has exactly one error packet among the combination of N
R packets in a particular retransmissionslot, he would be able to decode it correctly. We are interested in calculating the average number ofpackets that can be recovered using this strategy.
Similar to random retransmission scheme, suppose a user i, had not received x packets during thetransmission by the base station. During each retransmission, some of the packets among the xNACKedpackets for user i, will be the elements of the set consisting of NR packets being XORed in that slot. Weare interested in the distribution of these error packets among all the R retransmission slots. Let XR (x)
be the R-tuple that denotes the number of error packets which are among the constituents of packetsbeing XORed in each of retransmission slots, the sum total of which will be x.
Lemma 2 In the problem described, the joint distribution of the number of NACKed packets is given by
P (X1 = i1, . . . , Xr = ir) =
∏rj=1
(NRij
)(N−rN
Rx−
∑rj=1 ij
)(Nx
) for 1 ≤ r ≤ R (2.3)
proof : See Appendix A
By substituting r = R in (2.3) we get
P (XR (x) = {x1, x2, . . . , xR}) =
∏Rj=1
(NRxj
)(Nx
) (2.4)
WhereR∑i=1
xi = x
Then the expected number of packets that can be recovered for a given user is
E =∑XR(x)
P (XR (x) = {x1, x2, . . . , xR})R∑i=1
I (xi) , (2.5)
Where I (t) = 1 if t = 1 and 0 otherwise. Note that this condition is necessary since the user willbe able to decode the encoded packet only if it has exactly one error packet among the combination ofNR packets in a particular retransmission slot. The closed form solution of equation (2.5) is very difficultto calculate.
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We now consider the presence of an uplink feedback channel. In this section we propose two differenttypes of feedback channels ,each with a different amount of bandwidth which could be provisioned inthe 4G wireless Network. Then for these feedback schemes, we suggest retransmission algorithms toimprove the efficiency of the network.
2.3 Limited Feedback
Let {A1}, . . . , {AR} be the sets of packets that are to be coded and sent in each retransmissionslot. In each retransmission, the constituent packets of the set {Ai} are XORed to form a new packetwhich is then transmitted. As in the previous section, each packet Bi is the XOR of exactly ’NR ’ numberof original packets. All the sets {A1}, . . . , {AR} are disjoint which means that the constituent packetsare not repeated in any of the set. The idea is to distribute equally and uniquely all the packets in thegiven number of retransmission slots. This makes sure that all the packets have been retransmitted tothe users.
In this scheme, the users feedback the number of packet in errors among the R pre-determined re-transmission sequences. As in the previous scheme, a known scrambling sequence will be used to gen-erate the different packet combinations. This scrambling sequence is known to all the users in the cell.Every user j informs the number of errors in the constituent packets of the set {Ai} :N(j, Bi). Then thetotal number of errors to be corrected from the retransmitted packet Bi is N (Bi) =
∑Kj=1N(j, Bi).
The maximum uplink feedback overhead can be easily calculated as KRlogNR . There are K users,R retransmissions and since there are N
R packets in each transmission, logNR bits are required to denotethe number of packets.
LetC (j, Bi) = number of packets recoverable for user j after receiving packetBi. In factC (j, Bi) =
1 if N (j, Bi) = 1 and C (j, Bi) = 0 otherwise. Total number of packets recoverable for all users afterreceiving Bi is C (Bi) =
∑Kj=1C(j, Bi) which is deterministic given the user feedback.
Let Li, i = 1, 2, 3...R denote the coded packet (XOR of packets).
Let {ti} and {tj} be the original packets that constitute the coded retransmission packets Liand Ljrespectively.
Definition of interleave operation on Liand Lj
1. Interleave: Randomly spread 2 sequences {ti} and {tj} to form 2 equal-length packet sequences{t′i}, {t′j}
2. The interleaved constituent packets of {t′i} and {t′j} are XORed to form L′i and L′j respectively
3. Given the limited feedback at the base station, the value of the number of correctable errors forthe coded packets T ′i , T
′j is unknown (random) to the base station.
12
Below one-interleave algorithm for retransmission on the limited feedback channel is explained.
One-interleave Algorithm
1. Sort the proposed retransmission packets Bis in ascending order of C(Bi)N(Bi)
, ties broken withhigher number of reported errors i.e, N (Bi) . Sorted packet sequences are denoted as {Ti}, i =1, 2, . . . , R.
Note that each packet Li is still going to be made up of NR packets from the original packets.
2. Starting with L1, using equations (2.4) and (2.5) compute the set
S1 = {j‖(E[C(L′j) + C(L′1)] > C(Lj) + C(L1))} for j = 2, 3 . . . R
3. Then best packet sequence to interleave with L1is given by
j∗ = argmax j inS1E[C(L′j) + C(L′1)]
4. Then {t1} and {tj∗} are to be interleaved before transmission
5. The coded packets L1 and Lj∗ are transmitted in successive retransmission slots. Then L1 andLj∗ are eliminated for successive iterations.
6. If S1 is a null set, only L1 is sent as a retransmission
7. The procedure is repeated till no sequence is left.
In effect, in the one-interleave algorithm the decoding errors of the users are spread across the twonetwork-coded packet sequences, enabling more corrections for every user.
Figure 2.1, gives an example of the data sample for the partial feedback channel or the limitedfeedback channel. In this example there are 5 retransmission slots. There are 5 different pre determinednetwork coded packets. The first network coded packets consists of packets with sequence numbers 1-20 which are shown in yellow color in the above figure. Similarly packets with sequence number 21-40are shown in grey and are part of the second network coded packet. Different colors show constituentpackets of different network coded packet. The packet errors for all the users are shown on the left inthe figure above. The feedback information available is also shown on the right side in the figure above.
From the Figure2.1 we can see that N(1, P1) = 1, N (2, P1) = 0, N (3, P1) = 1, .. N (10, P1) = 3.
Then N (P1) =∑K
j=1N(j, P1) = 13. Similarly values for P2 P2, P3 and P5 are calculated. Also thevalues C (1, P1) = 1, C (3, P1) = 1 and C (5, P1) = 1 can be found out. And C (j, P1) = 0 for allother j. Then C (P1) =
∑Kj=1C (j, P1) = 3. Now to find out which packet should be interleaved with
the packet P1 the calculations are shown. Expected number of packets recovered when P1 is interleavedwith P2 is calculated as 6.75. Similarly values of expected packets recovered when P1 is interleaved
13
Figure 2.1 Example of one-interleave algorithm for Partial Feedback channel.10 users, 100 packets, 5retransmissions.Identity permutation for transmission
with P3, P4 and P5 are 8.4, 5.6 and 9.4 respectively. Without interleaving total errors that wouldbe corrected when packets P1 and P2 are transmitted would be 5. While the expected value of errorscorrected when interleaved packets P1 and P2 are transmitted is 6.75, which is an improvement of 1.75.Also from the figure we see that j∗ = 5 since the expected error corrected is the largest which is 9.4.
2.4 Perfect Feedback
In this scenario users NACK all their error packets. In this case the base station has complete informa-tion about the network. This would result in feedback overhead of O(KN) bits for every transmissioninterval, since there are K users and N packets are transmitted to them, which may be prohibitive forlarge number of users. Let E1, E2, . . . , EK be the sets containing NACKed packets of the K users.The optimal retransmission scheme will need maxi|Ei| slots to correct all errors. But devising such ascheme is known to be a NP−hard problem [4]. Here we suggest a sub-optimal algorithm, which wecall Disjoint Subset algorithm, which performs well.
The motivation behind this algorithm is that we want to code large number of packets in a particularretransmission slot. But if a particular packet has been NACKed by multiple users, higher priority isgiven to the user with the larger queue length and the packet is sent the next retransmission slot.
14
Disjoint subset algorithm
1. E1, E2, . . . , EK are sorted in descending orders of length. Say we call the sorted queues asS1, S2, . . . , SK . Let Sj(i) denote the ith packet in the sorted queue j
2. Initialize Active set AS and Active Packet Sequence AP. AS = {S1} and AP = {S1(1)}
3. As i varies from 2 to K
If for any j, Si(j) is in AP
Include i in AS
Else
Find j∗ = minj {Si (j) not in set SK} for all k in AS
If valid j∗ obtained, include i in AS and j∗ in AP
4. Remove AP from all queues S1, S2, . . . , SK are sorted in descending orders of length
5. Procedure repeats till no packet is left
In a given retransmission slot, Active User set, AU is the set of all users who have a packet addressedto, and Active Packet AP is the set denoting the specific packets that are coded(XORed). Packets arechosen from users in decreasing priority of their NACK queue lengths. The key idea is if any packet ofuser i is already in AP, include the user in AU. If not, select a packet from that user as long as it is notidentical to NACK requests of high-priority users. Finally network-code the packets in AP and transmit.
