Analysis of ARQ in a Fading Channel

F. Lorenzelli∗

The Aerospace Corporation, P.O. Box 92957, Los Angeles, CA 90009-2957, USA

In this paper we consider the performance of selective-repeat automatic retransmis-sion request (SR-ARQ) protocols over a fading channel. The matrix signal flow graph(MSFG) methodology is used to compute the probability-generating function of severalvariables, such as throughput, delay, error probability, etc. The MSFG framework is usedto describe different variants of the SR-ARQ scheme with the purpose of assessing theirrelative performance. The analysis is conducted from both the transmitter and the receiverperspective.

Nomenclature

Symbolsτ0 Coherence time

k Round-trip timeQ PersistenceT Time-out

SuperscriptsT Transpose

I. Introduction

The purpose of this work is to assess the performance of a selective-repeat automatic retransmission request(SR-ARQ) protocol1 for the transmission of packets over a fading channel. Many techniques have been

used in the literature for various retransmission schemes, such as the go-back-N (GBN) or variants of theSR-ARQ protocol,2–8 under different scenarios. To the author’s knowledge, the work that so far has shownhighest flexibility is Ref. 9. The mathematical tool used in Ref. 9 to analyze the SR-ARQ scheme is a matrixsignal flow graph (MSFG) approach. This methodology allows one to compute the probability generatingfunction10 of several variables (throughput, delay, error probability) under very general conditions. Amongthe design parameters that can be chosen are the packet error probabilities and the fading parameters onboth forward and feedback channels, a retransmission time-out, the round-trip time, and SNR. The forwardand backward channels are modeled as two independent hidden Markov models (HMMs), i.e., two-stateMarkov chains augmented with information on packet error rate in either state. Our contribution is theextension of that model to the case of finite persistence, i.e., finite number of allowed packet retransmissions,the inclusion of the decoder performance, as well as the analysis of delay and error rate from the receiver’sperspective. We also use the MSFG framework to describe different variants of the SR-ARQ scheme in orderto assess their relative performance.

II. SR-ARQ Schemes

The schemes we consider are summarized in table 1 on the following page. In the table we denote thedifferent states as follows. State 00 represents the case of successful reception of the information packet

∗Engineering Specialist, Comm. Systems Eng. Dept., The Aerospace Corporation

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26th International Communications Satellite Systems Conference (ICSSC)10 - 12 June 2008, San Diego, CA

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Copyright © 2008 by The Aerospace Corporation. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

by the receiver, followed by the transmission of the acknowledgment (ACK) packet, also correctly receivedby the transmitter. In state 01, the information packet is received, but its ACK is not. States 10 and 11represent the situations where the receiver detects an error in the information packet and sends a negativeacknowledgment (NAK) packet that is either received correctly (state 10) or not (state 11) by the transmitter.In the SR-NACK (“negative acknowledgment”) and SR-SACK (“select acknowledgment”) schemes, no actionis taken in states 10 and 11, which are then collapsed into a single state, state 1x. Note that in the SR-NACKscheme a retransmission occurs when a NAK is generated by an out-of-order packet or at time-out.

Table 1. Three SR-ARQ schemes.

SR-ACK SR-SACK SR-NACK

State 00 Send next packetState 01 Wait for next ACK or time-outState 10 Retransmit Wait for next NAK

or time-outState 11 Wait for time-out

These schemes can be described by matrix flow graphs11 whose edges are weighted by transition prob-ability matrices. These 4 × 4 matrices are the combination (Kronecker product) of the 2 × 2 transitionmatrices that correspond to the HMMs of the forward and feedback channels. Each channel is representedby a two-state HMM with “bad” and “good” states and the transition probabilities are calculated based onthe fading statistics of each channel (see the section on the Gilbert-Elliot model in Ref. 12). In each statea packet error probability per transmission is defined, which can be a function of the noise margin. As anexample, P01 is defined as P01 = P(f)

0 ⊗ P(b)1 , where ⊗ is the Kronecker product, P(f) and P(b) are the

transition matrices for the forward and feedback channels, respectively, P0 = Pdiag(1− ε), P1 = Pdiag(ε),and ε is the vector containing the error probabilities in each state (see the section on hidden Markov modelsin Ref. 12). We also use the shorthands P0x = P00 + P01, P1x = P10 + P11, etc.

