Statistical Analysis and Modeling for Optical Networks

Abstract

Optical networks serve as the backbone of modern communication, requiring statistical analysis and modeling to optimize performance, reliability, and scalability. This review paper explores statistical methodologies for analyzing network characteristics, dimensioning, parameter estimation, and cost prediction of optical networks, and provides a generalized framework based on the idea of convex areas, and link length and shortest path length distributions. Accurate dimensioning and cost estimation are crucial for optical network planning, especially during early-stage design, network upgrades, and optimization. However, detailed information is often unavailable or too complex to compute. Basic parameters like coverage area and node count, along with statistical insights such as distribution patterns and moments, aid in determining the appropriate modulation schemes, compensation techniques, repeater placement, and in estimating the fiber length. Statistical models also help predict link lengths and shortest path lengths, ensuring efficiency in design. Probability distributions, stochastic processes, and machine learning improve network optimization and fault prediction. Metrics like bit error rate, quality of service, and spectral efficiency can be statistically assessed to enhance data transmission. This paper provides a review on statistical analysis and modeling of optical networks, which supports intelligent optical network management, dimensioning of optical networks, performance prediction, and estimation of important optical network parameters with partial information.

1. Introduction

Optical networks are the foundation of modern communication infrastructure, enabling high-speed data transmission with minimal latency and signal degradation [1]. As global demand for bandwidth continues to surge due to the proliferation of cloud computing, 5G networks, and the Internet of Things (IoT), the need for efficient and reliable optical networks has become paramount. Unlike traditional copper-based transmission, optical fiber technology leverages light pulses to carry data across vast distances with superior spectral efficiency and lower attenuation [1]. However, maintaining optimal performance in such networks presents several challenges, including signal impairments due to attenuation, dispersion, and nonlinear effects. Moreover, unpredictable network traffic, environmental influences, and hardware limitations necessitate advanced analytical techniques to ensure seamless communication [2]. Statistical analysis and modeling play a crucial role in addressing these challenges by providing data-driven insights into network behavior, predicting failures, and optimizing resource allocation [3,4,5]. Through probabilistic modeling, statistical analysis, and machine learning approaches, researchers and engineers can develop strategies to enhance the performance, reliability, and adaptability of optical networks [4].

Statistical analysis and modeling are indispensable in network science and communication engineering [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. They provide the tools to understand and predict the behavior of complex communication systems and channels [5,6,7,8,9,10,11,12,13,14]. Statistical analysis helps characterize noise, interference, and signal variations, which is crucial for designing robust and reliable systems [1,2,3]. Modeling, using statistical frameworks, enables the visualization and performance evaluation of communication techniques before costly real-world implementation [1,2]. This includes channel modeling, traffic modeling, and network performance analysis, leading to optimized resource allocation and improved quality of service (QoS) [1,2,24,25]. Ultimately, these methods drive innovation and efficient design in modern communication technologies. Statistical analysis and modeling are equally critical in optical communication engineering for several reasons [3,4]. They are essential for characterizing and mitigating noise sources inherent in optical systems, such as shot noise and thermal noise, ensuring reliable signal detection. Statistical channel modeling helps researchers understand and predict the impact of various transmission impairments like attenuation, dispersion, and nonlinear effects on optical signals as they propagate through fibers or free space [1,2]. This allows for the design of effective compensation techniques. Statistical analysis and modeling are vital for designing and managing efficient optical communication networks. They enable the characterization of network traffic patterns, which is crucial for resource allocation and QoS management. Statistical modeling helps predict network performance under varying loads and potential failures, facilitating proactive network planning and optimization [7,10,11,12,13]. In optical communication network dimensioning, statistical analysis and modeling are paramount [1,3]. They enable accurate forecasting of future bandwidth demands based on historical data and trends. Statistical traffic models help determine the required capacity of network links and nodes, optimizing resource allocation and minimizing over-provisioning [2,3]. Furthermore, statistical analysis of network performance under varying traffic loads and potential failure scenarios informs the design of resilient and cost-effective network architectures, ensuring that the network can meet service level agreements while remaining economically viable [2]. Performance evaluation of optical communication systems heavily relies on statistical methods to analyze key metrics like bit error rate (BER), quality factor, and signal-to-noise ratio (SNR) [1,2,3]. These analyses guide the optimization of system parameters and the development of robust and high-capacity optical networks. Furthermore, analyzing network data statistically aids in identifying bottlenecks, improving network resilience, and ensuring the reliable delivery of high-bandwidth services in complex optical infrastructures. Statistical techniques offer powerful tools for evaluating and optimizing various network parameters [3,4,5,6,7,8,9,10,11,12,13,14,15,16]. For instance, probability distributions help model signal fluctuations and noise variations, allowing network operators to predict and mitigate signal degradation [2]. Stochastic processes, including Markov models and queuing theory, provide valuable insights into packet transmission dynamics, helping to optimize bandwidth allocation and reduce service delays. In addition to that, machine learning and regression models are increasingly being integrated into network management to enhance fault prediction and proactive maintenance [2]. These methods analyze large-scale datasets generated by optical networks to identify patterns and correlations that may indicate potential failures, allowing for preventive measures before critical disruptions occur. By leveraging statistical analysis, network operators can implement adaptive control mechanisms that dynamically adjust transmission parameters to maintain high performance in real-time scenarios [3].

Despite the advantages of statistical modeling in optical networks, several challenges remain. The sheer volume and complexity of data generated by high-capacity optical systems require efficient computational frameworks for real-time analysis [2]. Moreover, external environmental factors, such as temperature fluctuations, fiber bending, and mechanical stress, introduce uncertainties that must be accurately accounted for in predictive models. Integrating artificial intelligence (AI) with statistical techniques is a promising avenue for overcoming these challenges, enabling automated network monitoring and self-healing mechanisms [2]. Furthermore, emerging technologies, such as quantum statistical models and edge computing, are poised to revolutionize optical network management by enhancing security, reducing latency, and improving decision-making capabilities at distributed network nodes. As optical networks continue to evolve, the synergy between statistical analysis, machine learning, and intelligent automation will play a critical role in shaping the future of high-speed, resilient, and adaptive communication systems [2]. Statistical analysis and modeling are essential for ensuring the network requirements, reliability, efficiency, and scalability of optical networks [4]. By leveraging various statistical techniques, including time series analysis, probability distributions, stochastic processes, and machine learning, network operators can optimize performance, predict failures, and enhance overall network resilience [3]. As technology advances, integrating AI-driven and quantum statistical models will further revolutionize optical network management, paving the way for more intelligent and adaptive communication infrastructures [2].

In this paper, we show the importance of statistical analysis and modeling for optical networks, with the primary goal of outlining the current state-of-the-art in optical network analysis and modeling. We analyze the main statistical tools used for the optical networks, highlighting the key techniques and the topics explored. We show that these tools provide essential insights about the optical networks and their performance aspects. Its purpose is to explain the key statistical techniques used to analyze and model modern optical networks. It also demonstrates the utility of these techniques in the engineering aspects of modern optical networks. By leveraging statistical analysis and modeling, we aim to gain insights into the complex dynamics of optical networks, enabling the design of more efficient, reliable, and scalable communication systems.

The remainder of this survey has been arranged in six sections. Section 2 provides a literature review of the statistical analysis and modeling of networks. The purpose of Section 3 is to present the methods and materials central to this paper. The purpose of Section 4 is to lay out the fundamental concepts and vital components that underpin modern optical networks. Section 5 details the statistical analysis and modeling employed for optical networks. Section 6 presents the applications of statistical analysis and modeling in optical networks, followed by a discussion. Section 7 presents concluding remarks and the future research directions on the chosen topic. We have added a table below to make it easier for the readers to navigate through the rest of the paper.

2. Literature Review on Statistical Analysis and Modeling for Networks

Statistical analysis and modeling of networks is essential to understand their characteristics and nature. In recent times, it has emerged as a critical field for understanding complex systems across diverse domains, employing various methodologies to decipher underlying structures and dynamics. Researchers have focused on characterizing network topologies through statistical properties like degree distributions, clustering coefficients, and path lengths, revealing insights into different characteristics, such as small-world and scale-free phenomena [5]. In order to explain the formation of observed network structures, statistical models such as Exponential Random Graph Models and latent space models are utilized, inferring social processes and spatial influences on network evolution [6]. Community detection, crucial for functional understanding, employs statistical approaches like stochastic block models to identify densely interconnected node groups [7]. Furthermore, dynamic network analysis addresses the temporal evolution of real-world networks, developing models to capture growth, adaptation, and response to external stimuli [8]. Foundational texts and comprehensive surveys provide a robust framework for understanding these methodologies, covering topics from basic statistical network measures to advanced modeling techniques for both static and dynamic networks [9,10,11,12,13,14]. These advancements in statistical network analysis are vital for extracting meaningful information from the increasingly complex interconnected systems that define our contemporary world.