In Figure 2.2 above, the sample error scenario is shown. In the perfect feedback case the basestation knows the complete statistics associated with each user and each packet. In the above figurethe working of the algorithm is illustrated. For every iteration, the Active Set sequence and the Activepacket sequence are shown.
During the 5th iteration packet with sequence no 75 is not included in the Active packet sequencesince it also occurs in S1 which is given a higher priority. And therefore packet with sequence no 75is included in the 6th iteration. Similarly packet with sequence no 73 is included in the Active packetSequence during the 5th iteration even though the same has also been NACKed by the user S4 since ithas a lower priority than S2.
2.5 Simulation Results
We see the performance of the transmission schemes discussed in this chapter. These schemes areeffective in the presence of uplink feedback channel. From Figure2.3 we see that for 10 users showthat close to 100% errors (with perfect feedback) and 64.32% errors (with partial feedback) could becorrected using a conventional approach.
15
Figure 2.2 Example of Disjoint Subset algorithm for Perfect Feedback channel. For 10 users, 100packets, 5 retransmissions.Identity permutation for transmission
We compare the perfect feedback scheme and partial feedback scheme, which use an uplink feedbackchannel of varying bandwidth with the traditional ARQ scheme which also uses a feedback channel.We see considerable benefit with these schemes as compared to ARQ scheme. For a redundancy of5% , partial feedback scheme shows a improvement of 20% and perfect feedback scheme shows animprovement of 43% over ARQ in the number of packets recovered.
In Figure2.3, we see the benefit of network coding as compared to retransmissions without coding inthe absence of feedback channel. The gap between the two curves in the figure shows the improvementin the performance. The Random retransmission with coding curve indicates random network coding ofpackets at the base station without any feedback available. The Random retransmission curve indicatesa simple retransmission of packets without any network coding, when there is no feedback available atthe base station. Randomly any packet is picked up and transmitted in each of the retransmission slot.For a redundancy of 5%, there is a difference of 25% in the performance. This gap in the performance ofthese two curves goes on increasing as the number of redundant slots goes on increasing. With even littlefeedback available network coding further increases the performance as is shown by blue curve. For a5% redundancy, the difference between partial feedback and random network coding and no networkcoding is 8% and 35% respectively.
These results show that combining network coding and providing feedback information providesconsiderable benefit.
16
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
Redundancy(%)
Percen
tag
e o
f to
tal erro
rs c
orrecte
d
Perfect Feedback schemePartial Feedback schemeRandome retransmission with codingARQRandom retransmission
Figure 2.3 Performance of Retransmission schemes
2.6 Conclusion
In this chapter we studied random retransmission with network coding and random retransmissionwithout network coding which are schemes which do not use feedback information. Then we proposedpartial feedback channel and perfect feedback channel. For these type feedback channels we discussedretransmission algorithms which used coding. Finally the performance of these retransmission schemeswas discussed and their comparison plotted. The benefits achieved as compared to tradition ARQ is alsoshown.
17
Chapter 3
Shared Feedback Channel
In this chapter we propose a model of the feedback, which we call shared feedback. The base stationknows the number of users who have not received a particular packet, though it does not know theidentity of those users. Further, for this shared feedback scheme, we carry out analysis and developstatistical models.
3.1 System Model
The base-station broadcasts bursts of Nt packets sequentially to K users. The tolerable redundancyis assumed to be γ. In other words, the base-station shall take no more than Nt(1 + γ) slots to transmitthe burst. We also assume frequency-duplex division operation with limited uplink feedback bandwidth.In fact, the uplink feedback channel will be shared among all K users in the system. For every packetthat had not been acknowledged, the users in the system that had not received that packet inform thebase station via the uplink feedback slot corresponding to that particular packet. The base station countsthe number of users that have not received that packet from the feedback slot.In this manner the basestation is able to calculate the number of users which are in error for a particular packet.There willbe synchronization issues. But there is a proposal in 802.16m Multicast/ broadcast services group toprovision a similar mechanism. Ignoring noise gives an upper bound on performance of this scheme.Noise can be modeled into the current analysis to study exact degradation The identity of the users inerror is not known to the base-station. Therefore the base station can be said to have Blind statistics ofthe user errors. We note that the required feedback bandwidth is independent of the number of users.
Our work generalizes the example given in chapter 1, to a setting where there are several users andeach user has different packets in error. However the base station is assumed to have only partial feed-back information, namely the number of negative acknowledgements (NACKs) for each transmittedpacket. We also make no restrictions about the user distribution except that the locations and by ex-tension, the user errors are independent of each other. For the sake of simplicity in our modeling, thefeedback channel is assumed to be error-free. But we note that the effect of estimation error can befactored in easily. On the mobile user side, we do not pose any restrictions on the computational or
18
buffer requirements for the mobile user. If the user, at a particular instance of time is not able to decodea coded packet, it discards that packet.It does not store this packet for decoding at a time when moredecoding information is available. Thus we do only memoryless decoding of the network-coded packet.
3.2 Characterizing Error Correction
The users inform the base station how many users have a particular packet in error via the sharedfeedback channel, for all the Nt original packets transmitted.Feedback mechanism is based on voice-vote: Users with a particular packet in error will sound in the slot corresponding to the packet. The basestation will gauge the number of user errors by estimating the transmitted energy in the feedback. Inpractical systems this estimate will be coarse owing to quantization errors and effect of thermal noise.The analysis in the work provides an upper-bound to the performance of systems provisioned with sucha feedback scheme.
For example, base-station is aware that packet P1 was received in error by K1 users, packet P2 wasreceived by K2 users and so on. Let there be total N(= Nt) packets in error. The total number of errorsin the system is given by K =
∑Ni=1Ki. We are interested in studying a scheme which maximizes the
number of packets recovered out of K given the tolerable redundancy γ.
Designing the transmission scheme would involve selecting the packets in a given order and assessingthe optimal number of packets to be encoded during each retransmission slot. To do this, we firstcalculate the average number of packets recovered when an arbitrary number of packets are combined.Each of these packets has been requested by an independent set of users. The user is assumed toperform memoryless decoding during each retransmission slot. So, if more than one NACK request of aparticular user is encoded in a single XORed packet, the corresponding packets cannot be corrected forthe user. For instance, if a user requests packets P1, P2 and P3, and the base-station transmits P1
⊕P2
and P3⊕PX in the retransmission interval, the user will be able to correct only packet P3. Theorem
1, extends this idea to the general case when N packets are combined.The inclusion-exclusion principlestates that if A1, ....An are finite sets, then
∣∣∣∣ n⋃i=1
Ai
∣∣∣∣ = n∑i=1
|Ai| − · · ·+∑
i,j,k : 1≤i<j<k≤n|Ai ∩Aj ∩Ak| − · · · + (−1)n−1 |A1 ∩ · · · ∩An|
where |A| denotes the cardinality of the set A.
Theorem 1 Let the sets I1, I2, . . . ., IN contain the users who have NACKed packets P1, P2, . . . . , PN
respectively. The expected number of packets that can be recovered by sendingP1⊕P 2
⊕P3 · · ·
⊕PN
is given by
19
E (|I1| , |I2| , . . . .., |IN | ;K) =
N∑i=1
|Ii| − 2∑i,j
|Ii ∩ Ij |+ . . . . + (−1)N−1N
∣∣∣∣∣N⋂i=1
Ii
∣∣∣∣∣ (3.1)
Where |S| denotes the cardinality of the set S and E denotes the expectation operator.
Proof :We prove the theorem by first looking at the combination of small number of packets and thenlook at the general combination of N packets.Another proof of the same theorem can be found in Appendix B.
Combining 2-packetsThe union set U(I1, I2) contains the set of all users who have NACKed atleast one of the packets in
P = {P1, P2} . But this will also contain those users who NACKed both the packets. Because the userswill do memoryless decoding, all the transmission errors for a user with more than one NACK in P,cannot be corrected.
From the set union theorem we know that|U(I1, I2)| = |I1|+ |I2| − (|I1 ∩ I2|) .
