Convexity and Optimization in Deficit Round Robin Scheduling for Delay-Constrained Systems

TL;DR

This work tackles the challenge of configuring Deficit Round Robin quanta to meet delay constraints across multiple data flows. It introduces a modified delay bound to prove that the feasible set of quanta is convex, first for the two-flow case and then for general $n$ flows, enabling efficient optimization. The authors formulate convex optimization problems to maximize the total quanta served per DRR round while satisfying per-flow delay targets, and provide fixed-point algorithms with existence and uniqueness guarantees for computing the optimal quanta. Simulations using floor-rounded quanta demonstrate that the delay requirements remain satisfied under the conservative bound, highlighting practical applicability and pointing to future work on robust integer guarantees and tighter integration with network slicing QoS.

Abstract

The Deficit Round Robin (DRR) scheduler is widely used in network systems for its simplicity and fairness. However, configuring its integer-valued parameters, known as quanta, to meet stringent delay constraints remains a significant challenge. This paper addresses this issue by demonstrating the convexity of the feasible parameter set for a two-flow DRR system under delay constraints. The analysis is then extended to n-flow systems, uncovering key structural properties that guide parameter selection. Additionally, we propose an optimization method to maximize the number of packets served in a round while satisfying delay constraints. The effectiveness of this approach is validated through numerical simulations, providing a practical framework for enhancing DRR scheduling. These findings offer valuable insights into resource allocation strategies for maintaining Quality of Service (QoS) standards in network slicing environments.

Figures (4)

• Figure 1: The system as defined has $n$-sources, that are bounded by leaky bucket sources (parameterized with $r_i, b_i$), and a DRR-scheduler with constant rate $c$, such that the quanta are $q_i^{th}$ for $i^{th}$ queue.
• Figure 2: The shaded region in blue, red and green shows the points that belong to the sets $B_1, B_2$ and $A$. The blue, red and green lines denote the equations $q_2 = h_1^1(q_1), q_2 = h_2^1(q_1), a(q_1, q_2) = 0$ respectively, with the parameter values given by $b_1 = 10, L = 1, C = 25, d_1 = 1$. The Brown region denotes the set $\{A \cup B_1\}$ and the small pink region denotes the set $\{A^c \cup B_2\}$.
• Figure 3: Trajectory of $q_1$ under Algorithm \ref{['algo-two-flow-case']}, illustrating convergence to the optimal value $q_1^* = 3.181$. The parameters for this case are $b = [10, 15]$, $r = [1, 2]$, $d = [1, 0.5]$, $c = 50$, and $L = 1$, with an initial guess of $x_0 = 1$.
• Figure 4: Trajectory of $q_1$ under Algorithm \ref{['algo-two-flow-case']}, illustrating convergence to the optimal value $\theta^* = 20.312$. The parameters for this case are $b = [10, 15, 10]$, $r = [1, 2, 1]$, $d = [1, 1, 0.5]$, $c = 100$, and $L = 1$, with an initial guess of $\theta_0 = 0.5$.

Theorems & Definitions (29)

• Lemma 1: Modified Delay Bound
• proof
• Lemma 2
• proof
• Remark 1
• Lemma 3
• proof
• Lemma 4
• proof
• Remark 2