Learning EPON delay models from data: a machine learning approach

José Alberto Hernández; Amin Ebrahimzadeh; Martin Maier; David Larrabeiti

doi:10.1364/JOCN.437414

Journal of Optical Communications and Networking
Vol. 13,
Issue 12,
pp. 322-330
(2021)
•https://doi.org/10.1364/JOCN.437414

Learning EPON delay models from data: a machine learning approach

José Alberto Hernández, Amin Ebrahimzadeh, Martin Maier, and David Larrabeiti

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José Alberto Hernández, Amin Ebrahimzadeh, Martin Maier, and David Larrabeiti, "Learning EPON delay models from data: a machine learning approach," J. Opt. Commun. Netw. 13, 322-330 (2021)

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Abstract

There have been a large number of studies focused on the characterization of the upstream delay in time-division multiplexing passive optical networks (TDM-PONs). However, most of them focus on finding equations for the average delay and ignore other useful metrics like delay percentiles, which are of paramount interest in dimensioning PONs with delay guarantees. This work shows how to learn delay models from data using supervised machine learning (ML) techniques. Essentially, a nonlinear regression ML algorithm is trained with PON simulation data, showing that it can provide accurate equations for such metrics of interest. In particular, we obtain an ${R^2}$ score above 80% under Poisson traffic and above 65% under self-similar traffic, and we provide a general equation for any delay percentile in the upstream channel of a PON employing interleaved polling with adaptive cycle time. We further show its applicability in dimensioning Tactile Internet and 5G transport support scenarios.

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Data Availability

Both the datasets and R code are available at GitHub for the reader to replicate the experiments [42]

42. J. A. Hernández, “nls_for_EPONs,” GitHub (2020) [accessed 25 May 2020], https://github.com/josetilos/nls_for_EPONs/.

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Equations (23)

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Model	Equation $E (D)$	$R^{2}$	$RMSE$
Theo. Eq. (2)	$2 τ \frac{4 - 2 ρ}{1 - ρ} + 2.7 µ s$	0.952	151.5 µs
ML Model (Simple)	$1.07 \cdot 2 τ \frac{2 - ρ}{1 - ρ} + 2.7 µ s$	0.952	148.0 µs
ML Model (Full)	$2 τ \frac{4.45 - 0.51 ρ}{1.02 - ρ} + 2.7 µ s$	0.999	13.6 µs

	1G EPON		10G EPON
Model	Equation	$R^{2}$	Equation	$R^{2}$
ML Model (Simple)	$D_{.25} = 0.8 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.92	$D_{.25} = 0.633 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.823
ML Model (Full)	$D_{.25} = 2 τ \frac{3.9 - 0.36 ρ}{1.03 - ρ} + 2.7$ (µs)	0.999	$D_{.25} = 2 τ \frac{3.46 + 0.97 ρ}{1.07 - ρ} + 2.7$ (µs)	0.995
ML Model (Simple)	$D_{.50} = 1.04 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.95	$D_{.50} = 0.83 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.887
ML Model (Full)	$D_{.50} = 2 τ \frac{4.5 - 0.66 ρ}{1.02 - ρ} + 2.7$ (µs)	0.999	$D_{.50} = 2 τ \frac{4.13 + 0.24 ρ}{1.04 - ρ} + 2.7$ (µs)	0.998
ML Model (Simple)	$D_{.75} = 1.31 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.969	$D_{.75} = 1.03 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.917
ML Model (Full)	$D_{.75} = 2 τ \frac{4.97 - 0.65 ρ}{1.016 - ρ} + 2.7$ (µs)	0.999	$D_{.75} = 2 τ \frac{4.58 + 0.12 ρ}{1.03 - ρ} + 2.7$ (µs)	0.999
ML Model (Simple)	$D_{.90} = 1.60 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.972	$D_{.90} = 1.22 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.940
ML Model (Full)	$D_{.90} = 2 τ \frac{4.87 + 0.17 ρ}{1.014 - ρ} + 2.7$ (µs)	0.997	$D_{.90} = 2 τ \frac{4.85 + 0.01 ρ}{1.02 - ρ} + 2.7$ (µs)	0.999
ML Model (Simple)	$D_{.95} = 1.74 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.966	$D_{.95} = 1.31 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.95
ML Model (Full)	$D_{.95} = 2 τ \frac{4.72 + 0.94 ρ}{1.015 - ρ} + 2.7$ (µs)	0.996	$D_{.95} = 2 τ \frac{4.91 + 0.05 ρ}{1.02 - ρ} + 2.7$ (µs)	0.999
ML Model (Simple)	$D_{.99} = 1.98 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.95	$D_{.99} = 1.45 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.964
ML Model (Full)	$D_{.99} = 2 τ \frac{4.55 + 2.40 ρ}{1.018 - ρ} + 2.7$ (µs)	0.990	$D_{.99} = 2 τ \frac{5.3 - 0.32 ρ}{1.02 - ρ} + 2.7$ (µs)	0.9995