By retransmitting a combination of these two packets, we will not be able to correct those user errorswho have NACKed both the packets. As a result, we are interested in the number of users who haveexactly one packet in error. This can be easily obtained as
E (|I1| , |I2|) = {|U (I1, I2)| − (|I1 ∩ I2|)} .
= {|I1|+ |I2| − (|I1 ∩ I2|)− (|I1 ∩ I2|)}
= { |I1|+ |I2| − 2 (|I1 ∩ I2|)}
Combining 3-packetsThe union set U(I1, I2, I3) contains the set of all users who have NACKed at least one of the packets
in P = {P1, P2, P3} . But this will also contain those users who NACKed two packets, three packets.Because the users will do memoryless decoding, all the transmission errors for a user with more thanone NACK in P, cannot be corrected.
From the set union theorem, we know that
|U (I1, I2, I3)| = |I1|+ |I2|+ |I3| − (|I1 ∩ I2|+ |I1 ∩ I3|+ |I2 ∩ I3|) + (|I1 ∩ I2 ∩ I3|) .
In the Venn diagram in Figure3.1 , the set of all users who NACKed exactly two of the three packets,are shown in striped region. In fact, the three such sets are given by the set
−→I i,j = {Ii ∩ Ij} − {Ii ∩ Ij ∩ Ik} for all i, j ∈ {1, 2, 3} .
20
The set of users who NACKed all three packets are shown in solid in Figure 3-2. By transmitting acombination of these three packets, we will not be able to correct any of their reported errors. As aresult, we are interested in the number of users who have exactly one packet in error, which is shown inthe plain region.
Figure 3.1 Venn Diagram for N = 3
So the number of error corrections is calculated by removing all such users. This can be easilyobtained as
E (|I1| , |I2| , |I3|)
= {|U (I1, I2, I3)| − (|I1 ∩ I2|+ |I2 ∩ I3|+ |I1 ∩ I3|) + 2 (|I1 ∩ I2 ∩ I3|)}
= {|I1|+ |I2|+ |I3| − (|I1 ∩ I2|+ |I1 ∩ I3|+ |I2 ∩ I3|)
+ (|I1 ∩ I2 ∩ I3|)− (|I1 ∩ I2|+ |I2 ∩ I3|+ |I1 ∩ I3|) + 2 (|I1 ∩ I2 ∩ I3|)}
= {|I1|+ |I2|+ |I3| − 2 (|I1 ∩ I2|+ |I1 ∩ I3|+ |I2 ∩ I3|) + 3 (|I1 ∩ I2 ∩ I3|)}
Combining 4-packets
The union set U(I1, I2, I3, I4) contains the set of all users who have NACKed atleast one of the packetsin P = {P1, P2, P3, P4} . But this will also contain those users who NACKed two packets, three packetsand four packets. So the number of error corrections is calculated by removing all these users. From the
21
union of set formula we know that
|U (I1, I2, I3, I4)| = |I1|+ |I2|+ |I3|+ |I4| −
(|I1 ∩ I2|+ |I1 ∩ I3|+ |I2 ∩ I3|+ |I1 ∩ I4|+ |I2 ∩ I4|+ |I3 ∩ I4|)
+ (|I1 ∩ I2 ∩ I3|+ |I2 ∩ I3 ∩ I4|+ I1 ∩ I3 ∩ I4 + I1 ∩ I2 ∩ I4)−
(|I1 ∩ I2 ∩ I3 ∩ I4|)
The number of error corrections is given by
E (|I1| , |I2| , |I3| , |I4|) = {|U (I1, I2, I3, I4)| −
(|I1 ∩ I2|+ |I1 ∩ I3|+ |I2 ∩ I3|+ |I1 ∩ I4|+ |I2 ∩ I4|+ |I3 ∩ I4|)
+ 2 (|I1 ∩ I2 ∩ I3|+ |I2 ∩ I3 ∩ I4|+ I1 ∩ I3 ∩ I4 + I1 ∩ I2 ∩ I4)
− 3 (|I1 ∩ I2 ∩ I3 ∩ I4|)
= {|I1|+ |I2|+ |I3|+ |I4|
− (|I1 ∩ I2|+ |I1 ∩ I3|+ |I2 ∩ I3|+ |I1 ∩ I4|+ |I2 ∩ I4|+ |I3 ∩ I4|)+
(|I1 ∩ I2 ∩ I3|+ |I2 ∩ I3 ∩ I4|+ I1 ∩ I3 ∩ I4 + I1 ∩ I2 ∩ I4)− (|I1 ∩ I2 ∩ I3 ∩ I4|)
− (|I1 ∩ I2|+ |I1 ∩ I3|+ |I2 ∩ I3|+ |I1 ∩ I4|+ |I2 ∩ I4|+ |I3 ∩ I4|)+
2 (|I1 ∩ I2 ∩ I3|+ |I2 ∩ I3 ∩ I4|+ I1 ∩ I3 ∩ I4 + I1 ∩ I2 ∩ I4)− 3 (|I1 ∩ I2 ∩ I3 ∩ I4|)}
= {|I1|+ |I2|+ |I3|+ |I4|
− 2 (|I1 ∩ I2|+ |I1 ∩ I3|+ |I2 ∩ I3|+ |I1 ∩ I4|+ |I2 ∩ I4|+ |I3 ∩ I4|)+
3 (|I1 ∩ I2 ∩ I3|+ |I2 ∩ I3 ∩ I4|+ I1 ∩ I3 ∩ I4 + I1 ∩ I2 ∩ I4)− 4 (|I1 ∩ I2 ∩ I3 ∩ I4|)}.
General N packet combinationThe union set U(I1, I2, I3, . . . .IN ) contains the set of all users who have NACKed atleast one of thepackets in P = {P1, P2, P3, . . . PN}. But this will also contain all users who NACKed two packets,three packets and so on. Because the users will do memoryless decoding, all transmission errors fora user with more than one NACK in P, cannot be corrected. So the number of packet recovered iscalculated by removing all such users.
For the general case, we let Ci denote the set of all users who have NACKed atleast i packets in theset P . Then the number of users who have NACKed atleast one user is given by the set-union theorem:
UN = |C1| − |C2|+ . . . . +(−1N
)|CN | . (3.2)
All transmission errors of users in sets Ci for i ≥ 2 will not be corrected. As a result, these users willneed to be removed from UN in order to calculate the number of packets recovered. But we note thatsome of the users inCi will also be present inCj for all j > i. In order to accurately calculate the numberof users with exactly one packet in error, we remove all Cis (i > 1) from CN in (3.3) sequentially. Say,all the users in C2 (i.e. users with atleast 2 NACKs) are removed first. Next we remove all users who
22
have NACKed atleast three packets, i.e. |C3|. But C2 would have contained [(N
2 )(N−2
1 )(N
3 )] number of
users present in C3 and hence these users need not be removed again. As a result, we need to subtract
a3 ,
(1− (N
2 )(N−2
1 )(N
3 )
)users in C3.
Similarly since C2 and C3 would have contained [(N
2 )(N−2
2 )(N
4 )+ a3
(N3 )(
N−11 )
(N4 )
] number of users present
in C4 and hence these users need not be removed again. As a result we need to subtract a4 |C4| fromUN , where
a4 ,
1−(N2
)(N−22
)(N4
) − a3
(N3
)(N−11
)(N4
) (3.3)
By extending this argument, the subtraction of C2, a3C3, a4C4 . . . aN−1CN−1 would have con-tained
aN = [1−(N2
)(N−2N−2
)(NN
) − a3
(N3
)(N−3N−3
)(NN
) · · · − aN − 1
(NN−1
)(N−(N−1)N−(N−1)
)(NN
) ],
= 1−∑
N−1j−2 aj
(Nj
)(N−jj
)(NN
) (3.4)
Therefore the average number of packets recovered is given by
EN = E{|C1| − |C2|+ . . . . + (−1)N |CN | − a2 |C2| − a3 |C3| − . . . . − aN |CN |}. (3.5)
We define a2 , 1.
From definition and simple algebra operation we see that
a3 = −2, a4 = 3, a5 = −4, . . . . . . . aN = (−1)N (N − 1)
Substituting the above values into the equation (3.5), we get
EN = E[|C1| − |C2|+ . . . . + (−1)N |CN | −
(1) |C2| − (−2) |C3| − (3) |C4| . . . . −((−1)N (N − 1)
)|CN |].
