T-M" f 77'3 IK

NRL-Report 8140

SThroughput Performance of Data-Communicationj. Systems Using Automatic-Repeat-Request (ARQ)

Error-Control Schemes

JOEL M. MORRIS

Systemis Integration and Instrumentation BranchCommunications Sciences Division

September 8, 1977

CL_

'/DEC 12 1977NAVAL RtESEARCII LABORATORY

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10SUPPLEMENTARY NOTES

19. K EY WORDS fCantinus on feots,. oide, Itiecoes~n-an~ud identify by block nuipbet)

Automatic repeat request Data communicationsARQ Error detectionData throughput Error correctionThroughput efficiency Error control

T0. ASS CT (Continue on Pevere. oid* Of necessary end identify by block nuatbotJ

Nhe throughput performance of several ARQ schemes was evaluated. The purpose is to pro-vide data-communilcation design engineers with Information to help them choose error-controltechniques or to assess the performance of their proposed ARQ or hybrid designis. The scheeosconsidered Included the stop-and-wait, the go-back-N, and the selective-repeat schemes and somevariations of these, In addition, the performance of a hybrid 'scheme that uses the bedt propertiesof both the ARQ scheme and the forward-mror-correctlon (FEC) wchet-e was assessed. Most of,

OF I NOV(Continued)

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20. Abstract (Continued)

ýt e equations derived for computing throughput efficiency~ or optimal blocklength were obtainedas direct modifications or extensions or equations found In he published ilterature. The equationswere derived primarily for random-error channel models, although some burst-error conditions wereconsidered. Among Items yet to be studied are the optimal transmission rate and blocklength tomaximize the throughput (in bits per second).

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31CMNT CI"1FCTO MII rtW"D&W;

CONTENTS

INTRODUCTO A ....SCHEMES ...................................... 21

Basic Stop-and-Wait ARQ Scheme ................................ 2Variation 1 of the Stop-and-Wait ARQ Scheme ....................... 3Variation 2 of the Stop-and-Wait ARQ Scheme ....................... 3Continuous-ARQ Schemes ...................................... 3Go-Back-N Continuous-ARQ Scheme............................... 4

Variation of the Go-Back-N Continuous-ARQ Scheme.................. 4Selective-Repeat Continuous-ARQ Scheme........................... 5Hybrid ARQ Schemes ......................................... 5

THROUGHPUT PERFORMANCE OF THE VARIOUS ARQ SCHEMES ........ 5

Preliminary Notation, Definitions, and Assumptions.................... 6Throughput of the Basic Stop-and-Wait ARQ Scheme .................. 7Throughput of Variation 1 of the Stop-and-Wait ARQ Scheme........... 8Throughput of Variation 2 of the Stop-and-Wait ARQ Scheme........... 12Throughput of the Go-Back-N Continuous-ARQ Scheme ................ 12Throughput of the Variation of the Go-Back-N Continuous.ARQ

Scheme ................................................. 13Throughput of the Selective-Repeat Continuous-ARQ Scheme ............ 15Throughput Error of the Hybrid ARQ Scheme With Separate

Error Detection and Correction Encoding......................... 15

OPTIMAL BLOCKLENCTHS FOR ARQ SYSTEMS....................... 19

Optimal Blocklength for the Basic Stop-and-Wait ARQ Scheme...........20Optimal Blocklength for Variation 1 of the Stop-and-Wait

ARQ Scheme.......................... ................... 21Optimal Blocklength for Variation 2 of the Stop-mid-Wait

ARQ Scheme...................................... 26Optimal Blocklength for the Go-Back-N Continuous-ARlQ ch10eme....26

* Optimal Blocklength for the Variation of theGo-Back-N Continuous-ARQ Scheme ............................ 27

Optimal Blocklength for the Selective-Repeat Continuous-ARQ Schome ............................................. 29

SUMMARY AND CONCLUSIONS.................................... 29

REFERENCES ................................................. 33

THROUGHPUT PERFORMANCE OF DATA-COMMUNICATIONSYSTEMS USING AUTOMATIC-REPEAT-REQUEST (ARQ)

ERROR-CONTROL SCHEMES

INTRODUCTION

An automatic-repeat-request (ARQ) scheme [1-61 is a highly reliable error-controltechnique for the transmission of messages or blocks of data in data-transmission systemssuch as message or packet-switched networks. This high reliability is a result of the con-tinuous retransmission (repeats requested) of those messages or blocks found in error bythe receiver. Consequently, as the channel error rate increases, ARQ schemes yield lowerand lower throughput (received message or block acceptance rate) while maintaining highreliability.

In contrast, forward-error-correction (FEC) schemes [7], maintain a constant mes-sage or block acceptance rate at the receiver but suffer decreased reliability by acceptinglarger and larger numbers of uncorrected messages or blocks as the error rate increases.The FEC schemes however do not need feedback communications for the implementationof their error-correction techniques, and this absence of feedback channels is an operationalrequirement in many applications.

and Both ARQ and FEC schemes generally use coding to detect, correct, or both detectand correct some finite number of errors in the received message or block (a humanoperator may perform the operation mentally). The price paid for coding is reducedthroughput, since the ratio of information bits to transmitted bits is reduced. In addition,in any practical system, there is a nonzero probability - although usually acceptable withproper design - of some accepted messages or blocks being in error.

Generally the choice of ARQ over FEC schemes depends primarily on the availabilityof a feedback link and the desirability of constant high reliability rather than constantthroughput performance. Yet some combination of ARQ and FEC [6-8] schemes havebeen shown effective in moderate- mad high-error-rate environments. In this situation theerror-correcting properties of the code may considerably reduce the error rate experiencedby the error-detecting properties of the code if acting alone. The result is that the ARQcomponent of this scheme operates in an effective low-to-moderate error environment.Thus the scheme yields improved throughput performance overall.

In this report we evaluate the throughpý.tt performance of several ARQ schemes. Tilepurpose is to provide data-communication de~ign engineers with information to help themchoose error-control techniques or to assess the performance of their proposed ARQ orhybrid designs. The schemes considered include the stop-and-wait, tie go-back-N, and theselective-repeat schemes and some variations of these. In addition the performance of

.Mamustipt subtuilted May 31, 1977.

'1

4 JOEL M. MORRIS

ARQ-FEC hybrid schemes, which use the best properties of both, is assessed. Most of theequations presented for computing throughput efficiency 77 or optimal blocklength areobtained as direct modifications or extensions of equations found in the published literature.The equations are derived primarily for random-error channel models, although some burst-error conditions are considered.

In the next section of this report we discuss in general terms the different ARQ schemesconsidered. Hybrid schemes are also included. This section provides a heuristic presentationof these schemes for the interested reader who is not too knowledgeable about ARQ tech-niques. The respective throughput efficiency equations are developed and evaluated in thethird section. Throughput curves for some of the ARQ schemes are presented. The fourthsection contains the development of optimal-blocklength equations and algorithms for thoseARQ schemes that segment messages and data into blocks. Explicit results on optimalblocklengths are presented in this section for some of the considered schemes. The sum-mary, concluding remarks, and some recommendations for additional study are presentedin the final section.

VARIOUS ARQ SCHEMES

The various automatic-repeat-request (ARQ) schemes commonly considered are varia-tions of two basic ARQ error-control techniques: the stop-and-wait ARQ scheme and thecontinuous-ARQ scheme. However hybrid ARQ schemes which incorporate error-correctingtechniques are becoming popular in the current literature.

