[Paper Review] Optimization Results for 5G Slice-in-Slice Scheduling

Sharvari Ravindran, Saptarshi Chaudhuri+2 more|arXiv (Cornell University)|Jun 28, 2021

Software-Defined Networks and 5G4 citations

TL;DR

This paper presents novel optimization results for 5G slice-in-slice scheduling in Open RAN (ORAN) networks, leveraging graph coloring and probabilistic resource allocation to maximize network slice capacity. It proves that slice capacity tends to zero as relative resource allocation per user service approaches zero under bounded dependency and variance conditions, establishing foundational limits for SLA-compliant RAN slicing.

ABSTRACT

Open Radio Access Network (ORAN) Slicing for 5G and Beyond is an emerging architecture and feature that will facilitate challenging RAN Service Level Agreement (SLA) assurance targets. This could pave the way for operators to realize the benefits of network slicing efficiently. In this paper, we provide novel and detailed optimization results to achieve Slice-in-Slice Scheduling for 5G User Services in ORAN slicing.

Motivation & Objective

• To address the challenge of achieving reliable Service Level Agreement (SLA) enforcement in 5G Open RAN network slicing.
• To model and optimize slice-in-slice scheduling by maximizing user service capacity within constrained system resources.
• To analyze the theoretical limits of network slice capacity under diminishing per-user resource allocation.
• To apply graph coloring and probabilistic methods to model dependencies and resource allocation in multi-category slice-in-slice architectures.

Proposed method

• Formulates network slice capacity as the sum of maximum user services per slice-in-slice category, constrained by total system resource limits.
• Uses random variables for per-user resource allocation and defines relative allocation as $ A_i = R_i / r_{\text{max}} $, with expected value $ a $.
• Applies the Bienaymé–Chebyshev inequality and law of total variance to bound the variance of total relative resource allocation $ R $.
• Introduces dependency constraints via bounded mean $ a' $, variance $ \sigma_a^2 $, and covariance $ c_{a'} $ for inter-user service resource allocation.
• Models throughput using a non-homogeneous Poisson process with utility-based throughput function $ u_i(t) = \frac{f_d e^{\beta r_i s_i}}{\Delta t} $.
• Applies chromatic polynomial theory to $ k $-partite graphs, showing that $ k(k-1)^{n-1} $ colors are required for permutation coloring in $ n $-partite graphs.

Experimental results

Research questions

• RQ1 What is the theoretical limit of network slice capacity as per-user resource allocation tends to zero?
• RQ2 How do dependencies between user services affect the variance and stability of total resource allocation?
• RQ3 Can graph coloring models accurately represent and optimize resource allocation in slice-in-slice architectures?
• RQ4 What is the impact of bounded variance and covariance on the convergence of system resource allocation?
• RQ5 How does the chromatic polynomial of a $ k $-partite graph relate to optimal color (resource) assignment in multi-category slicing?

Key findings

• Network slice capacity $ \mathcal{C} $ tends to zero as the relative system resource allocation per user service $ \rightarrow 0 $, under bounded mean $ a' $, variance $ \sigma_a^2 $, and covariance $ c_{a'} $.
• Variance of total relative allocation $ V[R] \rightarrow 0 $ as $ a \rightarrow 0 $ and $ a' \rightarrow 0 $, provided $ \sigma_a^2 \cdot (g/a)^2 \rightarrow 0 $ and $ c_{a'} \cdot E[U^4 - U^2] \rightarrow 0 $.
• The system achieves non-zero SLA probability $ P[T \leq r_{\text{max}}] > 0 $ even at maximum capacity, due to bounded variance and dependency constraints.
• For an $ n $-partite graph, the permutation coloring strategy requires $ k(k-1)^{n-1} $ colors, where $ k $ is the number of available colors.
• The optimal resource allocation $ r_i $ is derived by maximizing the derivative of the probability distribution function with respect to SNR $ s_i $.
• The probability distribution function for user service throughput is expressed as a sum of exponential and polynomial terms involving $ \sum_{i=1}^n i e^{\beta r_i s_i} $, with closed-form expressions derived for conditional probabilities.