2.1. Essence of Statistical Analysis and Modeling for Communication Networks

In today’s world, telecommunications networks play a fundamental role in both personal and business communications. With the rapid expansion of the Internet, telecommunications now support essential global interactions [1,2,3,4]. The backbone of this connectivity is high-speed core networks, which are predominantly optical due to their unparalleled bandwidth capacity and ability to handle massive traffic loads [1,2,3,4]. Long-distance optical communication systems have evolved into various forms, supporting core networks, high-speed metro and access networks, and inter-cloud connectivity. Given their critical role, optical networks must be systematically analyzed in terms of both structure and behavior. Statistical analysis and modeling of network parameters are essential for understanding network performance, traffic patterns, and fault prediction [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. These techniques reveal key behavioral insights and enable accurate estimations of network parameters, even when complete data is unavailable. By leveraging statistical methodologies, researchers and engineers can optimize network design, resource allocation, and efficiency, ensuring that optical networks continue to support the ever-growing demands of modern digital communication [5,6,7,8,9,10,11,12,13,14,15].

Dimensioning and cost estimation of optical networks are crucial for design, planning, and optimization [3,4]. During the early planning stages, network upgrades, and performance optimization, accurate estimation of network parameters is essential for efficiency. However, in many cases, complete data is unavailable, and detailed estimations can be complex and time-consuming. Early-stage available parameters include coverage area and node count, but additional statistical insights, such as distribution patterns and first and second moments of network parameters, greatly enhance network planning and design [26,27,28,29,30]. These insights aid in determining modulation and demodulation schemes, compensation techniques, repeater and amplifier placements, and fiber length requirements. To achieve these goals, statistical models must be developed to accurately predict link length distributions and shortest path lengths [31,32,33]. In optical networks, fast and reliable statistical estimations are widely used due to their effectiveness in cases where gathering additional data is impractical. These estimations play a critical role in network planning, enabling efficient resource allocation, infrastructure development, and operational decision-making in modern optical communication systems.

2.2. Statistics for Networks

The indispensable role of statistics in network analysis and design stems from its capacity to decipher and optimize complex interconnected systems [34,35,36,37]. In the realm of network analysis, statistical methodologies are fundamental for characterizing network topology, discerning inherent patterns, and extracting actionable insights from extensive datasets. Descriptive statistics, for instance, quantify critical network properties, like degree distributions, clustering coefficients, and path lengths, thereby illuminating underlying organizational structures, such as small-world and scale-free networks. The evolution of statistical applications in network analysis and design parallels the progression of network theory and data analysis [34,35,36,37,38]. Initially, statistical methods were deployed in social network analysis to describe and quantify interpersonal relationships [35]. Sociological pioneers, starting with Jacob Moreno and continuing through sociometry in the 1930s, utilized statistical tools to map social interactions [36]. The mid-20th century witnessed the rise of graph theory, providing a mathematical scaffold for network analysis, with statistical techniques employed to assess network connectivity and structure [8,34]. The advent of computer networks in the latter half of the 20th century necessitated the use of statistical methods for analyzing network traffic, performance, and reliability. Optical networks became mainstream beginning in the 1990s, and their role became global. Optical communication techniques became popular, and all the core networks became optical [38,39,40,41,42]. Queueing theory, a probabilistic discipline, was applied to model network congestion and optimize resource allocation [29]. The internet’s exponential growth, coupled with the availability of vast network datasets in the late 20th and early 21st centuries, spurred the development of advanced statistical models, including exponential random graph models and stochastic block models, to capture intricate network structures and dynamics [12,15,43,44].

Statistical modeling, encompassing these techniques, facilitates the understanding of network formation mechanisms and the identification of community structures [15,19,20,21,22,23]. In network design, statistical analysis informs pivotal decisions regarding resource allocation, routing optimization, and network resilience [3]. By analyzing traffic patterns, connection probabilities, and network performance metrics, designers can enhance network efficiency and robustness. Moreover, statistical methods are crucial for predicting network behavior under varying conditions, enabling proactive strategies to mitigate congestion, failures, and security threats [44]. Overall, statistics provides the quantitative foundation for informed decision-making across all facets of modern network analysis and design.

Statistics for Network Analysis, Estimation and Dimensioning

In early-stage planning and dimensioning of optical networks, engineers often work with incomplete or partial information [3,30,31,32,33]. Typically, only basic parameters, such as coverage area and node count, are available. In such cases, statistical modeling becomes essential for network planning and design, enabling fast estimations even with limited data [4,31]. These models help predict compensation techniques at the receiving end, repeater and amplifier placements, and total fiber length requirements. A reliable statistical model of link lengths is crucial for making accurate optical network estimations with minimal information [31,32,33]. To ensure practical applicability, the model must be highly accurate so that its predictions remain trustworthy for real-world deployments. Therefore, rigorous testing on existing optical networks is necessary to validate its performance and reliability [25,45]. Statistical analysis of optical networks must integrate the characteristic behaviors observed in these systems [46]. By refining and validating these models, engineers can optimize resource allocation, improve network efficiency, and enhance overall system performance, ensuring that modern optical networks meet the growing demands of high-speed optical communication [47,48,49,50,51,52,53]. Early optical network design necessitates estimating link-dependent parameters, a challenge when network topology details are incomplete [53]. Consequently, estimation methods relying on partial network configuration knowledge are crucial. This demonstrates that employing a link length statistical model significantly enhances the accuracy of estimating key link-dependent network parameters compared to methods using average link length [31]. By leveraging statistical distributions of link lengths, more realistic and precise parameter estimations are achieved [31,32,33]. This approach addresses the inherent uncertainty in early-stage design, providing a robust foundation for subsequent network planning and optimization, particularly when detailed topology information is scarce [54].

Shortest path lengths are critical in communication network applications, like traffic management and routing, and for early-stage planning [33]. Realistic models of these lengths enable the estimation of CAPEX, OPEX, and MANEX, which are vital for network dimensioning [3,4,31,32,33]. In dynamic optical networks, fluctuating traffic necessitates statistical shortest path length models [55,56,57]. When complete network information is unavailable, as is often the case, these models facilitate parameter estimation [28,58]. They address the challenges of modern optical network complexities, demonstrating the utility of statistical models developed for shortest path length analysis. This approach is essential for effective network management and planning, especially in evolving network environments. Shortest path length calculations are fundamental in optical networks, driving several critical utilities and applications [33]. Primarily, they enable efficient routing of optical signals, minimizing latency and maximizing network throughput [59,60,61,62,63,64,65,66,67,68]. This is vital for time-sensitive applications, like high-frequency trading and real-time data streaming. Furthermore, these calculations aid in network design and optimization, allowing engineers to determine optimal fiber cable layouts and node placements [33]. They also support network resilience by enabling the identification of alternate routes in case of failures, ensuring uninterrupted service [60,61,62,63]. In addition, shortest path data is used in network management for traffic engineering, balancing loads, and preventing congestion [33,59,60]. In essence, it is a core tool for ensuring optical networks operate reliably and efficiently.

2.3. Statistics and Network Characterization

Complex networks are frequently observed across many sectors, such as biology, engineering, medical science, psychology, astronomy, and mathematics. The statistical analysis and modeling of complex networks has become a cornerstone in understanding diverse systems, from biological interactions to technological infrastructures, due to their inherent complexity and non-trivial topologies. These networks, composed of interconnected nodes and edges, exhibit emergent properties that necessitate statistical approaches for their characterization. Key statistical measures, such as degree distributions, clustering coefficients, and average path lengths, are employed to uncover fundamental structural properties, revealing phenomena like small-world effects and scale-free behavior, which deviate significantly from random network models [5]. Statistical modeling techniques are crucial for inferring network formation mechanisms and providing insights into the underlying processes that shape network evolution [6]. Further analysis includes the exploration of network dynamics, where temporal changes in network structure are modeled to understand network resilience, propagation phenomena, and adaptation to external stimuli [44]. Moreover, the application of statistical inference and machine learning techniques enables the prediction of network behavior and the identification of influential nodes, contributing to practical applications in network design and optimization [20,21,22,23,69]. The synthesis of these statistical methodologies allows for a comprehensive understanding of complex network behaviors, offering a powerful toolset for researchers across various disciplines to unravel the intricacies of interconnected systems [7].