Basic Stop-and-Wait ARQ Scheme

The stop-and-wait ARQ scheme [1-6] is probably the most popular error-controlscheme. This scheme is in widespread use because it is relatively easy to implement, itrequires only one block of temporary storage, and, since transmission of information isin only one direction at a time, half-duplex data links are appropriate.

In the stop-and-wait ARQ scheme the transmitter delays (stops) transmission of asucceeding block and temporarily stores the last block (waits) until receiving some formof acknowledgment from the receiver indicating the acceptability of the last transmittedblock. If the received block is error-free and therefore acceptable, the receiver indicatesthis by sending an ACK signal to the transmitter. The transmitter then proceeds to trans-mit the next block. If the received block contains errors and therefore is unacceptable,the receiver indicates this by sending a NAK signal to the transmitter. The transmitterresponds to this epeat request by retransmitting the last block.

The throughput efficiency of the stop-and-wait ARQ scheme is highly dependenton the channel quality and the waiting time between transmission of a block and receiptof the ACK or NAK. Clearly the quality of the channel determines the number of re.transmissions required. The waiting time consists of the round-trip propagation time,plus the link-and-modem turnaround times (if the channel operates in the half-duplex

2

NRL REPORT 8140

mode), plus the time duration of the forward (data) and return (ACK and NAK) signals.Waiting time can be so excessive on channels such as satellite channels that more time isspent waiting than in delivering good data to the receiver. Consequently throughput ef-ficiency may be poor when using the stop-and-wait ARQ scheme even on good-qualitychannels, although this efficiency can be optimized by proper blocklength choice, as shown

*' in later sections.

* Variation 1 of the Stop-and-Wait ARQ Scheme

In the basic scheme, whenever the receiver requests that a block be repeated, theblock is repeated one time. In a variation [6], referred to herein as variation 1, wheneverthe transmitter receives a repeat-request, m duplicates of the repeat-requested block aresuccessively transmitted before the transmitter stops and waits for the receiver acknowl-edgment. With this technique a repeat-request will be indicated again only when none of them successively transmitted duplicates are received correctly.

This technique spreads one normal waiting-time delay for stop-and-wait systems overm duplicates, with m being the expected number of duplicates that would ordinarily berequired for correct receipt in a given poor-quality channel. Thus improvement in through-put performance depends on the expected number of retransmissions per block and thesystem round-trip delay.

Variation 2 of the Stop-and-Wait ARQ Scheme

In another variation on the basic stop-and-walt ARQ scheme, referred to herein asvariation 2, a sequence of M blocks (a superblock) is transmitted before the transmitterstops and waits for receiver acknowledgment. When the receiver desires a repeat of someof the M blocks, it identifies their positions within the received sequence. The transmitterresponds by placing these repeat-requested blocks at the beginning of the next sequence(superblock). Only negative (NAK) acknowledgments by the receiver are needed oncecommunication is established; thus the feedback channel is used more efficiently.

This technique spreads the normal waiting-time delay (more accurately, the propaga-tion delay plus the average acknowledgment-message time) for the basic stop-and-wait sys-tem over M blocks. In other words, the time intervals are longer between the stop-and.wait points, but only those individual blocks within the received superblock which arefound in error are retransmitted. This variation of the technique should yield improvedthroughput performance over the basic stop-and-wait scheme for channels of all qualities.However the cost to implement this scheme is greater, due to increased storage and block-reordering requirements.

Continuous ARQ Schemes

The continuous-ARQ schemes reduce the waiting time between blocks and conse-quently are more efficient. In exchange for the increased efficiency these scheemes require

S"3

JOEL M. MORRIS

a simultaneous two-way data link, a more complex implementation, and larger temporarystorage for several transmitted but unacknowledged blocks of data. The number of tem-porarily stored unacknowledged blocks N is the number of blocks in transit or receivedbut unacknowledged at the transmitter. The two basic continuous-ARQ schemes are theselective-repeat scheme and the go-back-N scheme. Both schemes require the transmitterto continue transmitting subsequent blocks while awaiting receiver acknowledgment (ACK/NAK) for an unconfirmed block. Thus both schemes productively use their acknowledg-ment waiting time. These two schemes and variations of them differ in what is retrans-mitted.

Go-Back-N Contmuous-ARQ Scheme

The go-back-N ARQ scheme [1-6,81 is the easiest continuous-ARQ scheme to imple-ment but also, the least efficient. The transmitter sends blocks of data in sequence withoutwaiting for an acknowledgment (ACK or NAK). This "continuous" transmission is stoppedonly when a NAK is received at the transmitter. The NAK indicates that a block wasreceived in error and a retransmission is required. In response the transmitter stops sendingnew blocks and backs up to retransmit the NAK'ed block and all the following transmittedbut unacknowledged blocks.

The go-back-N ARQ scheme is efficient for good-quality channels. It requires no* . block reordering or resequencing logic at the receiver, but it retransmits N - 1 extra blocks

for each block found in error. Generally either these extra blocks or, usually, the originals areS * discarded without using them to improve error-control efficiency. However this may be a

good error-control technique for burst error environments. This scheme suffers from the' Isame type of delay or waiting time as the stop-and-wait ARQ when a block is received in

error. Thus the average waiting time depends directly on the block error rate (which isa function of blocklength), and the throughput efficiency can be optimized for specific

4: ,channels by proper choice of blocklength.

Variation of the Go-Back-N Continuous-ARQ Scheme

A variation is a modified go-back-N ARQ scheme [61 that differs from the basic go-back-N ARQ scheme in the content of the N retransmitted blocks. Instead of retransmit-ting the last N transmitted blocks, in this modified scheme the repeat-requested block is

* : .continuously retransmitted at least N times until received correctly (until accepted). Oncethe acceptance acknowledgment is received at the transmitter, the original block sequencetransmission picks up froan the block following the last accepted block. Thus N - 1 du-plicates of the repeat-requested block are received but ignored before the receiver seescontinuation of sequence.

The improvement in throughput provided by this variation of the go-back-N ARQschenle is gained by the reduction of time required for the total number of retransmissionsfor a repeat-requested block. In this instwice it is not necessary to waste time by trims.mitthig the N - 1 subsequent blocks, since they get ignored with each repeat-request

4

NRL REPORT 8140

retransmission. The amount of throughput performance improvement depends on theexpected number of retransmissions and is more significant in poor-quality, burst errorchannels, in which a large number of block retinnsmissions are likely to be required.

Selective-Repeat Continuous-ARQ Scheme

The selective-repeat scheme [1-4] is a more complex continuous-ARQ scheme withrespect to implementation but also is the most efficient in throughput efficiency. Thetransmitter sends blocks of data in sequence without waiting for an acknowledgment(ACK or NAK). This "continuous" transmission is interrupted only when a NAK isreceived at the transmitter indicating that a block was received in error and a retrans-mission is required. In response the transmitter stops the normal block-sequence flowlong enough to transmit the requested block. The selective-repeat ARQ scheme is ef-

* ificient for good- and moderate-quality channels. The additional implementation com-plexity is due to the block reordering or resequencing logic required, since some ac-

* icepted blocks will be out of their original order.