In the graphs, the variables z1, . . . , zn are used to enumerate the events of interest related to quantitiesη1, . . . , ηn (e.g., throughput, delay, etc.). Let z = [z1, . . . , zn]T. Once the matrix probability-generating func-tion φ(z) is computed by using traditional means, the (scalar) probability-generating function is computedas

φ(z) =πTIφ(z)1πTI 1

,

where πI is the initial-state distribution, which in turn can be computed from the steady-state distributionπ. The probability-generating function can be used to compute any moments of all the quantities of interest.In particular, the first moment (the mean) of ηi, the quantity associated with zi, can be obtained as follows

η̄i =d

dziφη(z)

∣∣∣∣z=1

,

and the variance is obtained as10

σ2ηi

=d2

dz2i

φη(z)∣∣∣∣z=1

− η̄i + η̄2i .

The quantities of interest are

• The throughput, or channel utilization, defined as the number of successful packets per number ofpackets sent. When retransmissions are allowed (i.e., the persistence Q is nonzero), then the throughputwill be generally less than one. When an unlimited number of retransissions is allowed, this is knownto be bound by (1 − εf )(1 − εb) (Ref. 13), where εf and εb are the packet error probabilities on theforward and feedback channels, respectively.

• The probability of incorrect reception of the packet, including all the Q retransmissions, not to beconfused with the probability of incorrect decoding of a packet, which is a design parameter and isuniquely defined by the decoder. As the persistence parameter grows, it is expected that the probabilityor error decreases to zero. Note that this quantity is defined from the receiver’s point of view.

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• The probability of correct packet communication, by which we mean the probability that the transmitterreceives positive acknowledgment of a successfully received packet. This quantity is defined from thetransmitter’s perspective.

• The delay, computed as the time between the transmission of the packet and its successful reception.The two first moments (mean and standard deviation, or jitter) are of primary interest, but additionalmoments can be calculated once the probability-generating function is obtained.

• The latency, defined here as the delay between the transmission of a packet and the acknowledgmentof its reception, and corresponds to the entire communication (defined as transmission plus positiveacknowledgment). The latency is computed only for packets successfully received.

The MSFGs for the analysis of throughput (see below for an explanation), delay (η2), latency (η3),probability of incorrect reception (η4), and probability of incorrect communication (η5) of the SR-ARQscheme are shown in figures 1 on page 5 and 2 on page 5. Figure 4 on page 6 shows the graph usedto compute the initial-state distribution. In order to compute the throughput, we enumerate the numberof packets (η1) that are sent (and retransmitted) per successful packet. The throughput is defined as thereciprocal of that number. It is important to remember that throughput, delay, and latency are definedfor successfully received packets and thus have to be adjusted by the probability of correct reception orcommunication, as defined above. The probability of incorrect communication is computed by counting thenumber of successful acknowledgments (0 or 1) received per transmitted packet, so it measures the error ratefrom the transmitter’s point of view. The probability of incorrect reception from the receiver’s point of viewis similarly computed by counting the packets correctly received, regardless of whether the acknowledgmentarrives at the trasnmitter.

In figure 1 on page 5 (the notation is similar in the other graphs) node I represents the transmission ofa packet. The acknowledgment is expected back by the transmitter k − 1 time slots later, at node A. If thepacket and its ACK are correctly received, then we transition to the final state O. If the packet is correctlyreceived, but the ACK is lost, then we move to node C, where the packet will be retransmitted after a delayd = T − k, given the round-trip time k and the time-out T . The packet will be retransmitted up to Qadditional times, Q being the persistence, or maximum number of retransmissions. If the packet was notcorrectly received, then there are two possibilities. The NAK is error-free and the transmitter retransmitsthe packet (again, up to a total of Q additional times). If the NAK is erroneous, then the transmitter waitsfor time-out, then resends.