By simple algebraic manipulation, we can reduce this to
EN = E{|C1| − 2 |C2|+ 3 |C3|+ . . . . . . . . . . +
(−1N
)N |CN |
}(3.6)
Hence equation (3.6) proves the theorem.
Let packet Pi be NACKed by Ki users, i.e, |Ii| = Ki for all i. Before we evaluate the overlap betweenthe NACK sets of the various users, we will recall the following definition which will be used repeatedlyin our analysis.
Next we characterize the number of common errors when two arbitrary packets are encoded.
23
Lemma 3 Let packets P1 and P2 be transmitted independently to K users. Say, K1 and K2 users
NACK packets P1 and P2 respectively. IfX is a random variable denoting the number of users reporting
a NACK on both the packets, then X ∼ fHGX (x;K1,K2,K) .
Proof: The number of ways in which K1 and K2 users (among K) can receive packets P1 and P2 inerror is given by
nK (K1,K2) =
(K
K1
)(K
K2
)The number of ways x common errors can occur in the two packets is given by
nxK (K1,K2) =
(K
x
)(K − xK1 − x
)(K −K1
K2 − x
)Now x can take all positive integer values up to min(K1,K2). Then the distribution of the number ofcommon errors is given by
fX (x) =nxK (K1,K2)
nK (K1,K2)=
(Kx
)(K−xK1−x
)(K−K1
K2−x
)(KK1
)×(KK2
) ∀ x = 0, 1, 2 . . . . . .min(K1,K2)
=K!K1! (K − x)!(K −K1)!
(K−K1
K2−x
)K!x! (K − x)!(K −K1)! (K1 − x)!
(KK2
)
=K!(K−K1
K2−x
)(K1 − x)!
(KK2
)
=
(K1
x
)(K−K1
K2−x
)(KK2
) ≡ fHGX (x;K1,K2,K)
The above equation is that of hypergeometric distribution with parameters K1,K2 and K. This impliesthat the expected number of common users among K1 and K2 NACKs is
E [K1,K2; ;K] =K1K2
K
Then from theorem 2, the average number of errors that can be corrected when the two packets arecoded, is given by
E [K1,K2; ;K] = K1 + K2 − 2(K1K2
K.)
24
3.3 Properties of Expected number of Packets recovered
Without loss of generality, we assume K1 ≥ K2. We can then observe the following propertieshold good
1. E[K2 ,K2;K
]= K
2 for all 1 ≤ K2 ≤ K1
2. E [K1,K2;K] ≤ K1 for all K1 > K2
3. E [K1,K2;K] is always increasing in K2 for fixed K1 < K2
Proofs of the above properties can be found in Appendix B at the end of this thesis.
The above properties show the behavior of expected number of errors when two packets with varioussizes are combined. Properties 1 and 2 are very important and indicate that when one of the packetsis greater or equal to K
2 , then it is not very useful to combine it with another packet since we don’tgain in anything in terms of expected number of packets recovered. Thus it is better to send that packetuncombined which has more than K
2 errors. Property 3 tells us that, when we are combining two packetseach of them having less than K
2 errors then it is better to combine those two packets which have themaximum number of errors.
The key observation from these properties is that whenever a packet is not received by more thanfifty percent of the users, it is optimal to retransmit the packet in its native form. Otherwise there arestrictly negative benefits from network-coding techniques. Next we extend these results to the generalcase when N packets are combined.
Theorem 2 LetK1,K2,K3 . . . . . . ,KN denote the number of NACK requests for packetsP1, P2, P3s . . . , PN
respectively. The expected number of common errors occurring among all the N packets is given by
E [K1,K2,K3,K4, . . . . . . .KN ;K] =
∏Ni=1Ki
KN−1 (3.7)
Proof:We know that the result in (3.7) is true for N = 2 as shown in the proof for lemma 1. We willuse an inductive argument to show this result holds for any N. First we assume the result is true forN = n− 1.
Let E(n− 1) denote the set of common errors who NACKed packets P1, P2, P3 . . . , Pn− 1. Say, xn−1is the random variable denoting the common errors among these n−1 packets. By induction assumptionwe have
E [xn−1] =
∏n−1i=1 Ki
Kn−2
Let Pn−1 be the Network-coded packet comprising of packets P1, P2, P3 . . . , Pn − 1. As before, Inisthe set of users who received packet Pn erroneously. From lemma 1 we know that the average number
25
of common errors in two packets Pn−1 and Pn with errors Xn−1and Kn is given by the conditionalexpectation
E [xn−1,Kn; ;K | Xn−1 = xn−1] =xn−1Kn
K.
Therefore, among the packets P1, P2, P3 . . . , Pn,the average number of common errors E[K1,K2,K3,K4, . . .KN ;K], is given by
Exn−1
(E [xn−1,Kn; ;K | Xn−1 = xn−1]
)=
Kn
KE [xn−1] =
∏nn=1Ki
Kn−1
From (3.1) and (3.7) the average numbers corrected by transmittingP1⊕P 2
⊕P3 . . . · · ·
⊕PN is given by
E [K1,K2,K3,K4, . . . . . .KN ;K]
=N∑i=1
Ki −2
K
∑i,j
KiKj +3
K2
∑i,j,l
KiKjKl + · · ·+ (−1)N−1 N
KN−1
N∏i=1
Ki
Where Ki is the number of users who received packet Ki is the number of users who received packetPi erroneously for all i.
3.4 Conclusion
In this chapter we looked at mathematical models to characterize error scenarios for the sharedfeedback channel. Some statistical tools were also developed for analyzing these models. Then theexpected number of errors when two packets are XORed was found out. Generalizing the result whenmultiple packets are coded, the expression for the expected number of errors in the system was derived.
26
Chapter 4
Design of Retransmission Scheme for shared feedback channel
In this chapter we exploit the statistical analytical models developed in the previous chapter to designefficient retransmission schemes for the Shared feedback channel scenario. In this chapter and the nextchapter we differentiate the function expected number of errors corrected E(n), with respect to n. Thevariable n in our problem statement is a discrete variable. For a given t and K, the function Et,1(n)can be shown to be unimodal i.e Et,1(n + 1) <= Et,1(n) and Et,1(n) > Et,1(n − 1) for some n. Wecan construct a continuous convex function Ect,1(n) which when sampled at integer points will yield ouroriginal discrete function Et,1(n). The maxima found for Ect,1(n) will map to the closest integer valuemaximizing Et,1(n). Our numerical results verify this conjecture as well
From the previous chapter we have the result that the average number of packets recovered by trans-mitting the encoded packet P1
⊕P2⊕P3⊕· · ·⊕PN is given by
E [K1,K2,K3, . . .KN ;K] =
N∑i
Ki −2
K
∑i,j
KiKj +3
K2
∑i,j,l
KiKjKl (4.1)
· · ·+ (−1)N−1 N
KN−1
N∏i=1
Ki
Where Ki is the number of users who received packet Pi erroneously for all i.
To get more insight into the above equation to be able to design simple and effective algorithms, weanalyze a few typical cases which simplify the analysis and provide mathematical tractability.
4.1 t-errors for each packet
We look at a situation when each packet has been NACKed by t(> 1) users each. For each of theNACKed packets P1, P2, . . . ..PN there are sets of t users corresponding to each of those packetsrespectively.
Therefore K1 = K2 = K3 = · · · = KN = t.
LetEt (N) denote the number of errors corrected or packets recovered when P1⊕P2⊕. . . ..
⊕PN is
27
transmitted, when each packet has been NACKed by t users each. Substituting K1 = K2 = K3 =
· · · = KN = t in equation (4.1) we get
Et (N) = Nt− 2
Kt2(N
2
)+
3
K2t3(N
3
)− . . . . + (−1N−1) N
KN−1NtN
(N
N
)=
N∑j=1
(−1K
)j−1
j
(N
j
)tj
= tN∑j=1
(−1K
)j−1
tj−1j
(N
j
)(4.2)
But (N
j
)=
(N)!