Hybrid ARQ Schemes

In the hybrid ARQ schemes [7-9] forward-error-correction (FELC) techniques are usedin conjunction with a basic ARQ technique to improve the throughput performance onpoor-quality channels. The purpose of the FEC component is to improve the channelquality (reduce error rate) experienced by the basic ARQ error-detecting-code component,thus reducing the high expected number of block retransmissions in high-error-rate channels.

There are two methods of incorporating the FEC. The first is to use an error-correctingcode to encode blocks which have been encoded with an error-detection code [8,9]. Thusthis method uses a two-step coding procedure. The second method uses a single code forboth error detection and error correction [7]. This dupl capability is exploited at thereceiver by the decoders. The error-correcting component can be selected for independentbit errors, burst errors, or a combination of both conditions. This capability may in someinstances be used or made operative only when the channel conditions warrant it.

THROUGHPUT PERFORMANCE OF THE VARIOUSARQ SCHEMES

In this section throughput efficiency equations for the various ARQ schemes discussedin the preceding section are developed and evaluated. Throughput performance curves forsome of these ARQ schemes are presented. Consequently a data.transmission-system designengineer is provided with information to help in choosing error-control techniques or inassessing the performance of proposed ARQ designs.

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JOEL M. MORRIS

Preliminary Notation, Definitions, and Assumptions

The throughput equations presented in this section for the various ARQ schemes areexpressed using common notation defined as follows:

q - Throughput efficiency expressed as a percent of the symbol trans-mission rate;

R - Symbol transmission (signaling) rate of a system in symbols per second;

D - System delay: the roundtrip propagation delays plus the transmitterand receiver response times for receipt of blocks or acknowledgementof messages;

B - Number of user information symbols per message block;

BS - Number of system information symbols per transmitted message block:B, plus the number of protocol or reader information symbols permessage block;

B - Number of total symbols per transmitted message block after encoding:Bs plus the number of error-control redundancy iymbols per messageblock;

E = EfE}- Expected number of transmissions of a given block with blocklength Bbefore correct reception;

"Pol{B - Probability of a given block with blocklength B being in error

•x - Transmission efficiency of a system: B divided by the average numberof symbols used for correct reception;

-- Background efficiency of a system: the throughput efficiency that aparticular system would have if the channel were error-free.

The derivations of throughput equations in this section are based on some commonsimplifying assumptions:

- The acknowledgment channel is error-free or is acceptable due to use of forward-error-correction (FEC) coding. Equivalently the probability of receiving an incorrectacknowledgment message is negligible.

* The error-detection-coding component of the various ARQ schemes essentiallydetects all errors. Equivalently the probability of not detecting errors in a received blockis negligible.

* System delay time is constant for a given connection, whether an ACK or NAKacknowledgment is required.

6

NRL REPORT 8140

9 Steady-state operation of the transmission system is the only mode considered.Block synchronization failure or acquisition are not considered, or they are no moredamaging to throughput than normal block errors.

* In most cases considered, except where noted, the block errors are independentfrom block to block, because the channel produces independent error bursts and becausethe error bursts do not appreciably overlap adjacent blocks.

* In most cases considered, except where noted, each block consists of real infor-mation; that is, each message length is exactly an integral number of blocklengths.

With these assumptions, the expected value E = EBB} of the number of transmissionsn is

BEB) = n(PefB})' 1 (1 Pe{BIl) = (1 - P,{B})" i (Pa(B))-', (1)

where n is the number of transmissions required for acceptance of an arbitrary block, and1 - Pe IB' = a JB is the probability of no errors detected in the block: the block'sacceptability.

Throughput of the Basic Stop-and-Wait ARQ Scheme

In the stop-and-wait ARQ scheme, for each correct reception of B, informationsymbols, B + DR symbol periods must be expended E times on the average. Thus, thethroughput [1-6] efficiency to the demand-access user is

I.'

B1W (B + DR)E(

The expression DR + (B + DR)(E - 1) can be interpreted as the expected wasted time,expressed as symbol periods, due to system delays and required block retransmissions.

Curves of i1s w for the aforemer'koned assumptions are exhibited in Fig. 1 for thecase of B1 = 200 bits and B - 255 bits. Throughput efficiency essentially attains itsmaximum value given by 0Sw = B1I(B + DR) when PL < 10-2. Also, asw -- R-1. Observethe dependence of 08W on DR for a given block length (B1 and B), or, conversely, observethe dependence of Psw on blocklength (BI and B) for given DR. The choice of optimalor "good" blocklengths with respect to throughput efficiency will be explored in the nextmain section.

: 7

JOEL M. MORRIS

0.8 S~DR=50.7

0.6

0.4

:!0.3

0.2S•"DR =1500

0.-d

10.6 104 10-3 10-2 10-1

P,

Fig. 1 - Stop-and-wait (SW) throughput efficiency versus block error ratefor the case of B1 200 bits and B = 255 bits, calculated using Eqs. (2)and (1)

Throughput of Variation 1 of theStop-and-Wait ARQ Scheme

In variation 1 on the basic stop-and-wait ARQ scheme, for each correct reception ofB1 information symbols, [(B + DR) + (mB + DR)Pe/(1 - P,, )] symbol periods on theaverage must be expended. This expression is obtained as follows. For each set of mduplicate blocks B (each containing the same B, information symbols) the probability

that each set is the last is (1 - Pem) and the probability that the nth set is required isp(n-1)n + 1. The expected number of sets (each of m duplicate blocks) transmitted is then

fe eE1 = (1 - pen n(Pl 1' Pe/(- Pe)"

nm1

Thus, 1B + DR symbol periods are expended on the first transmission of BI, and (mB + DR)E1additional symbol periods on the average are required for receiver acceptance of B1. Thethroughput efficiency (6) for this scheme is then

B1ns (B + DR) + (mB + DR)E"

8

NRL REPORT 8140

In this case DR + (mB + DR)E1 expresses the expected wasted time in symbol periodsdue to system delays and required block retransmissions; also,

F ~ 1-

Ow m1+ (DR 1B E and 0 SW, Bj/(B +DR).L + (DR/B)

F,.r rm 1 we have the basic stop-and-wait throughput efficiency s7w.

This variation yields better performance than the basic scheme ("sw1 > 17 sw) if msatisfies the inequality

nm + (DR/B) 1 + (DR/B)(3)

1 1P

The optimum value mo for given Pe., D, R, and B is obtained by minimizing the term(roB + DR)Pel(1 - Pro) with respect to m. This results in

•iDR Rn P

--.- eB (4)I + Qn ,'np0-m Pt-,m

"which defines the minimum over m > 1 as long as

-DR -- tB Pe 2+ ./

But the left-hand side is positive and the right-hand side is negative for all positive B andn0, since Pe < 1; therefore (4) does define the minimum, The scheme generally yields

effective improvenmnt over the basic stop-and-wait ARQ scheme for DRIB much largerthan Yn and for P. > 0.1. This implies relatively short blocklengths in systems charac-torized by long system delays and high block error rates. Thus situations may exist in

which -sw becomes so low that, even with the improvement gained by this suggestedvariation, implementation may not be cost effective. Table I illustrates the evaluation"of the left-hand side of the ir.equality (3) for DR - 1600 and B - 255.

