On the Optimal Delay Amplification Factor ofMulti-Hop Relay ChannelsDennis Ogbe, Chih-Chun Wang, and David J. LoveSchool of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana, USA

The relay channel is a classic information theory problemwhich has experienced a renewed surge of interest due toits applicability to a vast variety of modern communica-tion systems (IoT, integrated access & backhaul, etc.)

Many practical scenarios can be modeled as a simple sep-arated relay channel. When three or more channels areconcatenated, we speak of a multi-hop relay channel.Simple min-cut analysis shows that for multi-hop relaysC = min

({Cl}Ll=1

), which is achievable by the decode-&-

forward (DF) policy.

The recent push toward low latency communications(sub-1ms in IMT-2020 URLLC) motivates an investiga-tion into the delay-throughput tradeoff for fixed errorprobabilities—finite blocklength theory.

Our work in [1] devised a new relaying scheme calledtranscoding (TC) that substantially outperforms bothDF and amplify-&-forward (AF) in the finite blocklengthregime. However, while transcoding outperforms DF forshort block lengths, DF still has the upper hand in theasymptotic regime. To address this problem, this workpresents two new schemes based on transcoding whichoutperform DF in a delay-throughput sense in the regimeof asymptotically large, yet still finite block lengths.

The main tool in showing this performance increase is ourdefinition of the Delay Amplification Factor (DAF), a mea-sure of the multiplicative increase in delay when the fullnetwork is compared to the bottleneck link. An intuitiveexplanation of the DAF is that it is essentially the ratio ofthe end-to-end delay from source to destination to thedelay over the bottleneck hop alone.

We analyze those delays using error exponents. We obtainthe delay over the bottleneck hop from Gallager’s randomcoding error exponent. The end-to-end delay depends onthe specific relaying scheme used and is analyzed by de-riving the end-to-end error exponent from a descriptionof a relaying scheme. The DAF is then equivalently formu-lated as the limit of the ratio of the bottleneck and theend-to-end error exponents.

We show that the DAF for DF schemes is strictly largerthan 1. For AF schemes, the DAF does not apply, since theyare not capacity achieving.

In contrast to this, our first main result is an open-looptranscoding scheme that achieves DAF=1 when the bot-tleneck hop is the last hop (l∗ = L). The proof hingeson a careful construction of a concatenated code andmaximum-likelihood joint decoding at the destination

The second main result is a one-time stop feedbackscheme that achieves DAF=1 regardless of the position ofthe bottleneck l∗ 6= L. The proof builds on the idea thatthe relay knows which micro-blocks it decoded wrong af-ter forwarding all of them.

Using result #1, we can construct a sub-optimal(1 ≤ DAFΦ′ ≤ DAFDF) open-loop scheme thatoutperforms DF for l∗ 6= L. Our work in progress isinvestigating open-loop schemes that can do better.

s r d s r1 r2 · · · rL−1 s

DAFΦ , limR↗C

limε→0

Te2e(R, ε)

Tbn(R, ε)

DAFΦ , limR↗C

Erc,l∗(R)

EΦ(R)

Te2e(R, ε) & −1

EΦ(R)log(ε)

EΦ(R): “Error exponent of relayingscheme Φ”

I Coding schemeI Channel transition probabilitiesI Operation at relays

End-to-end error exponent

Main Result #1:DAFTC = 1 if l∗ = L

Main Result #2:DAFTC′ = 1 if l∗ 6= L

Thro

ughp

ut

Delay

DFAFTC

[1] C.-C. Wang, D. J. Love, and D. Ogbe, “Transcoding: A new strategy for relay channels,” Allerton 2017, Oct. 2017.[2] D. Ogbe, C.-C. Wang, and D. J. Love, “On the Optimal Delay Amplification Factor of Multi-Hop Relay Channels,” ISIT 2019, July 2019.

Motivation:

Delay Amplification Factor:

Main Results:

Publications:

separated separated multi-hop

IoT autonomousvehicles

MTC

URLLC: sub 1ms! {

end-to-end delay Te2e(R, ε)

bottleneck delay Tbn(R, ε)

C1 CLCl∗s r1 r2 · · · rL−1 s

Error exponent approximation

Tbn(R, ε) & − 1

Erc,l∗(R)log(ε)

ε ≤ exp [−nE(R)]

E(R) , limn→∞

− log(ε)

n

R

Erc(R)BSCC ≈ 0.5

Hop 1

Hop 2

Hop 3

Hop 4

pipelining!

Tdur

τ2

τ3

τ4

msgenc

msgdec

Feasible (T,R, ε): (i) T ≥ τL + Tdur, (ii)R ≤ log(|M|)Tdur

, (iii) ε ≥ P (M 6= M̂)

ThenEΦ(R) , lim supT→∞

supε:(T,R,ε)is feasible

− log(ε)

T

Cyclically shi�ed outer code (CSOC)M

Inner ENCInner ENC Inner ENC

∆Hop 1 ∆ ∆

∆ ∆ ∆

∆ ∆ ∆

DF DF DF

DF DF DF

Inner/Outer DEC

Hop 2

Hop 3

M̂

Transmit message usingK micro-blocks @ rateCL, length ∆

Correct errors from last hopusing concatenated code

{{

Cl>CL

CL

Inner ENCInner ENC Inner ENC

∆ ∆ ∆

DF DF DF

DF DF DFCorrection phaseuntil stop feedback

DF

τ2 = ∆

τ3 = 2∆

Sequential Random Permutation Outer Codes

· · ·

M

Hop 1

Hop l−1

Hop lHop l+1

Hop L

...

...

{{{

Cl>Cl∗

Cl>Cl∗

Cl∗

High error(error propagation)

Low error(Cl > Cl∗)

Low error(Relay can decode outer code,knows which blocks are in error)

(Message M ∈M)