Statistical distributions are fundamental tools for modeling and analyzing the complex behaviors, network requirements, and predictive modeling in optical networks [27,28,29,30,31,32,33]. The General Extreme Value (GEV) distribution in particular excels at characterizing link length variations, a critical aspect of network planning and optimization [28,31]. Gaussian distributions are commonly employed to represent additive noise in optical signals, while log-normal, Gamma-Gamma, and K-distributions accurately model intensity fluctuations, especially in free-space optical communication, where atmospheric turbulence plays a significant role [1,2,3,4,5,31,32,33]. The nodal degrees of several optical networks follow the Poisson’s distribution [30,46]. For resource allocation and reach-dependent modeling, statistical distributions help predict network demand patterns, enabling efficient resource allocation and capacity planning [70,71]. Furthermore, distributions like the exponential and Weibull are essential for reliability analysis, modeling the failure rates of optical components and systems, contributing to robust network design [31,32,33,55]. Understanding these distributions allows engineers to predict network performance, optimize resource allocation, and ensure network reliability, ultimately leading to more efficient and resilient optical communication systems. In [72,73,74,75,76], several statistical distributions and their parameter estimations have been presented along with their application domains. In [77], a repository of optical network topologies of real optical networks of many parts of the world have been presented. This repository of optical network topologies is used for several research purposes [27,28,31,32,33]. Several statistical and functional properties of optical networks have been presented in [78,79,80,81,82,83].

Recent advancements in optical network design and estimation are heavily leveraging artificial intelligence (AI), machine learning (ML), and other advanced computational tools to address the increasing complexity and demands of modern data traffic [84]. AI and ML algorithms are now crucial for optimizing various aspects, including traffic prediction, route optimization, and dynamic resource allocation (e.g., wavelengths and bandwidth) to ensure efficient network utilization and reduced latency [84,85,86]. For estimation, these tools enable highly accurate quality of transmission (QoT) prediction, which is vital for new connection provisioning and maintaining service level agreements. Furthermore, AI and ML significantly enhance network reliability through proactive failure prediction and anomaly detection by analyzing vast amounts of network data, allowing for pre-emptive maintenance and self-healing network capabilities [87,88]. The integration of these technologies facilitates self-configuring networks that adapt to changes, automates tedious manual inspections of fiber links, and optimizes modulation formats and power levels for adaptive transmission systems, leading to higher data rates and lower operational costs [85,86,87,88,89]. This shift towards intelligent, data-driven network management is paving the way for more autonomous and efficient optical communication systems.

3. Materials and Methods

This study employed a comprehensive literature review and analytical approach to demonstrate the importance of statistical analysis and modeling in optical networks with a focus on applications of convex area in link length and path length characterization for network design and planning. In the literature review, a thorough search of academic databases was conducted to identify relevant research papers, technical reports, and industry standards related to statistical analysis in communication systems, with a special focus on optical networks. The review focused on identifying commonly used statistical tools, their applications in optical network analysis, and the insights they provide regarding network performance and behavior. Based on the literature review, the primary statistical distributions and methods used in optical network analysis were identified. These included the following aspects of modern optical networks.

• Nodal analysis and modeling of optical networks,
• Classification of optical networks,
• Characterization of optical networks,
• Link analysis and modeling of optical networks,
• General extreme value (GEV) distribution for link length modeling,
• Use of different areas for link-related parameter estimation,
• Use of convex area for optical networks,
• Estimation of convex area,
• Estimation of link-related parameters from the GEV distribution for link lengths,
• Finding the distribution of shortest path lengths,
• Estimation of shortest path lengths,
• Estimation of the parameters of the Johnson S_B distribution for shortest path lengths,
• Applications of Johnson S_B distribution for shortest path lengths.

We elaborate how these concepts and techniques have been utilized in this review. This paper adopts a systematic and comprehensive research methodology to synthesize existing knowledge and identify emerging trends in the field. The investigation commences with a foundational exploration into the classification of optical networks, delineating their diverse architectures, operational principles, and technological underpinnings to establish a clear definitional framework. This will be followed by a detailed characterization of optical networks, focusing on key performance indicators, architectural components, and their inherent complexities that necessitate statistical analysis. A core component of the methodology involves an in-depth review of nodal analysis and modeling of optical networks, examining various approaches to characterize node behavior, capacity, and resource allocation strategies within optical infrastructures. Concurrently, the paper undertakes a rigorous link analysis and modeling of optical networks, investigating methodologies for understanding link characteristics, impairments, and their impact on network performance. A significant focus is placed on the application of statistical distributions for modeling optical network parameters. Specifically, the methodology will critically assess the widespread use of the GEV distribution for link length modeling, analyzing its theoretical underpinnings, empirical validation, and limitations across diverse network scenarios. This approach naturally leads to an examination of the use of different areas for link-related parameter estimation, exploring how geographical, topological, and population density factors influence the statistical distribution of link lengths and other relevant parameters. The review delves into the novel concept of the use of convex area for optical networks, investigating its relevance in network planning, resource optimization, and resilience. This involves an examination of methodologies for the estimation of convex area in various optical network configurations. Furthermore, the methodology also details the process of estimation of link-related parameters from the GEV distribution for link lengths, demonstrating how this distribution can be leveraged to infer critical network characteristics. Building upon the understanding of individual links, the research will proceed to explore methods for finding the distribution of shortest path lengths within optical networks, a critical aspect for routing and latency analysis. This involves a comprehensive review of statistical techniques employed for estimation of shortest path lengths in various network topologies. A dedicated segment of the methodology focuses on the estimation of the parameters of the Johnson S_B distribution for shortest path lengths, highlighting its applicability and advantages in accurately characterizing the often skewed and bounded nature of shortest path distributions in optical networks. Finally, the review comprehensively discusses the applications of the Johnson S_B distribution for shortest path lengths, demonstrating its utility in network design, performance prediction, and optimization strategies for future optical communication systems. The research primarily relies on a systematic literature review of peer-reviewed articles, conference proceedings, and reputable academic databases, employing keywords derived from the key phrases to ensure comprehensive coverage. The synthesis of findings involves a critical comparative analysis of different modeling approaches, their assumptions, limitations, and the practical implications of their findings for the design and operation of robust and efficient optical networks.

The research methodology of this review paper critically examines the application of various statistical techniques and models to understand and predict network requirements. For each identified tool, a systematic analysis has been conducted, starting with its mathematical formulation and underlying assumptions. This involves delving into the probabilistic or statistical principles governing each model, such as the parameters of the GEV distribution for link lengths (e.g., location, scale, and shape parameters) and the Johnson S_B distribution for shortest path lengths (e.g., shape, location, and scale parameters). Understanding these assumptions is crucial for evaluating the validity and limitations of applying a specific model to diverse optical network topologies and operational scenarios. Next, the applications in optical network analysis for each tool has been thoroughly investigated. This includes how these models are employed for tasks such as network planning, resource dimensioning, performance prediction, and cost estimation. For instance, the GEV distribution’s utility in characterizing the variability of link lengths due to geographical constraints or deployment practices will be explored, while the Johnson S_B distribution’s role in understanding shortest path length distributions, which directly impacts latency and routing efficiency, will be detailed. Subsequently, the insights gained regarding network characteristics and performance from the application of these statistical models have been explained. This involves analyzing how statistical analyses reveal hidden patterns, correlations, and critical dependencies within optical networks that might not be apparent through deterministic approaches. For example, insights into the “small-world” or “scale-free” properties of optical networks derived from nodal degree distributions, or the impact of link length variability on overall network resilience, will be highlighted. Finally, a crucial aspect of this methodology is the explicit analysis of the relation between the statistical models and the performance of the optical networks. This segment will bridge the gap between theoretical statistical constructs and their practical implications for network behavior. It explores how accurately estimated parameters from distributions like GEV and Johnson S_B translate into quantifiable improvements in network design or how they facilitate the identification of critical network bottlenecks and vulnerabilities, thereby guiding effective mitigation strategies. The aim is to provide a holistic understanding of how statistical rigor directly contributes to enhancing the efficiency, reliability, and scalability of optical communication infrastructures.

The findings from the literature review and statistical tool analysis were synthesized to demonstrate the overall importance of statistical analysis and modeling in optical network design and optimization. The study discussed how leveraging statistical insights can lead to the development of more efficient, reliable, and scalable optical communication systems.

4. A Brief Introduction to Optical Networks

We next introduce optical networks and their components in order to establish the relationship between statistical modelling and optical network design. Optical networks are high-speed communication systems that use light to transmit data through optical fibers, offering superior bandwidth, low latency, and minimal signal loss compared to traditional electrical communication networks [1]. These networks rely on wavelength-division multiplexing (WDM) and time-division multiplexing as the main technologies to transmit multiple data streams simultaneously over a single fiber by assigning different wavelengths (thus different frequencies) of light to each channel. Optical networks can be classified into backbone networks, metro networks, and access networks, serving global, regional, and local communication needs, respectively [2]. Key components include optical switches, amplifiers, and transceivers, which enable efficient routing and signal regeneration [1]. Their advantages make them essential for internet infrastructure, data centers, and telecommunications, supporting high-demand applications like 5G, cloud computing, and artificial intelligence [42]. Passive optical networks and elastic optical networks are advanced architectures which improve scalability and flexibility [1]. As data consumption grows exponentially, optical networks continue to evolve, driving next-generation connectivity and smart infrastructure.