Figure 2 illustrates the relationship between mt and P,, for H - 255 mid DR - 1500.The corresponding throughput efficiency niSty I is shown in Fig. 3. Figure 4 is a familyof curves for B versus -log (1 - P.) parameterized by in. Note that -log (1 - Pe) increaseswith P,.

9

...............- ' .''-.--

JOEL M. MORRIS

0.9

0.8

0.7

0.6

0.5

ý4-

0.4

0.3

0.2

0.1

0 I I I1 2 3 4 5 6 7

m o

Fig. 2 - Block error rate versusoptimal number of duplicateblocks for stop-and-wait variation1 for the case of B, = 200 bits,B = 255 bits, and DR = 1500

Table 1 - Values for the Left-Hand Side of Eq. (3)with DR 1500 and BI 255

(m + (DR/B)]/(1 -PPe 0.1 Pe 0.2 Pe 0.3 Pe -0. 4 Pe 0.5

1 7.647 8.603 9,832 11.471 13.765

2 7.962 8.211 8.662 9.384 10.51

3 - 8.954 9.129 9.49 10.151

4 - - 9.963 10.142 10.541

5 - - - 10,995 11.233

6 - - 11.931 12.071

7 - - -- 12.984

8 - = 13.937

10

4'

NRL REPORT 8140

0.12

0.10

F-. 0.08

0.06

0.04

0.0

0 I I I I I0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 09

P,"4'Fig. 3 - Throughput efficiency of stop-and-wait

variation 1 versus block error rate using m. for thecase of B1 -200 bits, B = 255 bits, DR = 1500

100

10

iNil

-04 10"

ii" Fig. 4 - Plot& of blocktenlgth vemoum -loj(1 -P,) relating• the families of curves B .. R0.1,

I',+ RAO o - e"'o rand PDe I - tB tor stop-and-wait vaiAstion I

•.".

•Id o

JOEL M. MORRIS

Throughput of Variation 2 of theStop-and-Wait ARQ Scheme

In variation 2 on the basic stop-and-wait ARQ scheme, for each correct reception ofBI information symbols, [B + (DRIM)]E symbol periods on the average must be expended.The term (DRIM) is the average fraction of total system delay assigned to an individualblock. Thus the throughput efficiency for this scheme is

SS2= BI MBI

[B + (DRIM)]E (MB + DR)E

In this case (DRIM) + [B + (DRIM)] (E - 1) expresses the expected wasted time in symbolperiods due to system delays and required block retransmissions. Also, atw 2 = E- and

PSW2 = MBI/(MB + DR). For M = 1 we have the basic stop-and-wait throughput efficiency.This variation yields better performance than the basic scheme if M > 1, as is clear fromEq. (5). Performance curves of flSW2 are the same shape as for nsw. Their correct inter-pretation in this case requires multiplication by the factor (B + DR)/(B + DRIM). Thusobservations made on the 71SW performance curves are directly applicable to nsw 2 per-formance. For instance, if DR = 1500 and M = 300, this variation would yield a perfor-mance displayed by the DR = 5 curve of Fig. 1 instead of the DR 1500 curve.

Throughput of the Go-Back-N Continuous-ARQ Scheme

In the go back-N continuous.ARQ scheme (B + DRPe)(1 - Pe)- 1 symbol periods mustbe expended on the average (B + DRPe symbol periods E *imes on the average) for eachcorrect reception of B1 information symbols. The DRPe term is the average delay experi-enced by a given transmitted block B: the delay incurred when a block error occurs timesthe probability of block error. Thus the throughput [1-6, 8, 9] efficiency for this schemeis

_ B,(B + DRPe)E a

which can also be expressed as

B, (6b)

z -1 ( - NP,) (6c)B

12

If

NRL REPORT 8140

for NPe/(l - Pe) NPe << 1, where N = 1 + (DR/B). The scheme receives its name fromthe fact that BIR is usually chosen so as to make N have integer values. For example, ifD = kB/R, then the transmitter goes back by N = k + 1 blocks when a repeat-request isreceived. Either the system delay must be essentially constant or N is variable and thusthe transmitter must know its current value (then the average N is used in Eqs. (6). Forthis scheme the expected wasted time is DRPe + (B + DRPe)(E - 1) or equivalently BENe(1 - Pe)-1 BNPeE =BN(E - 1). In addition OG B1/B and aCG 11 + [NPeI(li- Pe)]}.

Figure 5 displays flG as a function of Pe IB1 for our representative examples. Thecurves have the same value at low Pe, but the performance for N = 7 falls off sooner thanfor N = 1. This reflects the degrading effects of long delays.

0.8, -

0.7

0.6 8

0.5 N

S0.41

0.A

if " 051" 0310.2 10"1 1

'. P

Fig. 5 - Throughput efficiency of the go-back-N schemeversus block error rate for B, - 200 bits and B u 255

Throughput of the Variation of the Go-Back.NContlnuous-ARQ Scheme

The variation on the basic go-back-N scheme improves the throughput performance"by reducing the wasted time involved in retransmission of the repeat-requested block plusN - 1 subsequent blocks. He. B I I + Pe [E + 2(N - 1)]) symbol periods must be expendedon the average for each correct reception of B1 information symbols. Thus the throughputefficiency [6] in this instance is

B,S•o• = ~~B11 + PerL' + k(N - 1)11' •r•=2

The term 2(N - 1) Is the number of blocks transmitted but ignored either before the firstNAK or after the first ACK is received at the transmitter. The expression PCB[E + 2(N - 1)]

B(E - 1) + 2(N - 1)BPe is the expected wasted time due to system delays and requiredblock retrannislsions. Transmission and background efficiencies for this scheme are 0• I(1 + Pe(E + 2(N - 1)I) and PoI-.,.I.

-. 3

S+-'•~-- ,.

JOEL M. MORRIS

This variation yields better throughput performance than the basic go-back-N scheme0G 1>G) if

1 + Npe >e + 2 (N-l)Pe1 -Pe 1 -Pe

or

Pe PeIB} >•.

The implication of this requirement is that the basic go-back-N scheme is better than thisvariation under most practical conditions (Pe < 1/2). If the term 2(N - 1) is reduced bya factor of 2, then the resulting equations are

B,17G1 - fork= 1,

* NPo Po

1+ N > + P + (N -)Pe,

and

e > 0.

* !Thus, if the number of blocks transmitted but-ignored, 2(N - 1), can be reduced to N - 1,* • then this modified variation yields improved performance for any block error rate. How-

ever reducing the number of transmitted.but.ignored blocks requires additional processingand storage for the N - 1 blocks subsequent to the block received in error. For example,by the time the transmitter receives the first NAK for a repeat-requested block, the re-

* ceiver could have processed (detected errors in) the next N - 1 or less blocks from theoriginal sequence. Thus, when the repeat-requested block is received correctly, there isno need to wait an additional N - 1 symbol periods before processing and acknowledgingthe subsequent block in the original sequence. In addition, after each received ACK,transmission is initiated for the next block in the orginal sequence which has not as yetbeen transmitted.

Throughput efficiencies iG 1 and t?ý I are shown in Fig. 6 for the example of B1 ,200 bits, B - 255 bits, mad N - I and 7. Throughput for k w 1 is better or as good asthe throughput for k = 2 in this example. The gain in performance is mome pronouncedat high block error rates and long delays.

