[Paper Review] The Global Packing Number for an Optical Network

Yuan–Hsun Lo, Yijin Zhang+2 more|arXiv (Cornell University)|Sep 23, 2015

Advanced Optical Network Technologies5 references3 citations

TL;DR

This paper introduces the global packing number as a metric for minimizing wavelengths in WDM optical networks, focusing on ring and chain topologies. It derives exact formulas for the global packing number of cycles, characterizes ideal path systems, and proposes the LFP heuristic algorithm that reduces wavelength usage by 3–6% over random assignment, outperforming existing schemes in both uniform and random traffic models.

ABSTRACT

The global packing number problem arises from the investigation of optimal wavelength allocation in an optical network that employs Wavelength Division Multiplexing (WDM). Consider an optical network that is represented by a connected, simple graph $G$. We assume all communication channels are bidirectional, so that all links and paths are undirected. It follows that there are ${|G|\choose 2}$ distinct node pairs associated with $G$, where $|G|$ is the number of nodes in $G$. A path system $\mathcal{P}$ of $G$ consists of ${|G|\choose 2}$ paths, one path to connect each of the node pairs. The global packing number of a path system $\mathcal{P}$, denoted by $Φ(G,\mathcal{P})$, is the minimum integer $k$ to guarantee the existence of a mapping $ω:\mathcal{P} o\{1,2,\ldots,k\}$, such that $ω(P) eqω(P')$ if $P$ and $P'$ have common edge(s). The global packing number of $G$, denoted by $Φ(G)$, is defined to be the minimum $Φ(G,\mathcal{P})$ among all possible path systems $\mathcal{P}$. If there is no wavelength conversion along any optical transmission path for any node pair in the network, the global packing number signifies the minimum number of wavelengths required to support simultaneous communication for all pairs in the network. In this paper, the focus is on ring networks, so that $G$ is a cycle. Explicit formulas for the global packing number of a cycle is derived. The investigation is further extended to chain networks. A path system, $\mathcal{P}$, that enjoys $Φ(G,\mathcal{P})=Φ(G)$ is called ideal. A characterization of ideal path systems is also presented. We also describe an efficient heuristic algorithm to assign wavelengths that can be applied to a general network with more complicated traffic load.

Motivation & Objective

• To define and compute the global packing number, a metric that determines the minimum number of wavelengths required to support all pairwise connections in a WDM optical network.
• To characterize the global packing number for cycle graphs (ring networks), providing exact closed-form expressions.
• To identify ideal path systems—those achieving the minimum global packing number—and characterize their structural properties.
• To extend the framework to chain networks and general traffic patterns, including random and full-random connection models.
• To develop and evaluate the LFP (Longest First Packing) heuristic algorithm for efficient wavelength assignment in practical network scenarios.

Proposed method

• The global packing number Φ(G) is defined as the minimum k such that a path system P of all node pairs can be assigned wavelengths from {1, ..., k} without edge conflicts.
• For cycles, the paper derives explicit formulas: Φ(C_n) = n(n-1)/4 when n is odd, and Φ(C_n) = n(n-2)/4 when n is even.
• An ideal path system is one that achieves Φ(G, P) = Φ(G), and the paper characterizes such systems using graph packing and path selection strategies.
• The LFP algorithm is introduced as a greedy heuristic: paths are sorted by length (longest first), and each is assigned the smallest available wavelength not used on any shared edge.
• The method is extended to chain networks and tested under three traffic models: uniform, full-random, and quasi-random, with shortest paths selected for each pair.
• Performance is evaluated via simulations across ring networks of size n = 5 to 40, comparing LFP and RP (Random Packing) schemes in terms of average wavelength usage.

Experimental results

Research questions

• RQ1 What is the exact value of the global packing number for a cycle graph C_n, and how does it vary with the parity of n?
• RQ2 What structural properties define an ideal path system that achieves the global packing number in a cycle?
• RQ3 How does the LFP algorithm compare to random wavelength assignment (RP) in terms of wavelength efficiency under uniform and non-uniform traffic?
• RQ4 Can the LFP algorithm be generalized to chain networks and other topologies with arbitrary traffic patterns?
• RQ5 What is the performance gap between the optimal global packing number and practical heuristic assignments in real-world traffic scenarios?

Key findings

• For odd n, the global packing number of a cycle C_n is exactly n(n-1)/4; for even n, it is n(n-2)/4.
• The LFP algorithm requires only 3% to 6% more wavelengths than the optimal solution in the uniform traffic model, significantly outperforming the RP scheme.
• In full-random traffic, LFP uses less than 98% of the wavelengths required by RP for n ≥ 15, and less than 93% in quasi-random traffic.
• For n = 40, the optimal global packing number is 201, while LFP uses 209.02 on average, and RP uses 227.20.
• The LFP algorithm consistently outperforms RP across all tested network sizes and traffic models, demonstrating robustness and scalability.
• The paper establishes that ideal path systems exist and can be characterized, particularly in cycles, providing a foundation for optimal wavelength assignment in ring networks.