(j)! (N − j)!=
N (N − 1)!
j (j − 1)! (N − j)!=
N
j
(N − 1
j − 1
)(4.3)
Putting the value of (4.3) in (4.2)
Et (N) = Nt
N∑j=1
(−tK
)j−1(N − 1
j − 1
)
= NtN∑j=1
(−tK
)j−1(N − 1
j − 1
)1(N−1)−(j−1)
let m = j − 1
Then Et (N) = Nt∑N−1
m=0 (−tK )
m(N−1m
)(1)N−1−m which is a binomial expansion of the form
(1 + x)n. Therefore
Et (N) = N(1− t
K)N−1
. (4.4)
To find the number of packets to combine to maximize the number of packets recovered, we differentiateEt (N) with respect to N.
dEt (N)
dN= t
[(1− t
K
)N−1+N
(1− t
K
)N−1log
(1− t
K
) ]
Equating dEt(N)dN = 0.We have
t
[(1− t
K
)N−1+N
(1− t
K
)N−1log
(1− t
K
) ]= 0
Since t 6= 0 (1− t
K
)N−1+ N
(1− t
K
)N−1log
(1− t
K
)= 0
(1− t
K
)N−1 [1 +Nlog
(1− t
K
)]= 0
28
Since(1− t
K
)N−1 6= 0 for finite value of N and t < K.
N∗ =−1
log(1− t
K
) (4.5)
N∗ is the optimal number of packets to XOR which will maximize the expected number of packetscorrected where each packet has been NACKed by t users and the total number of users in the cell is K.Differentiation N∗ with respect to t
dN∗
dt= log
(1− t
K
)−2 1(1− t
K
) (−1K
)=
1
log(1− t
K
)2 1
(t−K)
Since (t− k) < 0dN∗
dt < 0, N∗ is a decreasing function of t, which means as t increases less number of packets shouldbe Network coded to maximize the error correction.
Lemma 4 The number of errors corrected Et (N) is maximum at N = N∗.
Proof:
Et (N) = Nt
(1− t
K
)N−1dEt (N)
dN= t
[(1− t
K
)N−1+N
(1− t
K
)N−1log
(1− t
K
) ]
d2Et (N)
dN2= t(
(1− t
K
)N−1log
(1− t
K
)+
log
(1− t
K
) {(1− t
K
)N−1+ N
(1− t
K
)N−1log
(1− t
K
) })
= tlog
(1− t
K
) (1− t
K
)N−1+
(1− t
K
)N−1+ N
(1− t
K
)N−1log
(1− t
K
)= tlog
(1− t
K
) (1− t
K
)N−1 [2 +Nlog
(1− t
K
) ](d2Et (N)
dN2
)N= −1
log(1− tK )
= t
(1− t
K
) −1
log(1− tK )
−1log
(1− t
K
)[2− 1]
= t log
(1− t
K
)(1− t
K)
−1−log(1− tK
)
log(1− tK
)
29
log(1− t
K
)< 0, since t
K < 1. And since (1− tK )
−1−log?(1− tK
)
log(1− tK
) is always positive. Therefore wehave
(d2Et (N)
dN2
)N= −1
log(1− tK )
< 0
which proves that Et (N) is maximum at N = N∗.
Figure 4.1 shows the variation in Et (N) with respect to N for different values of K and t. We see fromthe figure that the maximum value of the function Et (N) occurs at N = N∗.
The above result means that Xoring more than ’N∗’ packets in a retransmission is not the optimal wayto maximize the expected number of packets recovered.
4.2 one errors for each packet
The user errors are uncorrelated. As a result, we observe several packets which have been NACKedby only one user and fewer packets with multiple errors. In this section we carry out a brief analysis ofthe number of packets recoverd when multiple packets each with one user error are Network coded.
We look at a situation when each packet has been NACKed by t (= 1) users each. For each ofthe NACKed packets P1, P2, . . . PN there are sets of t users corresponding to each of those packetsrespectively. Therefore K1 = K2 = K3 = · · · = KN = 1.
Let Et (N) denote the number of errors corrected or packets recovered when P1⊕P2⊕. . . ..
⊕PN
are network coded and transmitted. Sub K1 = K2 = K3 = · · · = KN = 1 in the equation (4.1) weget
E1 (N) = N − 2
K
(N
2
)+
3
K2
(N
3
)− . . . . + (−1)N−1 N
KN−1N
(N
N
)=
N∑j=1
(−1)j−1(N
j
)tj
=
N∑j=1
j
(N
j
)−1K
j−1
We note that
j
(N
j
)= j
N (N − 1)!
j (j − 1)! (N − 1− (j − 1))!
= N
(N − 1
j − 1
)
30
0 5 10 15 20 25 30 35 40 45 500
2
4
6
8
10
12
14
16
Number of packets Network coded(N)
Exp
ecte
d n
um
ber
of
err
ors
co
rrecte
d
K=10,t=2K=20,t=3K=30,t=4K=40,t=5
Figure 4.1 Expected number of packets recovered Et(N) for different values of K and t
Therefore
E1 (N) = N
N∑j=1
(N − 1
j − 1
)(−1K
)j−1
1N−1−(j−1)
= NN−1∑m=0
(N − 1
m
)−1K
m
1N−1−m
= N(1− 1
K)N−1
(4.6)
The above result can also be obtained by substituting t = 1 in the equation (4.4). The optimumnumber of packets to be encoded is that value which maximizes E1 (N).
Figure 4.2 shows the plot between E1 (N) and N for different values of K, the total number of users inthe cell. From the figure we see that for a certain value of N∗ ∼= K the expected number of errors ismaximized.
31
0 5 10 15 20 25 30 35 40 45 500
5
10
15
Number of packets Network Coded(N)
Exp
ecte
d n
um
ber
of
err
ors
co
rrecte
d
K = 10K = 20K = 30K = 40
Figure 4.2 Expected number of packets recovered E1(N) for different values of K
4.3 Conclusion
In this chapter we discussed network coding to improve the retransmission algorithms. The networkcoding we discuss in thesis is very simple of form of Xoring. Thus coding of packets in this chapter isXoring the packets to form a new packet, which called the coded packet. The optimal number of packetsto Xor is given by the results discussed in this chapter.
32
Chapter 5
Algorithms for Retransmission for the Shared Feedback Channel
In this chapter we exploit the analytical models we developed in chapters 3 and chapter 4 to designefficient retransmission algorithms.
5.1 Serpentine Packet Selection scheme
The assumption of errors across packets being i.i.d was stated earlier. We observe several packetswhich have been NACKed by only one user or few users and fewer packets with multiple errors. Inorder to effectively address such error scenarios, we follow a serpentine order of packet selection. Togain more insight into this scheme, first consider n packets that have been incorrectly received by exactlyone user.
Let the expected number of packets recovered, when these n packets are XORed and retransmitted bedenoted by E1 (n) . From the previous chapter it is given by,
E1 (n) = n(1− 1
K)n−1
(5.1)
Let one of the n packets has t errors ( t > 1) and the remaining n − 1 packets have one erroreach. Let Et,1 (n) denote the expected number of packets recovered when these packets are encodedand retransmitted. Then Et,1 (n) is given by
33
Et,1 (n) = t+ (N − 1)− 2
K
[(N − 1
2
)+ t (N − 1)
]+
3
K2
[(N − 1
3
)+ t
(N − 1
2
)]. . . .(−1)N−2N − 1
KN−2
[(N − 1
N − 1
)+ t
]= N − 1− 2
K
(N − 1
2
)+
3
K2
(N
3
). . . (−1)N−1N − 1
KN−2
(N − 1
N − 1
)+ t
− 2
Kt (N − 1) +
3
K2(t
(N − 1
2
)) . . . .. (−1)N−1N − 1
KN−2 (t)
=N−1∑j=1
(N − 1
j
)j(−1K
)j−1
+ tN∑j=1
(N − 1
j − 1
)j(−1K
)j−1
(5.2)
Theorem 3 The Expected number of additional packets recovered when n − 1 one-error packets and
one t-error packet are XORed as compared to when n one-error packets are XORed is given by
δt,1 (n) = Et,1 (n)− E1 (n) = (t− 1)
(1− 1
K
)n−2 (1− n
K
)(5.3)
Proof : Refer to Appendix c
From (5.3) it is clear that the number of additional packets that can be recovered is increasing with t.In other words, it is beneficial to combine multiple packets with single errors with the packet having thelargest NACK requests.
Lemma 5 For a given t, the error correction is maximized when n∗ (t) packets can be XORed and
transmitted where n∗ (t) =(t−1)(1−K log(1− 1
K ))− (K−1)(K−t)log(1− 1
K ).