14

NRL REPORT 8140

0.80.7

N=1K= 1,21,

0.6

0.5 =

" 0.4

0.3

K=20.2

10-5 10 10.3 10.2 10"1 1

P,

Fig. 6 - Throughput efficiency of the variation of the go-back-N scheme versusblock error rate for B= 200 bits and B 255

Throughput of the Selective-Repeat Continuous-ARQ Scheme

In the selective-repeat continuous-ARQ scheme, which is the most efficient of thevarious ARQ schemes, the stop-and-wait-for-acknowledgment period and other systemdelays are essentially eliminated. Consequently, for each correct reception of B! infor-mation symbols, B symbol periods must be expended E times on the average. Thethroughput [1-41 efficiency to the demand-access user in this case is

SR BI

The expression B(E - 1) is the expected wasted time due to required block retransmissions.Transmission efficiency and background efficiency for this scheme are aSR = E-1 andi3SR = B,/B respectively. From the form of ??SR, throughput efficiency for the selective-repeat scheme is proportional to aSR = E-1 , the same as for the stop-and-wait scheme.Figure 7 depicts ilSR as a function of P. for our representative example with B1 = 200bits and B - 255 bits. Comparison of this curve with Fig. 1 shows that as expected thestop-and-wait ARQ performance approaches the selective-repeat ARQ performance as thedelays decrease.

Throughput Error of the Hybrid ARQ Scheme with SeparateError Detection and Correction Encoding

In the first hybrid scheme, the concern again is for the average number of symbolperiods expended in order for BI information symbols to be accepted at the receiver(correctly received). The error-correction capability of an error-correcting code is deter-mined by a measure of the distance between the code words. For block codes and con.-volutional codes this distance measure is the minimum Hamming distance dm in. (Accordingto Ref. 10, p. 104, the minimum distance for convolutional codes tend to be slightly higherthan the corresponding results for block codes for given values of n and k. Thus theminimum-distance bound for block codes will be used for both block and coawolutionalcodes in this analysis.)

15

JOEL M. MORRIS

0.8

0.7

0.6

0.5

0.3

0.2

0.1

0104 10.3 10.2 10"1 1

P,Fig. 7 - Throughput efficiency of the selective-repeat scheme versus block

error rate for B, 200 bits and B 255

The error-correction capability of a block code is given as

t (dm in 1)/2;

* that is, a code with minimum distance dm in can correct up to t symbol errors per code-word. Thus the maximum bit error rate per codeword for which the code provides com-plete protection is

Pmax t/BE,

where B0 = B plus error-correcting-code redundancy symbols. When independent symbolerrors are assumed, the probability P. that a received Be-tuple is not decoded into anerror-detecting-code codeword (thus requiring a repeat-request) satisfies

CB• .

P ( 1 P 1(c) Pej-(1 Pej-0

where BC = B/re, pe is the bit error rate, and r, is the least code rate for the best t-errorcorrecting block code given by the Varsharmov-Gilbert lower bound [10, section 4.11 onminimum distance. Thus for a given B and P. the optimum t and re cam be determinedwhich minimizes this upper bound on P, . More precisely, the term

(Blr',)

~~~ ( a/r~ P(e) pje(1 PC0 )(/e

16

•- :: , • i,•%:,•.• : -. . .. .. '• • '• • 'b•,•' • •:•' • :••" • •"L....... •'V • '"'r='•'':"'':: ..... "•""••"'•?•" ':.'."":.•'•:'•;"•" .:?t• -. •z! l' *.V'tV-4

NRL REPORT 8140

is minimized, subject to the constraint on t and rc

Z > 2 (l/r,"I)B

i=O

This inequality is a form of the Varsharmov-Gilbert lower bound [theorem 4.7 on page87 in Ref. 101. The minimum upper-bound-block-error probability Pe and the correspondingcode rate 10 (hence BO = B/rO) can be substituted in any of the throughput-efficiencyequations to determine a lower bound on the throughput efficiency of the various ARQschemes when the best error-correcting code is added. Figures 8a and 8b display a lowerbound on the stop-and-wait efficiency flsw versus Pe (BER) for DR = 5 and DR = 1500respectively. The five curves correspond to the code rates 1, 4/5, 3/4, 2/3, and 1/2. Inaddition the block error rate Pe versus Pe is depicted in Fig. 9 for the five code rates. Asthe code rate decreases, protection for high-error-rate conditions (high values of P.) isgained at the expense of lower low-error-rate performance. This is to be expected, sincelow code rates require additional redundant bits per transmitted information block. Inaddition the points of intersection in Fig. 8 for the r. = 1 curve and any of the othercurves indicate at what value of Pe error-correction coding should be added or dropped.

Since the minimum-distance lower bounds for block codes can be used for convolu-tional codos as well, the probability of erroneous decoding P. for the best (mB0, mB)convolutional code is bounded above by

j-(d/2] id-j h Bd 2-/\--2 ) 2-1

m " m

J-d

where B0, B/rt [d/21 denotes the integer part of d/2, and m is the constraint length ofthe code. Other bounds on P. have been derived which could be used here. For exampleViterbi t I ll has derived an upper bound on P, when Viterbi decoding is used. Againthese bounds can be used in the aforementioned throughput-efficiency equations to deter.mine performance with respect to the added convolutional coding.

17

.4

-. , ... ...... .. . . ......... . .. .. ..................... "."...,.............".........................".........' .. . . . . . . . .,,,,**'**r,'**'

JOEL M. MORRIS

08,

0.7=4/5

0.6 3/

rc=2130.5rc23

~~~ 04 r~1/2

0.3

0.2

0.1 0

(a) DR =5

0.12

0.03 2/

0 A

0.p.

.010

10- 10 b 10 10311.

NRL REPORT 8140

10-2

10

-44

•. 04__ __ __ __ __= _ __ __ __ _

i .1 0 1 102 10` 10"

Pe

Fig. 9 - Block-error-rate upper bound versus biterror rate of the best error-correcting-code forthe different code rates in Fig. 8 and B,1 200bits and Bl= 255

OPTIMAL BLOCKLENGTHS FOR ARQ SYSTEMS

As evident from the throughput and wasted-time expressions that were derived, themessage blocklength B is a prevalent factor of throughput efficiency. Consequently theknowledge of optimal blocklengths which maximize throughput efficiency (or converselyminimize the wasted time) is important in the efficient design of any of the ARQ systemsdiscussed earlier [5, 12, 13].

The message length of any message source generally varies from one message to thenext and can best be described by a probability distribution, Usually, for efficient pro.cessing and transmission, the random-length messages are partitioned into several fixed.lengthblocks. Admittedly the last partitioned block usually cannot be entirely filled by the mes-sage and is either filled with dummy information or is terminated by an end-of-messageindication. Thus the message blocklength must be chosen with these facts in mind [12, 13].In addition there is a tradeoff In selecting the optimal blocklength. On the one hand it isdesirable to select the largest blocklength to minimize the number of acknowledgments and

19

V•

JOEL M. MORRIS

the length of retransmissions. On the other hand it is desirable to select the smallest pos-sible blocklength to minimize the block-error probability and to minimize the wasted timedue to the last unfilled partitioned block (when an end-of-message indicator is not used).