In figure 1 on page 5, note the labels just to the left of node B (“0 ≤ ρ ≤ Q times”) and to the right of nodeC (“Q−ρ− 1 times”). These are meant as reminders that when Q <∞ the total number of retransmissionshas to be limited to Q (in addition to the original transmission of the packet). Retransmissions occur at bothnode B and node C. If ρ retransmissions have already been issued at node B, due to unsuccessful receptionof the packet, then a total of Q− ρ retransmissions are available at node C. Remember that the transmitterresends a correctly received packet if the corresponding ACK fails to be delivered. In order to reach state O,one of the retransmissions from state C has to be positively and successfully acknowledged, so there remaina total of Q−ρ−1 unacknowledged retransmissions. The way to compute the transfer function of this graphis to list all the possible paths from I to O, as follows (where EXY is the edge weight from node X to nodeY):

EIAEAO

+ EIAEAAEAO + EIAE2AAEAO + · · ·+ EIAEQAAEAO

+ EIAEACECO + EIAEAAEACECO + EIAE2AAEACECO + · · ·+ EIAEQ−1

AA EACECO

+ EIAEACECCECO + EIAEAAEACECCECO + . . .+ EIAEQ−2AA EACECCECO

+ . . .

= EIA

Q∑ρ=0

EρAA

EAO +Q−1−ρ∑ψ=0

EACEψCCECO

,

where we have used the shorthand EAA = EABEBA.The graph in figure 2 on page 5 contains the node R, which represents the state where packets are

successfully received but not acknowledged (corresponding to η6), whereas the state N in the graph of

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figure 3 on page 6 represents those packets that are never received (η7). These graphs are used to computethe overall number of packets sent and to adjust the average transmitted power (or signal-to-noise ratio).Note that the transfer functions EIR and EIN should be computed as follows:

EIR = EIAEACEQ−1CC ECR + EIAEAAEACEQ−2

CC ECR

+ · · ·+ EIAEQ−1AA EACECR + EIAEQAAEAR

= EIA

(Q−1∑k=0

EkAAEACEQ−1−kCC ECR + EQAAEAR

),

EIN = EIAEQAAEAN.

Figures 5 and 6 show the MSFGs for the SR-SACK and SR-NACK protocols, respectively. The differencesare in the branches A→B, which correspond to the situation where the packet is not correctly received, i.e.,state 1x. Graphs similar to figures 2 on the following page and 3 on page 6 are not shown for SR-SACK andSR-NACK, but are drawn in a similar fashion.

In figures 7 on page 8 through 9 on page 10 we show the performance of the three protocols with thefollowing assumptions:

• Forward and feedback channels both experience correlated Rayleigh fading, with an f−4 Dopplerspectrum12 and a coherence time τ0.

• Packets are transmitted at a rate equal to the reciprocal of the packet duration.

• The packet duration, Tp, is equal to 0.5τ0.

• The SNR varies from 5 dB to 15 dB.

• The curve that represents the decoder packet error rate vs. SNR is taken from a specific LDPC codeand corresponding decoder, and a given data rate.

• The round-trip time is set to be equal to 2τ0.

• The time-out exceeds the round-trip time by either zero (red curves) or four packet lengths (bluecurves).

• The persistence parameter Q is chosen to equal 2.

• Time is normalized with respect to the round-trip time.

Figure 7 on page 8 refers to the SR-ACK protocol. As expected, the throughput increases if the time-outis set to a value larger than the round-trip time, at the cost of a longer delay and a longer latency. Theprobability of incorrect reception also decreases. Similar results are encountered in the other two protocols.Figure 8 on page 9 shows the performance of SR-SACK. Because the SR-ACK protocol takes action assoon as an error is detected at the receiving end, its performance is superior to SR-SACK, both in terms ofthroughput and delay/latency. The plots regarding SR-NACK are given in figure 9 on page 10 and show aperformance that is very close to SR-ACK.

III. Conclusions

In this paper we have applied the MSFG methodology to compute performance measures of differentARQ protocols. MSFGs are very powerful tools in that they can incorporate HMM models of the fadingcommunication channels. The protocols can be evaluated in terms of a large number of design parameters,including the fading statistics, the decoder performance, and the retransmission parameters (round-trip time,time-out, persistence).

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I A

B C

Oz1(z2z3)

k−1Pk−1

z2z3z4P00 + z2z4P01

d−1∑i=0

zi+23 Pi

x1Px0

z2z3P10 + (z2z3)d+1P11P

d

z2zd+13 P01P

dx1

z1(z2z3)k−1Pk−1

z1zT3 PT

x1

z1z4

T−1∑i=0

zi+13 Pi

x1Px0

0 ≤ ρ ≤ Q timesQ − ρ − 1

times

Figure 1. MSFG for the SR-ACK protocol.