Proof:
Differentiation of Et,1 (n) with respect to n to maximize the number of errors corrected is
dEt,1 (n)
dn= 0
d[δt,1 (n) + E1(n)]
dn= 0
34
d[δt,1 (n) + E1(n)]
dn=d[(t− 1)
(1− 1
K
)n−2 (1− n
K
)+ n
(1− 1
K
)n−1]
dn= 0
= (t− 1)
[(1− 1
K
)n−2log
(1− 1
K
) (1− n
K
)+
(1− 1
K
)n−2(−1K
)]
+
(1− 1
K
)n−1+ n
(1− 1
K
)n−1log
(1− 1
K
)= 0
= (t− 1)
[log
(1− 1
K
) (1− n
K
)− 1
K
]+
(1− 1
K
)+ n
(1− 1
K
)log
(1− 1
K
)= 0
= (t− 1)
[log
(1− 1
K
)− n
Klog
(1− 1
K
)− 1
K
]+
(1− 1
K
)+ n
(1− 1
K
)log
(1− 1
K
)= 0
(t− 1)[Klog
(1− 1
K
)− 1]+ (K − 1)
K= nlog
(1− 1
K
) [t− 1
K−(1− 1
K
)]n∗ (t) =
(t− 1)(1−K log
(1− 1
K
))− (K − 1)
(K − t) log(1− 1
K
) (5.4)
For small values of x, using Taylor series approximation we have
log (1− x) ∼= −x
In fact, for large number of users, i.e higher K,
log
(1− 1
K
)∼= −
1
K
Then using the above approximation the optimal number of packets can be approximated is givenby n∗ (t) ≈ K(K+1−2t)
K−t .When t is one, this reduces to n∗ (1) = K.
In other words, when multiple packets with single errors are to be network-coded, then no more thanK packets are to be coded.
5.2 Serpentine Algorithm
The intuition behind the design of this algorithm comes from a few considerations.
• As we earlier observed, first is the fact that since the expected number of common errors when twopackets are coded depends on the size of the user errors corresponding to these packets. Thereforethe probability of correcting more errors is higher when a large error packet is combined with a
35
small error packet since there is smaller intersection between the two error sets corresponding tothe two packets.
• Second is the fact that the analysis of the problem becomes easier. Combining one large t−errorpacket with lots of single error packet gives us a closed form formula to analyze the expectednumber of packets recovered and gives insight into the problem.
• The third factor is the number of retransmission slots available. We want to send out as manylarge error packets as possible before the retransmission slots are exhausted. And to combinemultiple large user error packets would result in negative benefits in terms of packets recoveredsince the number of common errors in such a coded packet would be large.
• The fourth fact is presence of large number of single user error packets. To take into considerationthese multiple constraints, the Serpentine Algorithm is designed.
Say a burst of N ′ packets is transmitted to Kusers of which N packets were NACKed by atleast oneuser.Let P (N) = {P1, P2, . . . . . . , PN} be the sequence of N packets of descending order of their NACKrequests.Let Pi denote the set of packets that were network-coded and transmitted in slot i and Ki =
∑l∈Pi
Kl
be the corresponding number of errors for all those packets.Then the candidate set from which packets are to be drawn for the i+ 1th retransmission slot, is givenby
PCi+1 = P (N)−i⋃
j=1
Pj (5.5)
ThereforePCi+1 is simply the set of all those packets among the NACKed packets arranged in descendingorder of NACK requests which have not been retransmitted in the previous retransmission slots. For eachof retransmission slots we calculate the target number of errors which we expect the algorithm to correctin the ideal sense. This target is called as residual error in our algorithm. Because of the blind statisticsused in the algorithm we don’t achieve the desired residual error target but try to stay near to the residualerror value in each retransmission slot. The residual error value is calculated recursively during each ofthe retransmission slot. Similarly, the residual errors to be corrected in slot i+ 1 is defined as
eres (i+ 1) =K −
∑ij=1Kj
γ − i(5.6)
Where γ is the total number of retransmission slots.eres (i+ 1) can be interpreted as the maximum number of errors that is expected to be corrected duringthat slot.Now let PSPSi+1 denote the packet with the highest number of NACK requests in PCi+1, which is turn isgiven by KSPS
i+1 (> 1). In equation (5.4)the value of KSPSi+1 is substituted in place of t which gives the
optimal number of packets n∗(KSPSi+1 ) to coded.
36
From (5.4)and (5.6)the packet PSPSi+1 is to be network-coded with
nSPSi+1 = min(n∗(KSPSi+1
)− 1, eres (i+ 1) − KSPS
i+1 ) other packets selected in ascending order ofNACK request from PCi+1 and transmitted in slot i+1. This step says to network code the packet PSPSi+1
with nSPSi+1 other packets from the set PCi+1, selected in ascending order of NACK request. This order ofselection is performed since there are lots of packets in PCi+1 with single user error.
In the typical wireless broadcast networks, for packets which are uncorrelated across transmissionsand typical user error rate, there are lots of packets with only single user error. And since the number ofretransmission slots is very less as compared to the number of packets which have been NACKed, theabove scheme gives a good performance. This retransmission procedure is continued till all the γ slotsare exhausted (i.e. i > γ). Concisely summarizing this scheme, we note that during each slot, the packetwith the highest number of user requests is network-coded with an optimal number of packets with onlyone user request.
5.3 Incremental Error Correction Scheme
The average number of packets that can be recovered by sending Pn = P1⊕
P2 · · ·⊕
Pn isgiven by (4.1)
For notational convenience, we will use En to denote this result. If a packet Pn+1 with Kn+1 reportederrors, is to be network-coded with Pn, the average number of packets recovered is written as
En+1 =n∑i=1
Ki + Kn+1 −2
K
[KiKj + Kn+1
n∑i=1
Ki
]+
3
K2
∑i,j,l
KiKjKl+
Kn+1
∑i,j
KiKj +O1
kmm > 3. (5.7)
Therefore for large values ofK, the increment in error correction can be evaluated using the recursiverelation:
37
4n+1 = En+1 − En =
Kn+1
1− 2
K
n∑i=1
Ki +3
K2
∑i,j
KiKj + O(
1
Km
) (5.8)
= Kn+1[1−2
K(n−1∑i=1
Ki + Kn) +3
K2(n−1∑i,j
KiKj + Kn
n−1∑i=1
Ki
= Kn+1
1− Kn2
K+
3Kn
K2
n−1∑i=1
Ki −2
K
n−1∑i=1
Ki +3
K2
∑i,j
KiKj
=
Kn+1
Kn[Kn − 2
Kn
K
n−1∑i=1
Ki +3
K2Kn
∑i,j
KiKj − K2n
2
K+ 3
K2n
K2
n−1∑i=1
Ki]
=Kn+1
Kn(4n − K2
n
(2
K+
3
K2
n−1∑i=1
Ki
)) (5.9)
Where 40 , 0 and 41 , 1. This recursive evaluation significantly reduces implementationcomplexity.
5.4 Incremental Error Correction Algorithm
As in (5.5), PCi is the candidate set from which packets for the ith transmission slot is drawn.
Let KCi denote the corresponding set of number of user requests for those packets and KCi (n) denotethe first n users in the set KCi .Then the number of packets to be network-coded during t he retransmission slot i slot, nIECi is given by
nIECi ={maxn | 4n+1 ≥ 0 and Ki (n) ≤ eres (i)
}, (5.10)
whereKi (n) =∑n
l∈KCi and l=1Kl, is the sum total of user errors inKCi (n) and eres (i), the residual
errors to be corrected, is given by (5.6). Equation (5.10) suggests that for every retransmission slot, aslong as incremental error correction is positive and residual error target is not reached, packets from PCishould be network coded till any one of the condition is not satisfied.
In effect, during each retransmission slot, packets in descending order of NACK requests are succes-sively combined till there is no more positive error correction or optimal number of target errors havebeen reached.
P1 P2 P3 P4 P5 P6 . . . . . . .. P37
4 4 3 2 2 1 . . . . . . 1
38
Information available at the base station for the shared channel feedback scheme is shown in theabove table. The packets are arranged in descending order of number of errors reported. This table wasobtained using a simulation with 100 packets transmitted, 10 users in the system and with an averagepacket error rate of 5% for each of the user.