The goal of this section is to optimize the throughput efficiency tj with respect tothe blocklength B. The standard procedure is to obtain the derivative of t with respectto B, set it to zero, and determine if this leads to the global maximum; otherwise othertechniques must be applied. However, if the expected wasted-time expressions for the 4

various ARQ schemes are normalized by the blockleng'Ih B, it is clear from the form of17 for all schemes that it is sufficient to minimize this normalized expected-wasted-timeexpression with respect to B. This procedure ignores however the situation in which thelast block of the message may not be completely filled with useful data. Nevertheless agood approximation is sought without including that facet of the problem. The descriptionof block error rate Pe = Pe{BJ of particular concern is

"Pe 1 e-YB, (7)

where

= -Rn(l Pe)

for the independent bit error environment, with Pe being the bit error rate, and where

the mean block error rate for the Poisson-distributed block error environment.

Optimal Blocldength for the Basic Stop-and-Wait ARQ Scheme

The expected-wasted-time expression DR + (B + DR)(E - 1) for the basic stop-and-wait ARQ scheme has the following form when normalized by B:

f(B) = (E - 1) + DRB'E

=-.Pe DRP B- -- -P O ) ( 8 )

where E (1 - Pe) 1 . Differentiation of this expression with respect to B yields thestationary-point equation

dP DR(1 - PS)

dD B(B + DR)

20

S' '•. . , ...... . A

NRL REPORT 8140

from which the minimum can be determined. Under the assumption of Eq. (7) thestationary-point equation is

DRB(B + DR)

s'i This leads to the stationary-point expression [5]

B0 =DR + )1/2j

To determine whether or not the stationary point is the global minimum for this charac-terization, the behavior of [(B) must be better understood. If we substitute say Eq. (7)in (8) and differentiate, we find

['(B) yB2 + yDRB - DRS. ff '( B ) =.B 2 e-yB

The denominator on the right-hand side is positive. The numerator is quadratic in- B withpositive leading term and two real roots: one at BO and the other obtained by taking thesquare root in Eq. (9) with a negative sign. Therefore f' is negative between the roots ofthe numerator and poaitive elsewhere. But the second root is negative; therefore f' is

*1 negative on the interval (0, BO) and positive on (BO, -o). It follows that for positive block-lengths (the only ones of interest) f has its minimum at BO. Figure 10 displays f(B) for

¶ X,Pe = 10 -1, and 10-6, and DR = 5 and 1500, where B1 = 200 and B = 255. Note]: that BO is highly sensitive to variations in X and Pe; thus the choice of a single blocklength

for all error rates will not yield good uniform performance.

Optimal Blocklength for Variation 1 of theStop-and-Wait ARQ Scheme

The normalized. wasted-time expression to be minimized in the case of variation 1 ofthe stop-and-wait ARQ scheme is

h s n r (B) DRB"I(El + 1) + mE 1 ,

where ?n is the number of duplicate blocks per set transmitted (which may have beenchosen to satisfy Eq. (4)) and E1 - P,/(1 - Pe') is the expected number of sets tranmuitted.Thus

DPe / InP.f(B) D R (1 + 1~~ -" I Pa

21

"Y,

JOEL M. MORRIS

0.10

0.09 0.8

0.08 0.7

0.6

0.0.40 .08 - 0.6

•,0.05 - 0.4

S0.04 0.3

S.03 •0.2

0.02 -" i 0.1 ,0 1 2 3 4 5 0 10 20 30 40 50 60 70

B B

(a) DR = 5, fpe(B) atpe = 10', (b) DR =5, fp(B) at Pe = 10-3, B°e=B° 4.8, and f,() at k = 10-1, 68.2, and f\(B) at =10-3 , B"= 68.3Pe

BO 5.0

0.014 1400

0.013

.. •0,0| 12(00 ,

ot 001 11(00

0.010 100

0.Om6 800

0.007 700

0,004 400

W 0 G00 100 o0 240 0 2 4 6 8 W 12

f() I1 - , ) atpc 10" 1 B 0 . (d)DR - 1500, fp (B) at p, l 101,2219, and f4(D) at X - 10-,. Bi - 2234 -,, 9.4, and IX(B) at X, - 10-1,

-B 9.9

Fig. 10 - Stopand-wait wasted-tine functions versusblocklength for B, - 200 and B = 265

22

S¸=¸7 , o • • • -• -;=,,.• ;• • ,• •. •.. . . • • • •... .....

NRL REPORT 8140

150

140

130

120

110

100

90

80

70

60

60) I I I I ,I5 100 200 0 490 500 600 700 600

(e) DR 1500, fp (B) at Pe = 10-3, Bp°

685.9 and fx(B) at X = 10-3, BO - 686.1

0.8

0.7

0.0-•.. 0.6

• - 0.4

0.3

0.1

0 l0,ow 20.00 now 40.000

(f) DR 1-00, ff8) tp- 10-1-, i37732,5, aMt 4'(13) =t A. 10-0 I ,13 ,

37987A1

Pig. 10 (Continued) - Stop-and.wait wasted.tlme functlou.vo*u& blocklmngth for B, - 200 &Wd Bi 265

23

S•Y"

J JOEL M. MORRIS

The stationary-point equation is then

dPe 1 PTm+PedB i

+B (M +(m -

LDR(1 - Pe)

Under the assumption of Eq. (7) the stationary-point equation becomes

2 - e-tB -(1- e-^tB)m

mnB 2 - 1 (0e-yBfJB + ''+ (m - 1)(1 - 1

This nontrivial equation is only an implicit form of the stationary-point equation. Thusobtaining the solution with this scheme requires much more computation than with thebasic scheme. Although

dPe 1-Pe"+ Pe

dB mJ32

B DR (1~ - (m-1W

is a necessary and sufficient condition for f(B) to be mnonotonically nondecreasing, note* that

dPv 1 PC + P, ol- + PC

dB~ B[1 + (m - 1)Pe"J [B + (nB 2 IDR)J][I M -plw]

is a sufficient condition for f(B)'s nionotonicity, which may be exploited in determiningthe global mninimum-. The function f(B) from Eq. (10) is displayed in Fig. 11 for DR= 1500and -y -X or Rn(l - &a). These points are computed using the triple of points (in, 1) , B)

* -which are the simultaneous solutions to Eq. (4) and Pe 1 - e 1B (Fig. 4). In only oneintace(Fg ld it y=10-2-6) does the minimum of f(B) not occur for the largest

values of intersection poinits Pi.~ or B obtained from Fig. 4 (for a given value of -y). How-ever B21i1In in Fig. lIId is much larger than for any of the other examples. Again Bf)iappears highly sensitive to variations in the error rate -Y.

24

3-vm '"m.