I A

R

B C

O

R

z6Pk−1

P00 + P01

d−1∑i=0

Pix1Px0

P10 + P11Pd

P01Pdx1

P01Pdx1

z1(z2z3)k−1Pk−1

z6PTx1

z6

T−1∑i=0

Pix1Px0

z6PT+1x1

0 ≤ ρ ≤ Q timesQ−ρ−1

times

Figure 2. MSFG for the SR-ACK protocol for the computation of the nonacknowledged, received packets.

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I A

B

Nz7P

k−1 P1x

P10 + P11Pd z7P

k−1

Q times

Figure 3. MSFG for the SR-ACK protocol for the computation of the packets never received.

I

B

C

Pk−1(P10 + P11Pd)

Pk−1P01Pdx1

Pk−1(P10 + P11Pd)

Pk−1P01Pdx1

PTx1

Figure 4. MSFG used to compute the initial-state distribution.

I A

B C

Oz1(z2z3)

k−1Pk−1

z2z3z4P00 + z2z4P01

d−1∑i=0

zi+23 Pi

x1Px0

(z2z3)d+1P1xPd

z2zd+13 P01P

dx1

z1(z2z3)k−1Pk−1

z1zT3 PT

x1

z1z4

T−1∑i=0

zi+13 Pi

x1Px0

0 ≤ ρ ≤ Q timesQ − ρ − 1

times

Figure 5. MSFG for the SR-SACK protocol.

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I A

B C

Oz1(z2z3)

k−1Pk−1

z2z3z4P00 + z2z4P01

d−1∑i=0

zi+23 Pi

x1Px0

P1x

d−1∑i=0

(z2z3)i+2Pi

x1Px0+

P1xPdx1

z2zd+13 P01P

dx1

z1(z2z3)k−1Pk−1

z1zT3 PT

x1

z1z4

T−1∑i=0

zi+13 Pi

x1Px0

0 ≤ ρ ≤ Q timesQ − ρ − 1

times

Figure 6. MSFG for the SR-NACK protocol.

References

1Lin, S., Costello, D. J., and Miller, M. J., “Automatic Repeat-Request Error-Control Schemes,” IEEE Comm. Mag.,Vol. 22, No. 2, December 1984, pp. 5–17.

2Zorzi, M. and Rao, R. R., “Throughput Analysis of ARQ Selective-Repeat Protocol with Time Diversity in MarkovChannels,” Proc. IEEE GLOBECOM , November 1995, pp. 1673–1677.

3Zorzi, M. and Rao, R. R., “Bounds on the Throughput Performance of ARQ Selective-Repeat Protocol in MarkovChannels,” Proc. ICC , 1996, pp. 782–786.

4Zorzi, M., Rao, R. R., and Milstein, L. B., “ARQ Error Control for Fading Mobile Radio Channels,” TV , Vol. 46, No. 2,May 1997, pp. 445–455.

5Rossi, M. and Zorzi, M., “Analysis and Heuristics for the Characterization of Selective Repeat ARQ Delay StatisticsOver Wireless Channels,” TV , Vol. 52, No. 5, September 2003, pp. 1365–1377.

6Zhu, J. and Roy, S., “Improving Link Layer Performance on Satellite Channels with Shadowing Via Delayed Two-CopySelective Repeat ARQ,” JSAC , Vol. 22, No. 3, April 2004, pp. 472–481.

7Ausavapattanakun, K. and Nosratinia, A., “Analysis of Go-Back-N ARQ in Block Fading Channels,” IEEE Tr. WirelessComm., Vol. 6, No. 8, August 2007, pp. 2793–2797.

8Xiao, J., Qiu, J., and Cheng, S., “A Cross-Layer Adaptive Transmission Scheme Over Correlated Fading Channels,”Front. Electr. Electron. Eng. China, Vol. 2, No. 1, 2007, pp. 49–56.

9Ausavapattanakun, K. and Nosratinia, A., “Analysis of Selective-Repeat ARQ via Matrix Signal-Flow Graphs,” IEEETr. Comm., Vol. 55, No. 1, 2007, pp. 198–204.

