This chapter covers other canonical applications of network tomography that have been studied in the literature but fallen out of the scope of the previous chapters. This includes the inference of network routing topology (network topology tomography) and the inference of traffic demands (traffic matrix or origin-destination tomography). It also covers miscellaneous techniques used in network tomography that are not covered in the previous chapters (e.g., network coding). The chapter then concludes the book with discussions on practical issues in the deployment of tomography-based monitoring systems and future directions in addressing these issues.

Additive network tomography, which addresses the inference of link/node performance metrics (e.g., delays) that are additive from the sum metrics on measurement paths, represents the most well-studied branch in the realm of network tomography, upon which a rich body of seminal works have been conducted. This chapter focuses on the case in which the metrics of interest are additive and constant, which allows the network tomography problem to be cast as a linear system inversion problem. After introducing the abstract definitions of link identifiability and network identifiability using linear algebraic conditions, the chapter presents a series of graph-theoretic conditions that establish the necessary and sufficient requirements to achieve identifiability in terms of the number of monitors, the locations of monitors, the connectivity of the network topology, and the routing mechanism. It also contains extended conditions that allow the evaluation of robust link identifiability under failures and partial link identifiability when the network-wide identifiability condition is not satisfied.

This chapter completes the topic of measurement design for additive network tomography, started in Chapter 3, by discussing how to construct suitable measurement paths to identify additive link metrics using a given set of monitors. As in Chapter 3, the focus is on the design of efficient path construction algorithms that make novel use of certain graph algorithms (specifically, algorithms for constructing independent spanning trees) to find a set of paths that form a basis of the link space without enumerating all possible paths. The chapter also discusses a variation of the path construction problem when the number of measurement paths is constrained and each measurement path may fail with certain probability.

Chapters 7 and 8 are designated for network tomography for stochastic link metrics, which is a more fine-grained model than the models of deterministic additive/Boolean metrics, capturing the inherent randomness in link performances at a small time scale. Referred to as stochastic network tomography, these problems are typically cast as parameter estimation problems, which model each link metric as a random variable with a (partially) unknown distribution and aim at inferring the parameters of these distributions from end-to-end measurements. Chapter 7 focuses on one branch of stochastic network tomography that is based on unicast measurements. It introduces a framework based on concepts from estimation theory (e.g., maximum likelihood estimation, Fisher information matrix, Cramér–Rao bound), within which probing experiments and parameter estimators are designed to estimate link parameters from unicast measurements with minimum errors. Closed-form solutions are given for inferring parameters of packet losses (i.e., loss tomography) and packet delay variations (i.e., packet delay variation tomography).

This chapter introduces the definition of network tomography and the three branches of network tomography and provides an overview of the main issues addressed in the subsequent chapters.

Boolean network tomography is another well-studied branch of network tomography, which addresses the inference of binary performance indicators (e.g., normal vs. failed, or uncongested vs. congested) of internal network elements from the corresponding binary performance indicators on measurement paths. Boolean network tomography fundamentally differs from additive network tomography in that it is a Boolean linear system inversion problem in which each measurement path only provides one bit of information and hence deserves a separate discussion. This chapter introduces a series of identifiability measures (e.g., k-identifiability, maximum identifiability index) to quantify the capability of Boolean network tomography in uniquely detecting and localizing failed/congested network elements. As the definitions of these identifiability measures are combinatorial in nature and hard to verify for large networks, the discussion focuses on polynomial-time verifiable conditions and computable bounds, as well as the associated algorithms.

Based on the conditions for identifying additive link metrics presented in Chapter 2, this chapter addresses two network design questions: (1) Given an unbounded number of monitors, where should they be placed in the network to identify the metrics of all the links using a minimum number of monitors? (2) Given a bounded number of monitors, where should they be placed in the network to identify the metrics of the largest subset of links? The focus here is on the design of intelligent algorithms that can efficiently compute the optimal monitor locations without enumerating all possible monitor placements, achieved through strategic decomposition of the network topology based on the required identifiability conditions. Variations of these algorithms are also given to address cases with predictable or unpredictable topology changes and limited links of interest. In addition to theoretical analysis, empirical results are given to demonstrate the capability of selected algorithms for which such results are available.

In contrast to unicast measurements considered in Chapter 7, this chapter focuses on stochastic network tomography based on multicast measurements, where each probe is sent along a multicast tree from one source to multiple destinations, duplicated at each intermediate node with at least two outgoing links. Using loss tomography as an example, the chapter details how the correlations between the measurements at different destinations sharing links in the multicast tree can be utilized to infer link loss rates, while briefly discussing how this approach applies to other performance metrics. Moreover, this chapter further illustrates how correlated loss observations obtained from multicast probes can be used to reliably infer the topology of the multicast tree, which belongs to another branch of network tomography (network topology tomography) that will be formally introduced in Chapter 9, and complements the high-level discussions there.

This chapter covers preliminary materials required to understand the presentation in the following chapters, including selected definitions from graph theory, linear algebra, and parameter estimation. We also introduce a classification of routing mechanisms based on the controllability of the routing of probes by monitors generating the probes, which will facilitate the discussion in the following chapters.

Based on the identifiability measures for Boolean network tomography presented in Chapter 5, this chapter addresses the follow-up question of how to design the measurement system to optimize the identifiability measure of interest, with a focus on the placement of monitoring nodes.Depending on the mechanism to collect measurements, the problem is divided into (1) monitor placement, (2) beacon placement, and (3) monitoring-aware service placement, where the first approach requires monitoring nodes at both endpoints of each measurement path, the second approach requires a monitoring node only at one of the endpoints of each measurement path, and the third approach requires each measurement path to be the default routing path between a client and a server. As many of such problems are NP-hard, the focus is put on establishing the hardness of the optimal solution and developing polynomial-time suboptimal algorithms with performance guarantees. The chapter also covers a suite of path construction problems addressing how to construct or select measurement paths to optimize the tradeoff between identifiability and probing cost.