NRL REPORT 8140

1000 4-

9w 500

.- 70030

600 + 200 +

500 100 ++

400 40 1 2 3 4 5 010 20 30

Be

(a) y 10" (b) y = 10-1-5

90D.•

700

500 50

300 40

2w0 30

100 2D

0+ t1 +I 1 1+ 1 1 + '0 2o 4o eo OD IOD 120 10 0 100 200 300 400 500 00 700

(c) 10"2 (d) y 10 "s.5

Fig, 11 - Stop-and-wait varlation-1 wasted-timefunction versua blockleagth for DR t 1500

t*2i

•i: 25

JOEL M. MORRIS

Optimal Blocklength for Variation 2 of theStop-and-Wait ARQ Scheme

For variation 2 of the stop-and-wait ARQ scheme the normalized-wasted-time expres-4 sion to be minimized is

f(B) = (E - 1) + DR(MB)-lE

Pe DR

I Pe MB(1 -Pe

where M is the number of blocks transmitted in sequence before the transmitter stops andwaits for an acknowledgment. Differentiation of this expression with respect to B yieldsthe stationary-point equation

dPe DR (1 - Pe)

dB B(MB DR)'

For Ml 1, corresponding to the basic stop-and-wait scheme, we get the stationary-point equation for that scheme as expected. This stationary-point equation becomes

= DR/B(MB + DR), yielding

B DR [( + " (I1)S, k 4M 2

As stated previously, the results for the basic stop-and.wait scheme are directly applicablehere, with the substitution of DRIM for DR in the appropriate equations. This yields

['(B) = yB2 + y(DRIM)B - (DRIM):• B2e-yB

mad BO, given by Eq. (11), is the minimum by the argum=ent used for the basic scheme.

Optimal Blocklength for the Go-Back.N Continuous-ARQ Scheme

The normalized wasted-time expression to be minimized for the go-back-N continuous-AARQ scheme is

DR DRf(B) d+ (1 + P) (E -1)

(1DR

26

NRL REPORT 8140

where 1 + (DR/B) N is the number of blocks the transmitter goes back when a repeat-request is received. Note that f(B) - Pe(1 - Pe)-1 -c - as B -0 0; thus large values of Bare detrimental to throughput efficiency for this scheme.

The stationary-point equation is

dPe DR( Pe)

dB B(B + DR) Pe'

which is the same as for the basic stop-and-wait scheme except for the factor Pe. By useof Eq. (7) the stationary-point equation becomes

DR(1 - e-YB)B(B +DR)

For any block error distribution,

dPe DR(l Pe)dB B(B + DR) e

insures that f(B) is monotonically nondecreasing. When this inequality is not satisfied forall B and practical considerations dictate the minimum size of B, the behavior of f(B)should be examined in the region up to and including the last interval in which its deriva-tive is negative. However this inequality is satisfied on the interval [1, 00] for the errorcase Pe 1 - eY considered here, since for all B > 1 and -y> 0

B2 + BDR >DR(1 -eYB)

because the equality is satisfied at B 0 and

(B2 + BDR) 2B + DR >

Thus B 1 is the minimum point for f(B). In application the smallest B allowed underpractical considerations is the best choice for minimizing the expected wasted time.

Optimal Blocklength for the Variation of the Go-Back-NContinuous-ARQ Scheme

The variation of the basic go.back-N scheme and the modified variation are similarto the basic scheme and are treated together here. The variation (corresponding to to 1)has the normalized expected wasted time

27

S• ,•,•,,•-w.•€•..@. - - , • 'NY. . '

JOEL M. MORRIS

fi(B) (E -1) +k(N-1We ,k =2,

and the modified variation (corresponding to q~ 'G01) has

Pe DRf2(B) 4.- P,

It is sufficient to consider the behavior of

FB Pe + kD

The stationary-point equation is

dpe kDR(1 pe)2 pe

dB B[ + kDR(j Pe)2 ]

which yields the implicit-solution equation

kDR(e-'YB) (1- eY8 ).Y=B [~B + kRg2y~

Now ((B) is monotonically nondecreasing as long as

dPe kDR(1 - e2p

diB B[1B +kDR(1 pt) 2 1

This inequality is valid for the semi-infinite interval [BID, 00), where

with a Bsatisfying~

-yo 9n(l + 27a). (12)

28

NRL REPORT 8140

Thus for this case it is sufficient to evaluate f(B) on the interval [1, BO] to obtain thedesired minimum. In general the complexity of the stationary-point equation may requirethe plotting of f(B) directly or the use of an optimization algorithm. Figure 12 indicatesthe behavior of f(B), where the minimizing value of B does not occur at the endpointsof the intervals under investigation. The solution for Eq. (12) is up, = 1.2564247/Qn(1 - Pe)and aX = 1.256231765/X respectively. Table 2 lists the minimizing values of B for errorrates X and pe = 10-, where n - 1,..., 6. Only for DR = 1500 do we have Bmin * 1for all ?• and Pe considered. As a test consider

C eB) -y 1 e 2iB

27B yKDR

If C(1) > 1, then B n=. If C(1<1andC(B)1for B =BO, then B6in e (1,Bo]

As before, Bmin is highly sensitive to the error rate X or Pe.

Optimal Blocklength for the Selective-Repeat Continuous-ARQ Scheme

In the selective-repeat conthnuous-ARQ scheme the normalized expected-wasted-timeexpression to be minimized is

f(B) = E - =;• • 1Pe"

Note that f(B) is monotonically increasing in B, since Pe is monotonically increasing in B(that is, as the blocks get longer, the probability of a block error increases). Thus the

V/ smallest B allowed under practical considerations is the best choice for minimizing theexpected wasted time.

SUMMARY AND CONCLUSIQNS

This report provides the data-communication design engineers with information tofacilitate their choice of error control techniques or to assess the performance of theirproposed ARQ or hybrid designs.

In this report the performance of various ARQ schemes used for error control indata-transmission systems was described. The basic schemes, such as stop.and-wait, go-back-N, and selective-repeat, were described from a heuristic as well as a theoretical view-point. Recently reported variations and a hybrid (FEC plus ARQ) were also discussed.Specifically, the throughput performance of these ARQ schemes under certain assumptionsand their optimal blocklengths were sought.

29

...........................................

e JOEL M. MORRIS

160 29

140 28

130 27

120 26

S110 25

100 24

90 23 b,

8o 22

70 21

60 2D0 5 10 15 20 25 30 25 40

(a) k = 1, X = pe 10-1, where fpe(B) 181 1 1forBIe = 38, (pe = 12, and /X(B) for 10 2o 80 70 808•~B = , f39, aX =13!39 = (d) k = 2, X = pe = 10-2,_%e =

400 - 126, where B.e = 174, and BOe, 175

350

I-

3.04-

- 3.0 -

3.02

200

150 3.01

0 6 to 15 20 2( 30 3• 45 1 2.0 55,,

(b) k 2, X = Pe 10-1, where f4(B) for B o 54, 2,9

=12, and f.(B) for BO 55, ON 13 29u p N 2.98

15.0 -

14.5

2.941:, 4.0 F2.93I 13.5 0 10D 200 300 4 00 5 0

(el k - 2, X p

1 5. 1 26 6 , a nd B ; 0

I•.o - -BO - 653

*,. S, 2l 0Fig. 12 - Go.back-N-variation wanted-time function0: 20 0 W 0 10Do 120 UO versus blo1klength for DR 1500

U

-2i': ~~(C) k =1, -p,, _ 10"• O N••0

* 10, her f(B) for BO~, 123,i" "~~126, whereftpe( t'rJp lg3 :

and f4(B) for Bý m 124

30

'-.