4.1. Classification of Optical Networks

Optical networks can be categorized based on various factors, with transparency being a key classification criterion. In communication engineering, transparency refers to the ability of the physical medium—optical fiber in optical networks to support end-to-end data communication regardless of bit rates and signal formats [1]. Based on this, optical networks are classified as transparent, translucent, and opaque. Transparent optical networks allow signals to pass without any electrical conversion, maintaining full optical transmission. Translucent optical networks, or partially transparent networks, permit some electrical regeneration at certain nodes, balancing transparency with signal quality enhancement. Opaque optical networks, on the other hand, require full electrical conversion at each node, impacting scalability but ensuring signal integrity. These classifications are depicted in Figure 1.

This classification also relates to the types of repeaters used along network links, which are categorized as 1R, 2R, and 3R regenerators [79]. First, 1R repeaters use optical amplifiers for simple signal boosting. Next, 2R repeaters incorporate regeneration and reshaping, necessitating optical-electrical-optical (O-E-O) conversions. Finally, 3R repeaters perform regeneration, reshaping, and re-timing, further enhancing signal quality through multiple O-E and E-O conversions. These differences influence the data rates that OTNs can support across links. Despite these variations, modern traffic grooming techniques optimize data transmission rates, ensuring that overall network performance remains efficient and scalable [1].

4.2. Components of Optical Networks

Optical networks are high-speed communication systems that use light to transmit data over fiber-optic cables. The key components of these networks include optical fibers, optical transmitters, and optical receivers [1]. Optical fibers, typically made of silica or plastic, serve as the transmission medium, offering low signal attenuation and high bandwidth. They are classified into single-mode and multi-mode fibers, with single-mode being ideal for long-distance communication due to lower dispersion. Optical transmitters, usually laser diodes or LEDs, convert electrical signals into optical signals. These signals travel through the fiber and are received by optical receivers, which consist of photodetectors that convert the light signals back into electrical form [1]. These core components enable efficient data transmission, making optical networks a backbone for modern communication systems. In addition to the fundamental components, optical networks rely on several other components, which can be broadly divided into two types: node-related components and link-related components [3]. These components enable scalable and high-speed communication, making optical networks essential for long distance communications, data centers, and cloud computing [47].

Optical networks incorporate advanced control and management systems to optimize efficiency and reliability. Software-defined networking (SDN) enables dynamic control of network resources, ensuring optimal bandwidth allocation and reducing congestion [47]. Optical network monitoring systems continuously assess fiber health, detecting faults and signal degradation to prevent downtime. In addition to that, protection and restoration mechanisms, such as ring and mesh topologies, ensure network resilience against failures by rerouting traffic in case of disruptions [1]. As optical networks continue to evolve, emerging technologies, like photonic integrated circuits and quantum communication, promise even higher data rates, enhanced security, and improved energy efficiency [80]. These innovations solidify optical networks as the backbone of future high-speed global communication infrastructure. The node-related components are either placed in the nodes, or they directly depend on the features of the nodes [3]. The link-related components are either placed along the links, or they directly depend on the features of the links [3]. Both the node-related components and link-related components are equally important for the overall success of the optical networks.

4.3. Node-Related Components

Node-related components in optical networks include optical transmitters, receivers, switches, optical cross-connects (OXCs), electrical cross-connects (EXCs), and routers [1]. Optical transmitters, such as laser diodes and LEDs, convert electrical signals into optical signals for transmission. Receivers, equipped with photodetectors, convert incoming optical signals back into electrical form [1]. Optical switches and routers facilitate signal routing and switching without requiring conversion, reducing latency and power consumption. These components ensure efficient data processing, routing, and signal regeneration within the network nodes. Nodes in optical networks perform key functions, such as reception, transmission, signal processing, and routing, using various electronic and optical components. The crucial component is the OXC, which acts as a high-speed optical switch, providing interconnections between links, routing, wavelength conversion, signal regeneration, traffic provisioning, grooming, and restoration. OXCs can be transparent, translucent, or opaque, with internal switching fabric being either fully or partially optical [1]. EXCs on the other hand, operate in the electrical domain, handling traffic that is either dropped for local delivery or added to the core network. Before OXCs, EXCs were responsible for all node switching [1]. Another key component, transponders, receive, process, and transmit optical signals, adapting them between the client and the OTN based on specifications like bit rate and distance [2]. Transponders are often integrated into optical line terminals (OLTs) but can also exist separately. Depending on link length, they may be classified as short, medium, or long reach. OLTs, typically positioned at the ends of WDM links before and after OXCs, serve to multiplex and demultiplex wavelengths using transponders, wavelength multiplexers, and optical amplifiers. In addition, OLTs may manage optical supervisory channels, which monitor optical amplifier performance and other network management tasks.

Node-related components help manage traffic by directing optical signals to their appropriate destinations with minimal latency and power loss. Optical switches enable dynamic reconfiguration of the network, improving flexibility and fault tolerance [2]. Multiplexers and demultiplexers allow multiple signals to share a single fiber, increasing bandwidth utilization and reducing infrastructure costs. In addition to that, wavelength converters help to resolve wavelength contention issues, optimizing network performance in WDM systems [2]. Beyond signal management, node components enhance the resilience and scalability of optical networks. OXCs also facilitate intelligent traffic routing, ensuring efficient load balancing and network survivability in case of failures. These elements collectively support high-speed, high-capacity networks essential for modern applications such as 5G, cloud computing, and data centers [2]. The integration of software-defined networking (SDN) and elastic optical networking (EON) further enhances the adaptability of optical nodes, paving the way for future-proof, high-performance communication infrastructures. In modern optical networks, pre-compensation and post-compensation facilities are provided at the source and destination nodes [2].

4.4. Link-Related Components

Link-related components primarily consist of optical fibers, multiplexers, demultiplexers, amplifiers, optical performance monitors, nonlinearity compensators, and dispersion compensators [1]. Optical fibers, classified into single-mode and multi-mode types, serve as the primary transmission medium, offering high bandwidth and low signal attenuation. Wavelength division multiplexers allow multiple wavelengths to travel simultaneously over a single fiber, increasing network capacity. Demultiplexers separate these wavelengths at the receiving end. Optical amplifiers, like EDFAs, boost signal strength without electrical conversion, enabling long-distance communication. In addition to that, dispersion compensators correct signal distortions caused by fiber dispersion, maintaining data integrity over long distances [2]. These link-related components enable high-speed, long-distance communication, making optical networks a crucial backbone for modern telecommunications and data transmission systems.

A typical optical link comprises optical fiber and associated components that facilitate light signal communication. For short links, signal processing occurs at the nodes, whereas long and extra-long links require regenerators or repeaters along the fiber [1]. Common link elements include optical amplifiers, reshaping devices, and retiming devices. Optical add/drop multiplexers (OADMs) and reconfigurable optical add/drop multiplexers (ROADMs) are used when wavelength changes (new traffic is added or dropped) are needed. Optical amplifiers, also known as 1R regenerators, boost weak signals using pump-powered gain blocks, with popular types including EDFAs, Raman amplifiers, and semiconductor optical amplifiers (SOAs). While EDFA are widely used for amplification in the C-band, the other amplifiers are used for other bands and also for non-amplification tasks. Next, 2R regenerators enhance signal strength and correct distortion using optical amplifiers and wave-shaping circuits, requiring Optical-Electrical-Optical (O-E-O) conversion [79]. Finally, 3R regenerators further improve signal quality by correcting both shape and timing distortions through additional retiming circuits, also necessitating O-E-O conversion. OADMs, used in WDM systems, facilitate adding, dropping, forwarding, and routing of light paths, functioning similarly to OXCs but on a smaller scale. While OXCs are large-scale switching components placed at network nodes, OADMs operate within fiber links [2]. ROADMs, a more advanced form of OADMs, offer reconfigurability, enabling dynamic selection of wavelengths without prior planning [1]. This flexibility supports remote configuration, power balancing, and seamless network adjustments, making ROADMs highly valuable for modern optical networks.