	1G EPON
Model	Equation	$R^{2}$
ML Model (Simple)	$D_{.25} = 6.7 \cdot 2 τ \frac{2 - ρ}{1 - ρ} (µ s)$	0.76
ML Model (Full)	$D_{.25} = 2 τ \frac{- 1.8 + 12.5 ρ}{0.95 - ρ} + 2.7 (µ s)$	0.82
ML Model (Simple)	$D_{.50} = 12.0 \cdot 2 τ \frac{2 - ρ}{1 - ρ} (µ s)$	0.75
ML Model (Full)	$D_{.50} = 2 τ \frac{- 0.23 + 21.8 ρ}{0.96 - ρ} + 2.7 (µ s)$	0.78
ML Model (Simple)	$D_{.75} = 21.4 \cdot 2 τ \frac{2 - ρ}{1 - ρ} (µ s)$	0.66
ML Model (Full)	$D_{.75} = 2 τ \frac{7.31 + 32.0 ρ}{0.97 - ρ} + 2.7 (µ s)$	0.68
ML Model (Simple)	$D_{.90} = 28.5 \cdot 2 τ \frac{2 - ρ}{1 - ρ} (µ s)$	0.71
ML Model (Full)	$D_{.90} = 2 τ \frac{27.6 + 25.0 ρ}{0.97 - ρ} + 2.7 (µ s)$	0.72
ML Model (Simple)	$D_{.95} = 31.6 \cdot 2 τ \frac{2 - ρ}{1 - ρ} (µ s)$	0.75
ML Model (Full)	$D_{.95} = 2 τ \frac{41.5 + 17.8 ρ}{0.97 - ρ} + 2.7 (µ s)$	0.76
ML Model (Simple)	$D_{.99} = 37.4 \cdot 2 τ \frac{2 - ρ}{1 - ρ} (µ s)$	0.74
ML Model (Full)	$D_{.99} = 2 τ \frac{77.3 - 19.9 ρ}{0.97 - ρ} + 2.7 (µ s)$	0.76

Model	Equation $E (D)$	$R^{2}$	$RMSE$
Theo. Eq. (2)	$2 τ \frac{4 - 2 ρ}{1 - ρ} + 2.7 µ s$	0.952	151.5 µs
ML Model (Simple)	$1.07 \cdot 2 τ \frac{2 - ρ}{1 - ρ} + 2.7 µ s$	0.952	148.0 µs
ML Model (Full)	$2 τ \frac{4.45 - 0.51 ρ}{1.02 - ρ} + 2.7 µ s$	0.999	13.6 µs

	1G EPON		10G EPON
Model	Equation	$R^{2}$	Equation	$R^{2}$
ML Model (Simple)	$D_{.25} = 0.8 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.92	$D_{.25} = 0.633 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.823
ML Model (Full)	$D_{.25} = 2 τ \frac{3.9 - 0.36 ρ}{1.03 - ρ} + 2.7$ (µs)	0.999	$D_{.25} = 2 τ \frac{3.46 + 0.97 ρ}{1.07 - ρ} + 2.7$ (µs)	0.995
ML Model (Simple)	$D_{.50} = 1.04 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.95	$D_{.50} = 0.83 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.887
ML Model (Full)	$D_{.50} = 2 τ \frac{4.5 - 0.66 ρ}{1.02 - ρ} + 2.7$ (µs)	0.999	$D_{.50} = 2 τ \frac{4.13 + 0.24 ρ}{1.04 - ρ} + 2.7$ (µs)	0.998
ML Model (Simple)	$D_{.75} = 1.31 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.969	$D_{.75} = 1.03 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.917
ML Model (Full)	$D_{.75} = 2 τ \frac{4.97 - 0.65 ρ}{1.016 - ρ} + 2.7$ (µs)	0.999	$D_{.75} = 2 τ \frac{4.58 + 0.12 ρ}{1.03 - ρ} + 2.7$ (µs)	0.999
ML Model (Simple)	$D_{.90} = 1.60 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.972	$D_{.90} = 1.22 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.940
ML Model (Full)	$D_{.90} = 2 τ \frac{4.87 + 0.17 ρ}{1.014 - ρ} + 2.7$ (µs)	0.997	$D_{.90} = 2 τ \frac{4.85 + 0.01 ρ}{1.02 - ρ} + 2.7$ (µs)	0.999
ML Model (Simple)	$D_{.95} = 1.74 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.966	$D_{.95} = 1.31 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.95
ML Model (Full)	$D_{.95} = 2 τ \frac{4.72 + 0.94 ρ}{1.015 - ρ} + 2.7$ (µs)	0.996	$D_{.95} = 2 τ \frac{4.91 + 0.05 ρ}{1.02 - ρ} + 2.7$ (µs)	0.999
ML Model (Simple)	$D_{.99} = 1.98 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.95	$D_{.99} = 1.45 \cdot 2 τ \frac{2 - ρ}{1 - ρ}$ (µs)	0.964
ML Model (Full)	$D_{.99} = 2 τ \frac{4.55 + 2.40 ρ}{1.018 - ρ} + 2.7$ (µs)	0.990	$D_{.99} = 2 τ \frac{5.3 - 0.32 ρ}{1.02 - ρ} + 2.7$ (µs)	0.9995

José Alberto Hernández	https://orcid.org/0000-0002-9551-4308
Amin Ebrahimzadeh	https://orcid.org/0000-0002-1741-677X
Martin Maier	https://orcid.org/0000-0002-7035-3915
David Larrabeiti	https://orcid.org/0000-0003-4983-0243

Abstract

Data Availability

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Figures (5)

Tables (3)

Equations (23)

Journal of Optical Communications and Networking