NRL REPORT 8140

H- H- ,-4 H- rvI7- H HI H HHHH Hv H H VI-1 -Iv-H

00

co CD E

w coH 6 -H ~0 O Coir4 &,4qH~ H X H rHHHHH4r4 - riHH -4w- "-4 -oo eq CD~ ~ . -I H eq

to 00 CD0 -10m 0 Lco'-o v -i - q o oL- o Co 10 cqa r4N o a t*4 -I H- co L 1 - - cq o m 0 in - m C -

r4 H a D 11H H0 -I

H H w maLom -m o

v- LOI r-I 0 - 0 o q

0- C

H ~ ~ ~ ~ ~ ~ ~ -H- m- -~e qe ~u o ~ ~ o ~ 'J qe ~e

VH H. H eq e q T-1

V- 10 V-4 04 N0 N0 N0 N 0 ko to 1jtoW0 H

eq -qN~~~~~r H eq eq q v 4 q q eq q H V-4 V-4e

3141-

m o0

JOEL M. MORRIS

The results obtained are in the form of equations and algorithms for computingthroughput efficiency and optimal blocklengths. The more detailed results are forindependent-bit-error and Poisson-distributed block error channels.

In all cases the design engineer's choice is based on the common tradeoff betweenacceptable performance and implementation or complexity costs. The better performingschemes, such as selective-repeat ARQ, require additional storage and intelligence withinthe transmitter-receiver pair (not to mention separate feedback channel), which are notnecessary in the lesser performing stop-and-wait ARQ scheme. But microprocessor andmemory costs are dropping, so that the cost of additional control logic is becoming lessdominant in the choice of schemes. In general the optimal blocklength for a specificARQ scheme is sensitive to the error rate. Thus different acceptable blocklengths maybe required, for several corresponding ranges of error rate. Adaptivity is also implied forsome hybrid ARQ designs in which error-correction coding (FEC) is added at some pre-determined level of error rate to improive the throughput performance at high error rates.

Other conclusions can be drawn from the data in this report. If the variations ofimplementation and complexity cost among the ARQ and hybrid schemes are marginal orof minor importance, then the selective-repeat ARQ is the best choice at low-to-moderateblock error rates. Moreover performance falls off smoothly for high error rates (Pe > 10-2).This is true for all block sizes. For small delays DR, the stop-and-wait, stop-and-waitvariation-2, and go-back-N schemes yield less but comparable throughput efficiency. Thedifference becomes quite pronounced at moderate to high delays. From the example inwhich error-correction coding (FEC) was added, substantial improvement is possible overall the schemes here at moderate to high bit error rates (10. " P.). However, these gainsmay be at the expense of only low to moderate performance when used at low bit errorrates. The questions of interest then become: should FEC be switched on at some pre-determined error rate or should it be applied at all error rates, and what improvementsare possible for FEC when bit errors are not independent.

Under the assumption of Poisson block errors or independent bit errors we foundthat the go-back-N and selective-repeat ARQ schemes had optimal blocklengths of onebit (throughput efficiency improves as blocklength decreases) and that the stop-and-wait,stop-and-wait variation-2, and go-back-N-variation ARQ schemes have optimal throughputefficiency on their respective intervals (1, B0 1, where BO is the solution to the correspondingthroughput-efficiency stationary-point equation. However, for the go-back-N and selective-repeat schemes, practicality dictates some minimum blocklength to include block overheadinformation, such as redundancy bits for error detection or other block processing data.In addition the original assumption of all message blocks being the same length and con-taining no filler bits is thus justified for the go-back-N and selective-repeat cases. Theunsuitability of the assumption for the other ARQ schemes, for which the optimal block-lengths may be rather large, will have to be determined from further investigation.

The throughput results of this report are in terms of throughput efficiency 11 for afixed transmission rate R. In some applications the choice of transmission rates may beflexible. Thus it may be required to optimize overall throughput nR with repect to R.

32

. ~ *~*~ ~ * "..... ...-- -

NRL REPORT 8140

But because qi is also a function of R, both explicitly and implicitly (via the optimalblocklength dependence on R), determining the optimal transmission rate and theoptimal blocklength to maximize the throughput (in bits per second) is in general nottrivial. It is one of several topics for further investigation.

We briefly recommend some other extensions of this report:

* A comparison of optimal performances of the various ARQ schemes. These per-formances could be composed for both fixed and optimal transmission rates.

0 An evaluation of throughput performance, when various popular error-correctingcodes (FEC) are used at high error rates to augment the various ARQ schemes. The eval-uation should extend to dependent-bit errors as well. Likewise additional desirable infor-mation would include the error levels at which FEC is added, whether FEC should beapplied for the entire range of error rates (not just switched on at high error rates), andthe relationship between FEC and desired optimal blocklengths.

9 A determination of performance degradation due to the deviation of an actualapplication from the preliminary assumptions of this report. For example the assumptionof messages consisting of only an integral number of equal-length blocks is surely notrealistic. Yet how much this assumption improves, or possibly degrades, the throughputperformance is not known.

* An evaluation of the sensitivity of the optimal blocklengths for specific ARQschemes to the error rate environment. In addition the plausibility of optimal blocklengthsfor several error-rate ranges should be investigated.

REFERENCES

1. R.J. Benice and A.H. Frey, Jr., "An Analysis of Retransmission Systems," IEEETrans. Commun. Technol. COM-12 (No. 4), 135-145 (Dec. 1964'.

2. R.J. Benice and A.H. Frey, Jr., "Comparisons of Error Control Techniques," IEEETrans. Commun. Technol. COM-12 (No. 4), 146-154 (Dec. 1964).

3. E.X. Cacciamani and K.S. Kim, "Circumventing the Problem of Propagation Delay on

Satellite Data Channels," Data Communications, July/Aug. 1975, pp. 19.24.

4. D.R. Doll, "Calculating Throughput on Full-Duplex Data-Link Controls," Data Com-munications, Jan./Feb. 1976, pp. 25-28.

5. W.G. McGruther, "Thruput of High Speed Data Transmission Systems Using BlockRetransmission Error Control Schemes Over Voicebandwidth Channels," Int'l Commun.

Conf. Rec., 1972, pp. 15-19 through 15-24.

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6. A.R.K. Sastry, "Improving Automatic Repeat-Request (ARQ) Performance on SatelliteChannels Under High Error Rate Conditions," IEEE Trans. Commun. COM-23 (No. 4),436-439 (Apr. 1975).

7. A.J. Viterbi, "Error Control for Data Communication," Computer CommunicationReview, Jan. 1976, pp. 27-37.

8. A.R.K. Sastry, "Performance of Hybrid Error Control Schemes on Satellite Channels,"IEEE Trans. Commun. COM-23 (No. 7), 689-694 (July 1975).

9. A.R.K. Sastry and L.N. Kanal, "Hybrid Error Control Using Retransmission andGeneralized Burst-Trapping Codes," IEEE Trans. Commun. COM-24 (No. 4), 385-393(Apr. 1976).

10. W.W. Peterson and E.J. Weldon, Jr., Error-Correcting Codes, 2nd edition, Cambridge,Mass., MIT Press, 1972.

(, 11. A.J. Viterbi, "Error Bounds for Convolutional Codes and an Asymptotically OptimumDecoding Algorithm," IEEE Trans. Inform. Theory IT-13, 260-269 (Apr. 1967).

12. W.W. Chu, "Optimal Message Block Size for Computer Communications with ErrorDetection and Retransmission Strategies," IEEE Trans. Commun. COM-22 (No. 10),1516-1525 (Oct. 1974).

13. A. Shaheen and J.C. Majithia, "Comparison of Overhead Factors in Message-Switchednmd Packet-Switched Networks," Electron. Lett. 12 (No. 1), 20-21 (8 Jan. 1976).

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