Link-related components in optical networks are crucial for ensuring seamless, high-speed data transmission over long distances with minimal signal degradation. These components include optical fibers, optical amplifiers, dispersion compensators, and fiber Bragg gratings, each playing a vital role in maintaining signal integrity and optimizing network performance [1]. Optical fibers serve as the primary transmission medium, offering high bandwidth and low attenuation, making them ideal for high-speed data communication. However, as signals travel through fibers, they experience losses and distortions due to attenuation, dispersion, and nonlinear effects [1]. Optical amplifiers, such as EDFAs and Raman amplifiers, counteract these losses by boosting signal strength, allowing data to travel over extended distances without requiring costly OEO conversions. Dispersion compensators and fiber Bragg gratings address chromatic dispersion and polarization mode dispersion, which can distort signals over long-haul links [2]. These components ensure that optical signals remain clear and synchronized, reducing BER and enhancing data reliability. Advanced link technologies, such as coherent transmission and WDM, optimize spectral efficiency, enabling higher data rates and greater network scalability. The integration of these link-related components enhances the overall efficiency, resilience, and capacity of optical networks, making them indispensable for supporting modern applications, such as 5G, cloud computing, and hyper-scale data centers [2]. As optical networks continue to evolve, innovations in link technologies will further improve their performance, ensuring they meet the ever-growing demands of global data communication.

5. Statistical Analysis and Modeling for Optical Networks

Statistics are indispensable in science and engineering, underpinning studies, measurements, characterization, analysis, modeling, and applications [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. As networks and systems become increasingly complex, statistical methods offer crucial tools for understanding their intricate behavior. Statistical modeling of networks, particularly for characterization and practical estimation, are vital across diverse fields like science, engineering, and social science [5,6,7,8,9,10,11,12,13,14,15]. However, the specific motivations for statistical analysis and modeling vary depending on the context and application. The application of statistical methods in communication systems has a long-standing history, beginning alongside the development of communication engineering itself [59].

Networks are closely related to the graphs and their physical parameters [5,6,7,8,9,10,11,12,13,14,15,16]. Mathematicians solve graph-related problems using graph theory, which originated accidentally when Euler attempted to solve the famous ‘Seven Bridges of Königsberg’ puzzle [34]. Today, graph theory plays a crucial role in network analysis, often incorporating statistical modeling for problem-solving. Physicists study the statistical mechanics of networks, focusing on their origin, growth, and time-dependent degree distributions, along with structural changes like percolation, which has applications in physics, chemistry, and material science [11,12,13,14,15,16,17,18,19,20,21,22,23]. In computational biology, statistical network analysis helps model the spread of epidemics, bacterial growth, viral infections (such as HIV, COVID-19, and Ebola), and protein–protein interactions in human metabolism [10,17]. Communication networks also rely heavily on network statistical models, from early telegraph systems to modern Internet advancements, helping engineers understand network complexities [29]. Sociology and psychology were among the first disciplines to apply network concepts, with studies on ‘small-world’ behavior predating modern social networks like LinkedIn and Facebook [8,35,36]. The ‘six degrees of separation’ theory, popular in the 1990s, emerged from small-world research, which also explains how brain cells are interconnected. In business and international trade, small-world models enhance trade efficiency and logistics, demonstrating the widespread influence of network analysis across multiple fields, from biology and physics to communication, social sciences, and global commerce [36]. In optical networks, statistical analysis and modeling has several applications, from network planning and dimensioning to design [3,4].

5.1. Characterization of Optical Networks

Statistical characterization of optical networks involves analyzing their performance, reliability, and efficiency using probabilistic models and statistical techniques. Statistical distribution models, stochastic models, and Gaussian noise models, help predict performance under varying traffic loads and environmental conditions. Monte Carlo simulations and machine learning techniques further refine network optimization. Statistical methods enable the assessment of link failures, congestion patterns, and quality of signal, aiding in the design of resilient, high-capacity optical communication systems. By leveraging statistical analysis, network operators can enhance fault tolerance, optimize resource allocation, and improve overall network efficiency, making optical networks more adaptive to dynamic data demands.

5.1.1. Random Network

Nodal degrees (the number of links connected with a node) of these networks follow Poisson distribution. When the average nodal degree of these networks becomes high, their distribution takes a bell shape (commonly known as the bell curve). The typical characteristics of random networks, often modeled by the Erdős–Rényi model, means that most nodes have a similar number of connections. This Erdős–Rényi model describes random networks where N nodes connect with probability p [G(N, p)] or through M randomly chosen edges [G(N, M)]. It helps analyze network properties like connectivity, clustering, and phase transitions in random networks. They exhibit a short average path length, where any two nodes are likely connected by a relatively small number of steps. In addition to that, they tend to have a low clustering coefficient, indicating that neighboring nodes are unlikely to be connected themselves. These networks lack the hubs or central nodes found in many real-world networks, and their structure arises purely from random connections. The nodes in random networks are typically not very well connected. The optical networks which are similar to the random networks are called ‘random-like’ optical networks. They exhibit a Poisson distribution-like pattern for the nodal degrees though they are not completely random. These topologies follow these ‘random-like’ patterns. In Figure 2, we show the nodal degree distribution of vBNS network (a Pan-USA core optical network), which is a random-like optical network.

A large fraction of optical networks, like many complex networks, exhibit a scale-free nature, meaning their degree distribution follows a power law, where a few nodes have many connections, while most nodes have few. This characteristic impacts network robustness and vulnerability. Random networks exhibit key characteristics, such as degree distribution, clustering, and path length. In Erdős–Rényi networks, the degree distribution follows a Poisson distribution, while in scale-free networks (e.g., Barabási–Albert), the degree distribution follows a power law, with a few highly connected hubs. Clustering coefficients in random networks are typically low, indicating weak local connectivity. The average path length tends to be short, following the “small-world” effect in some models. Connectivity and robustness depend on the probability of link formation; above a critical threshold, the network forms a giant connected component, while below it, fragmentation occurs. Percolation properties influence resilience—random failures affect networks differently than targeted attacks. In addition to that, information flow and diffusion dynamics in random networks depend on topology, impacting fields from epidemiology to social media spread.

5.1.2. Scale-Free Network

The degree distribution in scale-free networks follows a power law, meaning that the probability of a node having a specific degree decreases proportionally to a power of that degree. This power law distribution of nodal degrees resembles a rectangular hyperbola. In a scale-free network, the number of connections (degree) a node has is not evenly distributed. Instead, a small number of nodes have a very high degree (acting as “central nodes”), while the majority of nodes have a relatively low degree. While the Internet serves as a classic example, optical networks, due to their survivability requirements, deviate from a strict scale-free structure. However, their nodal degree distributions often approximate a power law, leading to their classification as scale-free-like (SF-like). In Figure 3, we show the nodal degree distribution of Germany, which is an SF-like network.

Scale-free networks have a few central nodes (also known as hubs), which have a disproportionately high number of connections, while most nodes have only a few. This uneven connectivity pattern leads to resilience against random failures, as the majority of low-degree nodes can be removed without significantly disrupting the network. However, scale-free networks are highly vulnerable to targeted attacks on hubs, which can fragment the system. Their self-organizing nature and preferential attachment mechanism, where new nodes are more likely to connect to highly connected nodes, drive their formation. This structure enhances efficiency in information flow and robustness in many real-world systems, including the internet, social networks, biological networks, and citation graphs. The presence of hubs allows for rapid communication and efficient resource distribution but also introduces hierarchical organization. Scale-free networks are a fundamental framework in complex systems, shaping dynamics in both natural and engineered networks.

5.1.3. Small-World Network

Small-world (SW) networks are characterized by a logarithmic relationship between the average path length and the number of nodes in the network (i.e., network size), N [13,78]. Specifically, the typical number of steps required to connect two random nodes scales as log (N), meaning even large networks can be traversed with few links. Optical networks that exhibit this property are referred to as small-world-like (SW-like). Most optical networks exhibit SW-like characteristics. In Figure 4, we show an SW-like optical network (i.e., Via Network) which has nodes at London, Paris, Amsterdam, Frankfurt, Berlin, Munich, Lisbon, Madrid and Barcelona. Just below it, we show its SW traits.

SW networks exhibit a unique structure characterized by high clustering and short path lengths, striking a balance between local cohesion and global reachability. In these networks, most nodes are connected to nearby nodes, forming tightly knit clusters, while a few long-range connections significantly reduce the average shortest path between any two nodes. This structure enhances information flow, robustness, and efficiency, making SW networks common in social interactions, biological systems, and technological infrastructures. The defining properties of SW networks are high clustering coefficient and small average shortest path, which enable rapid communication and resilience against random failures. SW networks emerge naturally in systems like neural networks, power grids, and the Internet, where efficiency and redundancy are crucial. The Watts–Strogatz model mathematically captures these characteristics, demonstrating how rewiring a small fraction of links in a regular network can transform it into a small-world structure [13]. Their prevalence highlights their fundamental role in complex systems across disciplines.

5.1.4. Complex Network

Complex networks consist of numerous interconnected nodes with diverse interactions, displaying non-trivial topological features. These features often resemble those found in ‘small-world’ or ‘scale-free’ networks. Examples include the Internet, neural networks, social networks, metabolic networks, and genetic networks. It is noteworthy that the concept of complex networks encompasses random networks, small-world networks, and scale-free networks. These three types are often discussed as fundamental models or archetypes within the broader field of complex networks. In reality, random, small-world, and scale-free networks are specific models or classes that demonstrate different structural properties which fall under the umbrella of “complex networks.” The study of complex networks involves understanding these models, their formation mechanisms, and how they contribute to the behavior and function of real-world systems.