10Leon-Garcia, A., Probability and Random Processes for Electrical Engineers, Addison-Wesley, 2nd ed., 1994.11Mason, S. J., “About Such Things as Unistors, Flow Graphs, Probability, Partial Factoring and Matrices,” IRE Tr. on

Circ. Th., 1957, pp. 90–97.12Lorenzelli, F., “Modeling the Scintillation Process,” ICSSC , AIAA, 2008.13Yao, Y.-D., “Performance of ARQ and NAK-Based ARQ on a Correlated Fading Channels,” Proc. VTC , 1999, pp.

2706–2710.

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10 12 14SNR

0.7

0.8

0.9

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(a) Channel utilization vs. SNR.

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8 10 12 14SNR

1.0

1.5

2.0

Delay and Latency

(b) Delay (solid), and latency (dashed) vs. SNR.

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10 12 14SNR

1.0

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Delay and Latency Jitter

(c) Delay jitter (solid) and latency jitter (dashed) vs. SNR.

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8 10 12 14SNR

1 ´ 10-4

5 ´ 10-4

0.1000.050

0.0100.005

0.001

Tx and Rx logPHeL

(d) Probability of incorrect communication (Tx, solid) andincorrect reception (Rx, dashed) vs. SNR.

1.0 1.5 2.0 2.5Time

0.1

0.2

0.3

0.4

0.5

Delay PMF

(e) Probability mass function of delay for SNR=5 dB and theshortest time-out.

1.0 1.5 2.0 2.5Time

0.05

0.10

0.15

0.20

0.25

0.30

Latency PMF

(f) Probability mass function of latency, for SNR=5 dB andthe shortest time-out.

Figure 7. Performance curves for the SR-ACK protocol. The red curves correspond to time-outs equal to theroundtrip time, the blue curves to a time-out exceeding the roundtrip time by four packet lengths.

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0.7

0.8

0.9

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(a) Channel utilization vs. SNR.

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1.0

1.5

2.0

Delay and Latency

(b) Delay (solid), and latency (dashed) vs. SNR.

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1.0

1.5

2.0

Delay and Latency Jitter

(c) Delay jitter (solid) and latency jitter (dashed) vs. SNR.

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ò

ò

ò

8 10 12 14SNR

1 ´ 10-4

5 ´ 10-4

0.1000.050

0.0100.005

0.001

Tx and Rx logPHeL

(d) Probability of incorrect communication (Tx, solid) andincorrect reception (Rx, dashed) vs. SNR.

1.0 1.5 2.0 2.5Time

0.1

0.2

0.3

0.4

0.5

Delay PMF

(e) Probability mass function of delay for SNR=5 dB and theshortest time-out.

1.0 1.5 2.0 2.5Time

0.05

0.10

0.15

0.20

0.25

0.30

Latency PMF

(f) Probability mass function of latency, for SNR=5 dB andthe shortest time-out.

Figure 8. Performance curves for the SR-SACK protocol.

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0.7

0.8

0.9

Η

(a) Channel utilization vs. SNR.

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ì

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8 10 12 14SNR

1.0

1.5

2.0

Delay and Latency

(b) Delay (solid), and latency (dashed) vs. SNR.

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0.4

0.6

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1.6

Delay and Latency Jitter

(c) Delay jitter (solid) and latency jitter (dashed) vs. SNR.

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8 10 12 14SNR

1 ´ 10-4

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0.1000.050

0.0100.005

0.001

Tx and Rx logPHeL

(d) Probability of incorrect communication (Tx, solid) andincorrect reception (Rx, dashed) vs. SNR.

1.0 1.5 2.0 2.5Time

0.1

0.2

0.3

0.4

0.5

Delay PMF

(e) Probability mass function of delay for SNR=5 dB and theshortest time-out.

1.0 1.5 2.0 2.5Time

0.05

0.10

0.15

0.20

0.25

0.30

Latency PMF

(f) Probability mass function of latency, for SNR=5 dB andthe shortest time-out.

Figure 9. Performance curves for the SR-NACK protocol. The red curves correspond to time-outs equal tothe roundtrip time, the blue curves to a time-out exceeding the roundtrip time by four packet lengths.

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