5.5 Simulation results of shared feedback channel
We numerically study the performance of the two retransmission algorithms presented in this chapter:Serpentine Packet selection and Incremental Error Correction. We then compare the error correctionbenefits with the following baseline schemes:
1. ARQ(Automatic Repeat Request).
2. Simple Network coding: All packets that were NACKed, are retransmitted via network-coding.Equal number of packets go out in each retransmission slot.
We see in Figure 5.1 , the benefits of all schemes based on network-coding is quite appreciable comparedto the naive retransmission scheme. Overall there is about 10 percent additional error correction withthe proposed retransmission schemes as compared to baseline network coding scheme. Also we observethe advantage of our algorithms over the traditional ARQ scheme. For small redundancy values, thereis smaller benefit in performance. Typically at 5% redundancy, there is a benefit of approximately 15%over the ARQ scheme, which is considerable.
We discussed a practical retransmission scheme in broadcast/multicast applications using network cod-ing, assuming feedback which is independent of the number of users. We develop several analyticalresults to design the optimal algorithm for constructing the retransmission packets. In the first scheme,the optimal number of packets that are combined in a serpentine manner, based on statistical obser-vations: the packet with the highest number of user requests is network-coded with several packetswith only one user request to construct each retransmission packet. The second scheme is based on thestraight-forward intuition that in a single retransmission packet, several packets can be combined till thepoint where there is no positive benefit in error correction. The performance of these schemes whencompared with the two baseline algorithms showed considerable benefit.
Although both Serpentine packet selection and the Incremental error correction scheme achieve similarerror correction performance, the Serpentine packet selection scheme performs a little better especiallyaround the 5% redundancy region. This performance may be explained due to the fact that for ouranalysis we have assumed a small number of users in the cell (in the above plot, the number of users are10). For this type network setup there are lot of packets with single user errors and very few packets withmultiple user errors. The Serpentine packet selection scheme optimizes the algorithm taking this factinto consideration and hence achieves a slight improvement as compared to Incremental error correctionscheme for the smaller number of users in the cell setup. In fact the intuition to combine one multiple
39
user error packet with lots of single user error packets for the Serpentine algorithm has come partiallyfrom the simulation of the network setup and partially from the statistical tools developed for errorcharacterization.
Though both the algorithms are designed for blind statistics available at the base station, the Serpentinepacket selection has been designed in particularly for the error statistics which occur typically whenthere is smaller number of users in the network. When the number of users in the cell increases con-siderably, for the uncorrelated and independent errors occurring the packets and users, we would expectlarger multiple user errors packets and few single user error packets. And in such a scenario, the In-cremental error correction scheme would perform much better since it is designed for more generalstatistics.
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
Redundancy(%)
Pe
rce
nta
ge
of
To
tal
err
ors
co
rre
cte
d
Serpentine packet selectionIncremental correction schemeSimple Network codingARQ
Figure 5.1 Performance of Algorithms for Shared feedback channel
5.6 Comparison across schemes
To get an idea of the performances of various retransmission algorithms for different feedback band-width we plot a graph in Figure 5.2. As expected Perfect feedback scheme achieves the best performance
40
by far. For a redundancy of 5% there is a difference of approximately 30% between perfect feedbackand other schemes. And the partial channel feedback and the shared channel feedback schemes showalmost equal performance. For redundancy values lesser than 4%, we see that serpentine and incre-mental algorithms perform equal to partial feedback scheme. At lesser redundancy values they slightlyoutperform the partial channel feedback scheme.For higher redundancy values, at redundancy of 6% there is a difference of 10% in the number of packetsrecovered between the partial feedback scheme and shared channel schemes. For redundancy valueof 7% this difference in percentage packets recovered increases to 15% between the partial feedbackscheme and the shared channel scheme. Therefore we notice that as the number of retransmission slotsincreases partial channel feedback scheme tends to exploit the advantage much better than the sharedchannel feedback schemes. This suggests that for higher values of redundancy better optimizationtechniques for design of these algorithms could be designed which take into factor the larger numberretransmission slots.
41
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
Redundancy(%)
Percen
tag
e o
f T
otal E
rro
rs c
orrected
Perfect FeedbackPartial Feedback Serpentine Packet selectionIncremental Error correct
Figure 5.2 Performance analysis for different Algorithms
42
Chapter 6
Conclusions
Using network coding based retransmission algorithms gives considerable benefit over the simpleretransmission schemes which do not employ network coding. These coding based feedback schemescan be used in the 4G Network being designed to efficiently address the problem of recovery of packetloss. Since these schemes are also not computationally intensive, they can be easily implemented at thebase station or the transmitter in a broadcast setting.
The benefits achieved with the schemes designed are the following.
1. For different bandwidth different retransmissions are possible. There is efficient trade-off betweenfeedback bandwidth and retransmission overhead
2. Savings in the retransmission slots or retransmission packets leads directly to considerable savingin the amount of bandwidth, transmitted power and the data rate of the system being considered.
3. All the schemes proposed in this thesis employ memoryless decoding which means low Mobile-side complexity.
4. The schemes can be implemented for the base station HARQ(Hybrid Automatic Repeat Request)transmissions.
5. These schemes are effective in the presence of limited feedback channel.
6. Typically in the real world scenario, 5% redundancy is available. For such type of networks wecould have a shared feedback channel which will perform as well as partial feedback channel ifnot better.
7. Good benefits are achieved even when file sizes are small or when streaming applications demandshort playback buffer.
*
43
Appendix A
Lemma 1:In the problem described, the joint distribution of the number of NACKed packets is given by
P (X1 = i1, X2 = i2, . . . , Xr = ir) =
∏rj=1
(NRij
)(N−rN
Rx−
∑rj=1 ij
)(Nx
) for all 1 ≤ r ≤ R. (1)
Proof : Follows from a recursive argument. First, consider the distribution of error packets in thefirst retransmission slot . This would be similar to selecting i error packets without replacement from N
balls. This is given by
P (X = i) =
(N−xNR−i
) (xi
)(NNR
) ,
=
(NRi
)(N−N
Rx−i
)(Nx
) ,
Which satisfies the expression to be shown.
Next consider the case where the joint distribution of the error packets in the first two retransmissionslots are to be computed. Then
P (X1 = i, X2 = j) = P (X2 = j |X1 = i) P (X1 = i)
=
(N− N
R− x + i
NR− j
)(x−ij
)(N − N
RNR
)(N−xNR−x
) (xi
)(NNR
)
=
(NRi
)(NRj
)(N−2N
Rx−i−j
)(Nx
)Which is again consistent with the expression to be shown. Now we assume that (1) is true for the
case for the joint distribution of the first R− 1 retransmission slots, i.e
44
P (X1 = i1, X2 = i2, . . . , XR−1 = iR−1) =
∏R−1j=1
(NRij
)( NR
x−∑R−1
j=1 ij
)(Nx
)The joint distribution of the first R retransmission slots is then given by
P (X1 = i1, . . . , XR = iR) = P (XR = iR |XR−1 = iR−1) P (X1 = i1, . . . , XR−1 = iR−1)
=
(N − (R−1)N
R− iR
NR− iR
)(iRiR
)(NRiR
)∏R−1j=1
(NRij
)(Nx
)=
∏Rj=1
(NRij
)(Nx
)where iR = x−
∑R−1j=1 ij .
*
45
Appendix B
1.Proof of Theorem 2 For notational ease, we use the symbol K for the set notation as well as thecardinality of the set depending on the context in which it is used. The used of the symbols in theequation will make it clear as in what context it is used.
P1, P2 . . . PN are the sequence numbers of the packets which are in error.
let K1K2 denote the intersection of two sets K1 and K2.
Similarly K2K3 would denote the intersection of sets K2 and K3 and so on.
Similarly let K1K2K3 denote the intersection of sets K1, K2 and K3 and so on.
Let K1K2K3 . . . . . . . . . . . . ..KN ′ denote the intersection of all the
sets K1,K2,K3, . . . . . . .,KN ′ .
Let (K1K2)′ denote the set which is common to both the sets K1 and K2 but
not common to set K1, K2 and K3.
Therefore we can write K1K2 = (K1K2)′ ∪ K1K2K3.
The same notations apply to higher number of combinations of sets.First we show why this theorem istrue by taking smaller number of packets and then extend the result to general N.Combining 2 packets
K1 and K2 are the sets corresponding to packets P1 and P2 respectively.
Number of errors correctd = K1 + K2 − 2(K1K2).
where K1K2 is the intersection of sets K1 and K2.
Combining 3 packets
K1,K2 andK3 are the sets corrsponding to packets P1, P2 andP3 respectively.