Optical networks exhibit several complex network characteristics, making them a fascinating subject for study within complex network theory. These characteristics arise from their scale, connectivity patterns, and the dynamics of information flow. Here, we present some key complex network characteristics observed in optical networks. The complex network characteristics are very common in the optical access networks. However, in the core optical networks, these behaviors are less common. However, the optical access networks in large cities and metropolitan areas exhibit many features of complex networks.

While not always a perfect power law, many large scale optical networks, particularly backbone and core networks, show a highly heterogeneous distribution of connections (node degrees). This means that some nodes have a disproportionately high number of links compared to the majority of nodes. These highly connected nodes act as “hubs” for traffic aggregation and distribution. Due to the presence of hubs, optical networks can be robust against random failures (e.g., a minor fiber cut). However, they can be highly vulnerable to targeted attacks or failures of these critical hub nodes, as such failures can significantly disrupt large portions of the network. Optical networks, especially core and metropolitan networks, are designed for efficient data transmission over long distances. This often results in a relatively small average path length between any two nodes. This “small-world” property means that information can travel across the network in a surprisingly few hops, even for geographically dispersed networks. In some segments of optical networks, particularly within metropolitan areas or data centers, there can be a high clustering coefficient. This implies that if two nodes are connected to a third node, there is a higher probability that they are also directly connected to each other, forming local “communities” or highly interconnected clusters. Real-world optical networks are often complex combinations of these basic topologies (e.g., a meshed core connected to ring-based metropolitan networks, which then connect to star-like access networks). This creates a multi-layered complex network. The growth of optical networks (e.g., adding new fiber links, expanding to new cities) can sometimes follow principles similar to preferential attachment, where new connections are more likely to be made to existing, well-connected nodes (hubs) due to economic or deployment reasons. Modern optical networks often incorporate intelligence (e.g., SDN and NFV) to dynamically reconfigure paths, manage traffic, and recover from failures, adding a dynamic layer to their complex network behavior. Overall, optical networks are not simple, regular structures. They are designed for high capacity, speed, and resilience, along with organic growth, which leads to emergent properties that align well with the concepts of random, scale-free and small-world networks, making them a prime example of real-world complex systems.

Network characterization relies heavily on statistical analysis, a tool now integral to network science and engineering, far surpassing traditional graph and circuit theory [5,6,7,8,9,10,11]. Optical networks, the core of global communication, demand robust reliability and security [1]. Statistical analysis is crucial for performance evaluation, fault prediction, and optimization. Over the last decade, statistical modeling has become vital in optical network research, especially during early planning, when data is limited. This facilitates informed decisions and efficient resource use. Understanding parameter distributions reveals network properties. Beyond optics, statistical analysis and stochastic modeling are essential in the Internet, cellular, and trade networks, enabling physical characterization and practical applications. However, statistical models for optical communication networks remain underdeveloped.

5.2. Dimensioning of Communication Networks

Dimensioning a communication network involves determining the necessary resources to meet service demands while optimizing cost and performance. This process focuses on calculating the required capacity of network elements, such as links, switches, and servers, to ensure that optimal performance parameters, like latency, throughput, and packet loss, are within acceptable limits. It involves analyzing traffic patterns, predicting future demand, and applying mathematical models and simulation tools to estimate resource requirements. Key factors considered include the number of users, the types of applications being used, and the desired level of reliability. Dimensioning aims to prevent congestion, over-provisioning, and under-provisioning, thereby ensuring efficient utilization of network infrastructure.

However, the typical dimensioning of optical networks is associated with the cost estimation of the network [3,4]. This is required at the planning stage of the optical network design and deployment [4]. In these processes, all the network requirements are analyzed, and based on that, the costs are estimated using different cost estimation parameters. Very often, the statistical analysis and design are essential for such exercises as they exploit the key characteristics of the optical networks. Several similar techniques have been developed in recent years [3,4,27,28,31,32,33].

5.3. Dimensioning in Optical Networks

Dimensioning of optical networks is a complex process as it is associated with several components and functions. In order to overcome that, the dimensioning tasks or the costs associated with the optical networks are divided into node-related and link-related parameters. The total costs associated with the optical networks are thus divided into two parts. The first is the costs associated with the nodes (C_N), and the second is the costs associated with the links (C_L), so the total cost of optical networks is the sum of these two parts.

C_T = C_N + C_L

(1)

5.3.1. Estimation of Node Related Parameters

In modern optical networks, nodes function as both sources and sinks of information while also serving as key signal processing hubs. They facilitate traffic routing between links, and in large OTNs, network topologies are often designed to be survivable—meaning each node is connected to at least two links. Survivability is quantified through nodal degree, which represents the number of connections per node. A survivable OTN topology ensures a nodal degree of at least two for all nodes. Extensive research has been conducted on nodal degrees across various network types, particularly in complex networks like the Internet, where nodal degree analysis provides critical insights into network structure, stability, and resilience. Similarly, in OTNs, nodal degree statistics offer valuable information about network features beyond survivability. The mean nodal degree, for instance, helps define network structure; when the nodal degree equals two, the network forms a simple ring. Networks with central nodes, where a single node connects to most others, typically exhibit higher mean nodal degrees. The overall statistical distribution of nodal degrees characterizes network connectivity patterns. Historically, OTNs followed a Poisson distribution for nodal degrees, but recent measurements indicate a shift towards scale-free characteristics, reflecting evolving network architectures and dynamic connectivity patterns.

The nodal degree of a node in an optical network is the number of links connected with that node. The average nodal degree is quantified as 2L/N, where L and N denote the number of links and the number of nodes in the network, respectively. Estimation of node parameters in optical networks from network statistics is based on utilizing observed traffic patterns, connection demands, and performance measures. Based on connection establishment/release rates, holding times, and routing paths, node processing abilities, switching rates, and buffer sizes can be estimated [30]. Statistical analysis of latency and blocking probabilities provides insights into node latency contributions and contention levels [1]. In addition to that, analysis of signal degradation along routes assists in inferring node-caused impairments, such as crosstalk and filtering [2]. Sophisticated statistical techniques, such as machine learning and optimization algorithms, can be utilized for relating observations across networks to specific node properties, so that even in advanced, dynamic optical networks, accurate parameter estimation is possible. Restoration and protection in optical networks ensure resilience against failures. Protection is proactive, using pre-established backup paths like 1 + 1 (dedicated) or 1:1 (shared) protection, switching traffic instantly upon failure [1]. Restoration is reactive, dynamically finding alternate paths post-failure, using protocols like GMPLS and ASON [2]. Techniques include optical-layer (wavelength rerouting) and network-layer (IP/MPLS rerouting) recovery. Fast-switching mechanisms, like Automatic Protection Switching and self-healing rings, enhance reliability [2]. Machine learning and SDN improve adaptive restoration, optimizing resource utilization. Hybrid approaches balance speed and cost, ensuring high availability in critical applications like telecom, data centers, and smart grids.

5.3.2. Statistical Modeling of Links

Links in optical networks serve as crucial connectors between nodes via fiber channels, with link lengths representing the fiber distances over which information travels. The total fiber length in a network is the sum of all link lengths, influencing several link-dependent parameters, such as the number of amplifiers required, the selection of modulation and demodulation schemes at nodes, the necessity of compensation techniques, and the number of preamplifiers needed. Despite their significance, statistical models for OTN link lengths were previously unavailable. The link length statistical model developed in [28] fulfils that gap and helps estimate link-dependent parameters just from the coverage area and the number of nodes. The developed model for the USA100 (a Pan-American optical network with 100 nodes) optical network is in Figure 5. It shows that the link lengths of optical networks follow GEV distribution. In traffic routing and forwarding, the shortest path lengths between nodes are critical, particularly in transparent OTNs, where they determine appropriate modulation-demodulation schemes and compensation methods at nodes. It has been observed that shortest path lengths influence various network estimations, aiding both data and control plane operations. However, like link lengths, statistical models for shortest path lengths in OTNs were historically lacking, highlighting a gap in network analysis and optimization.

Typically, the link length statistics in optical networks are estimated from their average link lengths [3]. The average link lengths can be estimated from the coverage areas of the optical networks. The coverage areas have been defined in three different ways: convex area; exact area; and geographical area [28]. The convex area is the area of the convex polygon that takes all the nodes of the optical network into consideration (an example is shown in Figure 6). The exact area is the area confined by the outer-most links of the optical network and covered by the topology of the optical network. The geographical area is the area of the countries or states in which the optical network is confined. From the measurements, and subsequent analysis, it was confirmed that the convex area is the most accurate areas for the estimation of the average link lengths of the optical networks.

