Joint Network-and-Server Congestion in Multi-Source Traffic Allocation: A Convex Formulation and Price-Based Decentralization

TL;DR

This work tackles a joint network-and-server congestion problem where end-to-end delay is driven by rate-dependent access delays $D_{ij}(\lambda_{ij})$ and load-dependent server delays $D_j(\Lambda_j)$. It proves the problem is convex and derives a Wardrop-type optimality condition using total marginal costs $C_{ij}(\lambda_{ij})$ and $C_j(\Lambda_j)$, enabling a light-weight, price-based decentralization in which each service node broadcasts a scalar price $p_j$ and sources solve simple separable convex subproblems. The proposed distributed algorithm converges to the centralized optimum under standard stepsize conditions, reproducing the same load allocations and marginal-cost equalization as the centralized solution. The framework applies to diverse distributed systems (e.g., HetNets, Cloud-RAN/vRAN, pub-sub brokers, edge-to-cloud offloading) and provides a provable performance benchmark with interpretable signaling, facilitating evaluation of heuristic or data-driven approaches in practice.

Abstract

This paper studies an important rate allocation problem that arises in many networked and distributed systems: steady-state traffic rate allocation from multiple sources to multiple service nodes when both (i) the access-path delay on each source-node route is rate-dependent (capacity-constrained) and convex, and (ii) each service node (also capacity-constrained) experiences a load-dependent queueing delay driven by aggregate load from all sources. We show that the resulting flow-weighted end-to-end delay minimization is a convex program, yielding a global system-optimal solution characterized by KKT conditions that equalize total marginal costs (a path marginal access term plus a node congestion price) across all utilized routes. This condition admits a Wardrop-type interpretation: for each source, all utilized options equalize total marginal cost, while any option with strictly larger total marginal cost receives no flow. Building on this structure, we develop a lightweight distributed pricing-based algorithm in which each service node locally computes and broadcasts a scalar congestion price from its observed aggregate load, while each source updates its traffic split by solving a small separable convex allocation problem under the advertised prices. Numerical illustrations demonstrate convergence of the distributed iteration to the centralized optimum and highlight the trade-offs induced by jointly modeling access and service congestion.

Figures (8)

• Figure 1: Illustration of the general formulation (paper notation). Each source $i$ with rate $\lambda_i$ splits traffic into $\lambda_{ij}$ over servers $j$. Each path has capacity $\mu_{ij}$ and experiences a convex increasing access delay $D_{ij}(\lambda_{ij})$. Each server $j$ experiences load $\Lambda_j=\sum_i \lambda_{ij}$ and a convex increasing delay $D_j(\Lambda_j)$, with capacity $\mu_j$.
• Figure 2: Distributed flow-weighted: objective value $F(\lambda^{(t)})$ versus iteration.
• Figure 3: Distributed pricing iteration: per-SN utilization trajectories $\Lambda_j^{(t)}/\mu_j$ over iterations (dashed = distributed final reference).
• Figure 4: Distributed pricing iteration: per-source mean delay trajectories versus iteration.
• Figure 5: Wardrop (KKT) check via total marginal costs $M_{ij}=C_{ij}(\lambda_{ij})+C_j(\Lambda_j)$. For each source, all used routes (filled markers) align closely in total marginal cost, while unused routes (hollow markers) have no smaller cost, consistent with \ref{['eq:wardrop_structure']}. Top: centralized optimum. Bottom: distributed fixed point, showing the same marginal-cost equalization pattern.