Number of errors corrected = K1 +K2 +K3−
2(K1K2)′ + (K2K3)
′ + (K1K3)′ − 3(K1K2K3)
= K1 +K2 +K3 − 2K1K2 −K1K2K3 +K1K3 −K1K2K3 +K2K3 −K1K2K3−
K1K2K3
= K1 +K2 +K3 − 2K1K2 +K2K3 +K1K3 + 3(K1K2K3).
46
Combining 4 packets
K1,K2,K3 andK4 are the sets corresponding to packets P1, P2, P3 andP4respectively
Number of errors corrected
= K1 +K2 +K3 +K4 − 2((K1K2)′ + (K2K3)
′ + (K1K3)′ + (K1K4)
′ + (K2K4)′+
(K3K4)′)− 3(K1K2K3)
′ + (K2K3K4)′ + (K1K3K4)
′ + (K1K2K4)′ − 4(K1K2K3K4)
= K1 +K2 +K3 +K4 − 2(K1K2 − (K1K2K3)′ − (K1K2K4)
′ −K1K2K3K4+
K2K3 − (K1K2K3)′ − (K2K3K4)
′ −K1K2K3K4 +K1K3 − (K1K2K3)′−
(K1K3K4)′ −K1K2K3K4 +K1K4 − (K1K2K4)
′ − (K1K3K4)′−
K1K2K3K4 +K2K(4)− (K2K3K4)′ − (K1K2K4)
′ −K1K2K3K4+
K3K4 − (K1K3K4)′ − (K2K3K4)
′ −K1K2K3K4)− 3((K1K2K3)−K1K2K3K4+
(K2K3K4)−K1K2K3K4 + (K1K3K4)−K1K2K3K4 + (K1K2K4)−K1K2K3K4)−
4K1K2K3K4
= K1 +K2 +K3 +K4 − 2(K1K2 +K2K3 +K1K3 +K1K(4)
+K2K(4) +K3K4) + 2(K1K2K3)′ + (K2K3K4)
′ + (K1K3K4)′+
(K1K2K4)′ + (K1K2K3)
′ + (K2K3K4)′ + (K1K3K4)
′ + (K1K2K4)
+ (K1K2K3)′ + (K2K3K4)
′ + (K1K3K4)′ + (K1K2K4)
′ − 3((K1K2K3)
+ (K2K3K4) + (K1K3K4) + (K1K2K4)) + 20K1K2K3K4
47
= K1 +K2 +K3 +K4 − 2(K1K2 +K2K3 +K1K3 +K1K4
+K2K(4) +K3K4) + 2((K1K2K3)−K1K2K3K4 +K2K3K4 −K1K2K3K4
+ (K1K3K4)−K1K2K3K4 + (K1K2K4)−K1K2K3K4 + (K1K2K3)−K1K2K3K4+
K2K3K4 −K1K2K3K4 + (K1K3K4)−K1K2K3K4 + (K1K2K4)−K1K2K3K4+
(K1K2K3)−K1K2K3K4 +K2K3K4 −K1K2K3K4 + (K1K3K4)−K1K2K3K4+
(K1K2K4)−K1K2K3K4)− 3(K1K2K3) + (K2K3K4) + (K1K3K4) + (K1K2K4)+
20K1K2K3K4
= K1 +K2 +K3 +K4 − 2K1K2 +K2K3 +K1K3 +K1K(4) +K2K(4) +K3K4+
3(K1K2K3) + (K2K3K4) + (K1K3K4) + (K1K2K4)− 4(K1K2K3K4)
Similarly extending the above argument for N packets proves the theorem
2.Proofs of Properties of the Expected number of errors correctedWithout loss of generality, we assume K1 ≥ K2. We can then observe the following properties holdgood.Property1:E
[K2 ,K2;K
]= K
2 for all 1 ≤ K2 ≤ K1
Proof:
E
(K
2, K2;K
)=K
2+ K2 − 2
K2 K2
K
=K
2+ K2 − K2 =
K
2.
Therefore E(K2 , K2;K
)= K
2 for all values of K2
property 2: E [K1,K2;K] ≤ K1 for all K1 > K2
proof:
E (K1, K2; ;K) = K1 + K2 − 2K1K2
K−K1
= K2 − 2K1K2
K
= K2(1− 2k1K
)
Since K1 ≥K
2,
(1− 2
K1
K
)≤ 0 .
Therefore E (K1, K2; ;K)− K1 ≤ 0
which means E (K1, K2; ;K) ≤ K1 for all K1 ≥K
2
48
property 3: E [K1,K2; ;K] is always increasing in K2 for fixed K1 < K2
proof
E (K1, K2;K) = K1 + K2 − 2K1K2
K
Differentiating w.r.t to K2;
d
dK2E (K1, K2; ;K) = 1− 2fracK1K
Since K1 <K
2; ; 1 − 2
K1
K> 0
Therefore E (K1, K2; ;K) is increasing with K2 for any fixed K1 <K
2
*
49
Appendix C
Proof of Theorem 4
δt,1 (n) = Et,1 (n)− E1 (n)
= t
n∑j=1
j
(n− 1
j − 1
)(−1K
)j−1
+
n−1∑j=1
(n− 1
j
)j(−1K
)j−1−
n∑j=1
j
(n
j
)(−1K
)j−1
=n−1∑j=1
j(−1K
)j−1 [(n− 1
j
)−(n
j
)]− n(
−1K
)n−1
+ tn∑j=1
j
(n− 1
j − 1
)(−1K
)j−1
(2)
But
(n− 1
j
)−(n
j
)=
(n− 1)!
j! (n− 1− j)!− n!
j! (n− j)!
=(n− 1)! (n− j)− n!
j! (n− j)!
=(n− 1)!(n− j − n)
j! (n− j)!
= − (n− 1)!
(j − 1) (n− j)!
= −(n− 1
j − 1
)(3)
Substituting equation (3) in the equation (2) above
50
δt,1 (n) = −n−1∑j=1
j
(−1K
)j−1 (n− 1
j − 1
)−− n
(−1K
)n−1+ t
n∑j=1
j
(n− 1
j − 1
)(−1K
)j−1
= −
n−1∑j=1
j
(−1K
)j−1− n
(−1K
)n−1 + tn∑j=1
j
(n− 1
j − 1
)(−1K
)j−1
= −
n∑j=1
j
(−1K
)j−1 (n− 1
j − 1
)+ tn∑j=1
j
(n− 1
j − 1
)(−1K
)j−1
= (t− 1) [n∑j=1
j
(−1K
)j−1 (n− 1
j − 1
)]
= (t− 1) [n∑j=1
(j − 1)
(−1K
)j−1 (n− 1
j − 1
)+
n∑j=1
(−1K
)j−1 (n− 1
j − 1
)] (4)
We know(j − 1)
(n− 1
j − 1
)= (n− 1)
(n− 2
j − 2
)(5)
Substituting the above equation (5) in equation number(4) we have
δt,1 (n) = (t− 1) [(n− 1)
n∑j=2
(n− 2
j − 2
)(−1K
)j−2 (−1K
)+
n∑j=1
(−1K
)j−1 (n− 1
j − 1
)]
= (t− 1) [(n− 1)(1− 1
K)n−2(−1
K
)+
(1− 1
K
)n−1]
= (t− 1)
(1− 1
K
)n−2 [(n− 1)
(−1K
)+ 1− 1
K
]= (t− 1)
(1− 1
K
)n−2 (1− n
K
).
51
Related Publications
1. Mohammad Shaheer Zaman and Naveen Arulselvan, Network coding-based retransmission schemesfor 4G multicast/broadcast networks with shared feedback channel, Proc. UKIWCWS-2009, NewDelhi, India.
2. Mohammad Shaheer Zaman and G. Rama Murthy, Clusetered and leveled Disjoint MultipathRouting Algorithm for Wireless Sensor Networks, Proc. UKIWCWS-2009, New Delhi, India.
3. Mohammad Shaheer Zaman and G. Rama Murthy, A new Degree distribuiton for LT Codes forBroadcasting in ad-hoc Networks using network coidng , Proc. UKIWCWS-2009, New Delhi,India.
4. Mohammad Shaheer Zaman and G. Rama Murthy, Disjoint Multipath Routing Algorithm forWireless Sensor Networks , Proc. RACE, 2008. Hyderabad, India.
52
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