The link lengths of optical networks follow GEV distribution. The parameters of the GEV distribution of optical networks can be estimated from their average link lengths [28]. The average link length (<l>) of an optical network can be estimated from its convex area (A_C) with good accuracy, as shown below.

$$<l>={\displaystyle \frac{0.97\sqrt{A_c}}{\sqrt{N}-1}}$$

(2)

The GEV distribution has three parameters: α, β, and $\xi$. The three parameters can be estimated from the average link length, as shown below.

$$\alpha =0.6577<l>+8.67$$

(3)

$$\beta =0.441<l>-12.37$$

(4)

$$\xi ={\displaystyle \frac{0.0887<l>-1.557}{0.5297<l>-13.927}}$$

(5)

With the knowledge of these parameters, the probability distribution function (PDF) of the GEV distribution of optical networks can be estimated using the following expression.

$$f\left(l,\alpha , \beta , \xi \right)={\displaystyle \frac{1}{\beta }} {\left[1+{\displaystyle \frac{\xi (l-\alpha )}{\beta }}\right]}^{\left(-1/\xi \right)-1}exp\left\{-{\left[1+{\displaystyle \frac{\xi (l-\alpha )}{\beta }}\right]}^{-1/\xi }\right\}$$

(6)

From this PDF of the GEV distribution of link lengths, several link-dependent parameters of the optical networks can be estimated with very good accuracy. The convex area of the optical networks, which is a prime parameter for these estimations, can be estimated from the network topology and its circumferential ellipse [32].

5.3.3. Convex Area of Optical Networks

Coverage areas in optical networks can be classified into three types: exact area (bounded by the topology’s boundary links), convex area (the largest convex set formed by the network nodes), and geographical area (determined by the physical region where the network operates). Among these, the convex area is the most suitable for link-related estimations, as it enables accurate calculations of average link lengths. Using the probability density function of link lengths, various link-related parameters can be derived. While geometric measure theory provides a method for estimating a graph’s approximate area, it requires numerous inputs and does not guarantee exact results. Additionally, such methods rely on complete knowledge of the network topology, making them impractical for optical networks in which exact areas fail to yield accurate estimations. Instead, coverage areas facilitate rapid estimation of optical network parameters, even without full network details. The eccentricity of a network’s ellipse (or ellipticity) is particularly useful in studying network properties, aiding in the determination of parameters such as diameter, radius, and shortest paths. These estimations play a crucial role in fields like networking, distributed computing, and graph theory, where rapid calculations are essential. Furthermore, rapid estimation methods assist in determining network diameters, shortest paths between nodes, and cost-related factors such as capital expenditure (CAPEX) during optical network planning.

For regular structures like squares, rectangles, or regular polygons, a circumcircle can always be identified. However, in irregular graphs, which are common in optical networks, finding such a circle is not always feasible. It is very much the same for all large optical networks [32]. Instead, circumferential ellipses (CEs) are used, which is possible for all planar networks (meaning when a planar topology of a network is available, its CE can be estimated). The CE of an optical network is the smallest possible ellipse that encloses the entire network, touching at least three nodes (it is possible that the CE may touch more than 3 points). The only points of intersection between the CE and the optical network topology are these finite nodal points. If the CE is reduced in size, it will inevitably intersect at least one link (or at least two links in a survival optical network), ensuring that every network topology has a unique CE. These ellipses serve as valuable tools in analyzing network topology, optimizing link estimations, and improving network design efficiency. In Figure 6, we have shown the convex area (the regular pentagon ABCDEA) of a star-shaped network APBQCRDSETA. In Figure 7, we have shown the topology (in blue) and CE (in red) of the Bulgarian Research and Education Network (BREN) [32,77]. It has been found that the convex area and the area of the CE of optical networks follow a linear trend.

5.3.4. Statistical Modeling of Shortest Path Lengths

Statistical modeling of shortest path lengths in optical networks is important for understanding and optimizing network performance. This analysis involves characterizing the distribution of distances between node pairs, which significantly impacts factors like routing, latency, resource allocation, and overall network efficiency. Researchers employ statistical techniques to analyze real optical transport networks, often finding that distributions, like the Johnson S_B distribution, effectively model these path lengths [33,90]. These models enable the estimation of key network parameters, such as average path lengths and variations, from fundamental network information like node locations. This allows for the prediction of path-length-dependent system behaviors, including the selection of appropriate modulation formats, enhancing network design and planning. By statistically analyzing shortest path lengths, engineers can improve network reliability, predict potential bottlenecks, and optimize routing algorithms, ultimately leading to more robust and efficient optical communication systems.

The shortest path lengths in optical networks are found to follow the Johnson S_B distribution [33,90]. This distribution is one of the transformed distributions of the ‘Johnson System’ family [58,72,73]. It is also a bounded distribution. Johnson S_B is similar to the normal distribution in many aspects. It can be regarded as a modified lognormal distribution [33]. This distribution has four parameters ($\gamma ,\delta ,\lambda ,\zeta$), and its PDF is shown below.

$$f \left(z;\gamma ,\delta ,\lambda ,\zeta \right)={\displaystyle \frac{\delta }{\lambda \sqrt{2\pi } z(1-z)}}exp\left[-{\displaystyle \frac{1}{2}}{\left\{\gamma +\delta log\left({\displaystyle \frac{z}{1-z}}\right)\right\}}^2\right]$$

(7)

Its parameters can be estimated from the convex area of the optical networks (A_c) and their number of nodes (N). These estimations are shown below.

$$\lambda =\left\{\begin{array}{c}10.67{\left(\sqrt{A_c}\right)}^{0.7564}, if \sqrt{A_{c }}<1000\\ 0.0016A_c-4.076\sqrt{A_c}+5600, if \sqrt{A_c}\ge 1000\end{array}\right.$$

(8)

$$\delta =\left\{\begin{array}{c}cosec \left({\displaystyle \frac{t+30}{40}}\right)+0.25cos\left(62.46t\right)-0.1cos\left(112.62t\right), if t \left(={\displaystyle \frac{\sqrt{A_c}}{N}}\right)<70 \\ 0.8\mathit{sin}\left({\displaystyle \frac{t+126.5}{162}}\right)+0.17cos\left(50.258t\right)+0.12cos\left(149.226t\right), if t \ge 70\end{array}\right.$$

(9)

$$\xi =\left\{\begin{array}{c}0.1\sqrt{A_c}, if \delta <0.82 \\ -0.11\sqrt{A_c}, if \delta \ge 0.82\end{array}\right.$$

(10)

$$\gamma =\left\{\begin{array}{c}0.12+\left|1.02-0.06t\right|+0.03sin\left(4t\right),if t\left(={\displaystyle \frac{\sqrt{A_c}}{N}}\right)<29\\ 0.235+sinc\left(2t-112.4\right),if 29\le t\le 96\\ 0.24-{\displaystyle \frac{1}{t-95}}+0.45sinc\left(2t-253\right),if t>96\end{array}\right.$$

(11)

The Johnson S_B distribution is well-suited for modeling shortest path lengths in optical networks due to its ability to capture both unimodal and bimodal probability density functions, frequently observed in these networks. Notably, bimodality arises in networks with large convex areas and sparse node distribution. We have demonstrated that the mean, median, and standard deviation of shortest path lengths can be linearly estimated from these convex areas. This estimation, along with the upper bound of the maximum path length, is achievable without complete network topology knowledge, as the convex area can be calculated directly from node coordinates. This offers a practical approach to characterizing path length distributions, leveraging the Johnson S_B distribution’s flexibility and the simplicity of convex area computation. The knowledge of this distribution helps in the estimation of several parameters of the optical networks which depend on the shortest path lengths.

As shown in Figure 8, the distribution of the shortest path lengths can be used to identify the optical modulation and demodulation techniques at the end of the node pairs. It can also be used to determine the shortest path related parameters, such as the compensation techniques at the nodes for the transparent optical networks.

5.3.5. AI and Machine Learning for Statistical Modeling of Optical Networks

The relentless growth of data traffic and the increasing complexity of optical network architectures necessitate advanced tools for performance prediction, fault detection, and resource optimization. Traditionally, statistical modeling of optical networks has relied on analytical approaches and empirical measurements. However, the sheer volume of data generated by modern optical systems, coupled with the intricate interplay of physical layer impairments (e.g., chromatic dispersion, polarization mode dispersion, non-linear effects), demands more sophisticated techniques. This is where AI and machine learning (ML) emerge as transformative forces [84]. AI/ML algorithms, particularly supervised and unsupervised learning, offer an unparalleled ability to identify complex patterns, build predictive models, and classify anomalies within vast datasets that would overwhelm conventional methods. For instance, neural networks can be trained on historical network performance data to predict future quality of transmission (QoT) for various light-paths under dynamic traffic conditions, factoring in the impact of impairments and cross-talk [85]. Similarly, clustering algorithms can group similar network states, aiding in proactive fault identification and root cause analysis. Reinforcement learning (RL), on the other hand, presents a promising avenue for autonomous network control, allowing intelligent agents to learn optimal routing and resource allocation strategies through interaction with the network environment, continuously adapting to changing demands and degradation [86]. The shift towards AI/ML-driven statistical modeling marks a paradigm shift, enabling optical networks to become more resilient, efficient, and self-managing. RL offers a powerful framework for statistical modeling of optical networks by enabling intelligent agents to learn optimal decision-making policies through trial and error. Unlike traditional supervised learning, RL agents interact directly with the network environment, receiving feedback for their actions, and iteratively refining their strategies to maximize long-term performance. This makes RL particularly well-suited for dynamic problems in optical networks, such as real-time resource allocation (e.g., routing, spectrum assignment, and modulation format selection), proactive fault management, and power consumption optimization [86]. By learning from the statistical properties of network states and the consequences of various actions, RL models can adapt to unpredictable traffic patterns, evolving network conditions, and the stochastic nature of physical layer impairments, leading to more resilient, efficient, and self-optimizing optical networks.

The application of AI and ML to statistical modeling of optical networks extends beyond prediction and classification; it fundamentally enhances operators’ ability to understand and control these complex systems [87]. Traditional statistical methods often fall short in managing complex, multi-vendor, multi-layer optical networks because they struggle to capture the full range of operational intricacies. As AI/ML models learn from real-time and historical operational data, they can provide a holistic view of network health and performance. This includes developing highly accurate statistical models for SNR estimation, predicting the likelihood of link failures based on various environmental and operational parameters, and optimizing power consumption across the network [85]. Furthermore, AI/ML can facilitate the creation of digital twins for optical networks, allowing for the simulation and testing of different network configurations and failure scenarios in a virtual environment before deployment, thereby reducing risks and accelerating innovation. The integration of these advanced computational techniques empowers network operators with prescriptive insights, moving beyond reactive troubleshooting to proactive optimization [87]. As optical networks evolve towards disaggregation and open architectures, the role of AI/ML in statistical modeling will become even more critical, providing the intelligence required to navigate increasing complexity and unlock the full potential of future optical communication systems. AI and ML are also transforming optical network management by enabling sophisticated statistical modeling of traffic and QoT [88,89]. These techniques predict network congestion, identify anomalies, and optimize routing dynamically, moving beyond traditional rule-based systems [88]. By analyzing vast datasets of network performance, AI/ML models can proactively mitigate issues like signal degradation and crosstalk, ensuring high QoT and efficient bandwidth utilization [89]. This intelligent automation leads to more reliable, resilient, and cost-effective optical networks, crucial for handling ever-increasing data demands [91].

6. Discussion and Applications

Statistical characterization of optical networks plays a critical role in analyzing performance, reliability, and efficiency. Methods such as statistical distribution models, stochastic processes, and Gaussian noise models help predict network behavior under varying traffic loads and environmental conditions. Monte Carlo simulations and machine learning algorithms further refine network optimization by enabling precise fault prediction and adaptive resource allocation. Statistical methods assess link failures, congestion patterns, and signal quality, supporting the development of resilient, high-capacity optical communication systems. As optical networks form the backbone of global communication, their reliability and security demand robust statistical modeling. Over the last decade, statistical analysis has become integral to optical network research, particularly during early planning phases with limited data. Understanding parameter distributions helps reveal network properties and optimize resource utilization. Despite advancements, statistical models for optical communication networks remain underdeveloped, warranting further exploration for improved efficiency and scalability.

The robust design of optical networks critically relies on the meticulous dimensioning of node-related parameters, a process profoundly informed by statistical analysis and sophisticated modeling techniques. Architectural choices, such as the number of ports in optical switches and their overall switching capacity, are not arbitrary; they are derived from traffic demand forecasts and network growth projections, often requiring statistical modeling to predict future loads and optimize resource allocation. The design of high-capacity optical switches, including ROADMs, which enhance dynamic wavelength routing and fault tolerance, benefits immensely from statistical simulations that evaluate their performance under various traffic patterns and failure scenarios.

Furthermore, the integration of wavelength conversion capability, a key design decision for efficient spectrum utilization in WDM networks, necessitates statistical analysis to quantify its impact on mitigating wavelength contention and improving network throughput. The design of optical regeneration mechanisms, be it 3R (Re-amplification, Re-shaping, Re-timing), 2R, or purely optical techniques, is guided by statistical models of signal degradation over distance, ensuring seamless data transmission with minimal latency and packet loss by predicting and counteracting impairments. Beyond core switching and signal management, effective node design incorporates traffic load balancing, redundancy, and energy efficiency, all of which are optimized through statistical modeling of traffic flow and failure probabilities. The strategic placement of optical amplifiers, a crucial design aspect in long-haul and backbone networks, is determined by power budget calculations and statistical models of signal attenuation. The integration of SDN in network design introduces intelligent control and adaptive resource allocation, with real-time adjustments in node dimensioning made possible by statistical analysis of dynamic traffic demands. As optical networks evolve to accommodate emerging technologies, like 5G, IoT, and cloud-based applications, the precise design and dimensioning of node parameters, underpinned by rigorous statistical analysis and predictive modeling, become increasingly vital in crafting high-performance, cost-effective, and future-proof network infrastructures. This continuous advancement necessitates ongoing research and development to refine dimensioning strategies through sophisticated statistical methodologies, ensuring robust, scalable, and next-generation communication capabilities.

Intelligent management of optical networks heavily relies on the estimation of link-related parameters, a critical process deeply rooted in statistical analysis and modeling. Precisely understanding factors like fiber length, attenuation, dispersion, and nonlinearity is not just about measurement; it is about predicting their impact on signal integrity and network capacity through sophisticated models. For instance, statistical models of signal degradation over varying fiber lengths inform the strategic placement of optical amplifiers like EDFAs and Raman amplifiers, ensuring power compensation and extended reach without costly OEO conversions. Dispersion management, including chromatic and polarization mode dispersion, relies on statistical estimation to deploy compensation modules or advanced digital signal processing techniques effectively, preserving signal clarity across the network. Furthermore, modeling the effects of fiber nonlinearities, such as self-phase modulation and four-wave mixing, is crucial for maintaining spectral efficiency and network reliability, as these impairments are often stochastic in nature.

Beyond physical impairments, intelligent link dimensioning leverages statistical analysis to optimize spectral efficiency and network scalability. This involves using statistical models to determine proper wavelength spacing in WDM and assess the benefits of coherent detection techniques, maximizing data rates while minimizing interference. Maintaining optical SNR within acceptable limits is a continuous estimation and control problem, solved through optimized amplifier placement and power control strategies derived from statistical network performance models. Emerging technologies like EONs and SDN enable dynamic link adaptation, driven by real-time traffic demands and intelligent algorithms that use statistical analysis to predict and respond to network changes. As optical networks evolve to support high-speed applications, the accurate estimation of link parameters through robust statistical analysis and modeling remains paramount for achieving high performance, reliability, and cost-effectiveness.

7. Conclusions

Statistical analysis and modeling are essential for optimizing the performance, reliability, and scalability of optical networks in an era of increasing data demand. By leveraging techniques such as probability distributions, signal quality analysis, stochastic processes, and machine learning, researchers and engineers can gain deep insights into network behavior, predict potential failures, and enhance resource management. These methods allow for real-time monitoring and proactive maintenance, reducing downtime and ensuring seamless data transmission. In addition to that, statistical models help mitigate challenges posed by signal impairments, environmental factors, and unpredictable traffic patterns, enabling adaptive control mechanisms that optimize quality of signal and network efficiency. However, the complexity and volume of data generated by optical networks necessitate advanced computational frameworks, including AI-driven automation and edge computing, to process and interpret vast datasets efficiently. The integration of intelligent statistical modeling with emerging technologies like quantum communication and decentralized network architectures holds great promise for the future of optical networking. As networks continue to evolve to meet growing demands, statistical analysis will remain a fundamental tool in developing innovative, high-capacity, and resilient communication systems that support global digital transformation. Future optical network research will prioritize pervasive AI/ML integration for fully autonomous, self-optimizing, and self-healing networks. Key areas include advanced digital twins for predictive maintenance and real-time control, intelligent resource orchestration across multi-vendor and multi-layer domains, and quantum communication for ultra-secure transmission. Emphasis will also be on sustainable “green” photonics to drastically reduce energy consumption, alongside novel fiber technologies like hollow-core and multi-core fibers to push capacity beyond current limits, addressing the relentless demand for bandwidth driven by AI, 6G, and immersive technologies.