The rapid progress of the Internet of Things (IoT) has continued to offer humanity numerous benefits, including many security and safety-critical applications. However, unlocking the full potential of IoT applications, especially in high-consequence domains, requires the assurance that IoT devices will not constitute risk hazards to the users or the environment. To design safe, secure, and reliable IoT systems, numerous frameworks have been proposed to analyse the safety and security, among other properties. This paper reviews some of the prominent classical and model-based system engineering (MBSE) approaches for IoT systems’ safety and security analysis. The review established that most analysis frameworks are based on classical manual approaches, which independently evaluate the two properties. The manual frameworks tend to inherit the natural limitations of informal system modelling, such as human error, a cumbersome processes, time consumption, and a lack of support for reusability. Model-based approaches have been incorporated into the safety and security analysis process to simplify the analysis process and improve the system design’s efficiency and manageability. Conversely, the existing MBSE safety and security analysis approaches in the IoT environment are still in their infancy. The limited number of proposed MBSE approaches have only considered limited and simple scenarios, which are yet to adequately evaluate the complex interactions between the two properties in the IoT domain. The findings of this survey are that the existing methods have not adequately addressed the analysis of safety/security interdependencies, detailed cyber security quantification analysis, and the unified treatment of safety and security properties. The existing classical and MBSE frameworks’ limitations obviously create gaps for a meaningful assessment of IoT dependability. To address some of the gaps, we proposed a possible research direction for developing a novel MBSE approach for the IoT domain’s safety and security coanalysis framework.

The widespread adoption of Internet of Things (IoT) systems emphasises the need for dependable performance. However, challenges emerge due to the inadequacy of informal failure analysis models in evaluating the complex and distributed nature of IoT designs. These challenges can lead to inconsistencies, complicated system modifications, and extended development timelines. This paper presents a Model-Based Systems Engineering (MBSE) approach to create an integrated quantitative analysis framework for assessing the reliability of IoT systems. This approach simplifies the modelling and analysis of failure behaviours in IoT design, linking failure models to system design using standard engineering frameworks. Based on descriptions of IoT systems, the integrated approach establishes a traceability link between the system description and the analysable artefacts generated, such as stochastic Fault Tree Analysis (FTA) and Bayesian network (BN) models, which are essential for effective failure analysis of IoT systems. The approach demonstrates how to quantify system-level failures, identify and prioritise the weakest links in the design, and show how targeted modifications can improve the overall reliability of IoT systems. An illustrative case study on IoT is included to highlight the effectiveness and practical applicability of the proposed framework.

A multipartite system comprised of n subsystems, each of which is described with ‘local variables’ in $${{\mathbb {Z}}}(d)$$Z ( d ) and with a d-dimensional Hilbert space H(d), is considered. Local Fourier transforms in each subsystem are defined and related phase space methods are discussed (displacement operators, Wigner and Weyl functions, etc). A holistic view of the same system might be more appropriate in the case of strong interactions, which uses ‘global variables’ in $${{\mathbb {Z}}}(d^n)$$Z ( d n ) and a $$d^n$$d n -dimensional Hilbert space $$H(d^n)$$H ( d n ) . A global Fourier transform is then defined and related phase space methods are discussed. The local formalism is compared and contrasted with the global formalism. Depending on the values of d, n the local Fourier transform is unitarily inequivalent or unitarily equivalent to the global Fourier transform. Time evolution of the system in terms of both local and global variables, is discussed. The formalism can be useful in the general area of Fast Fourier transforms.

Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $$D=d^n$$D = d n with d an odd integer, and is inspired by the Cooley-Tukey formalism. The ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (together with some other transforms) in quantum systems with d-dimensional Hilbert space. Limitations of the method are discussed. In some special cases, the n Fourier transforms can be performed in parallel. The second method is for systems with dimension of the Hilbert space $$D=d_0...d_{n-1}$$D = d 0 . . . d n - 1 with $$d_0,...,d_{n-1}$$d 0 , . . . , d n - 1 odd integers coprime to each other. It is inspired by the Good formalism, which in turn is based on the Chinese reminder theorem. In this case also the ‘large Fourier transform’ is expressed as a sequence of n ‘small Fourier transforms’ (that involve some constants related to the number theory that describes the formalism). The ‘small Fourier transforms’ can be performed in a classical computer or in a quantum computer (in which case we have the additional well known advantages of quantum Fourier transform circuits). In the case that the small Fourier transforms are performed with a classical computer, complexity arguments for both methods show the reduction in computational time from $$\mathcal{O}(D^2)$$O ( D 2 ) to $$\mathcal{O}(D\log D)$$O ( D log D ) . The second method is also used for the fast calculation of Wigner and Weyl functions, in quantum systems with large finite dimension of the Hilbert space.

Two fractional Fourier transforms are used to define bi-fractional displacement operators, which interpolate between displacement operators and parity operators. They are used to define bi-fractional coherent states. They are also used to define the bi-fractional Wigner function, which is a two-parameter family of functions that interpolates between the Wigner function and the Weyl function. Links to the extended phase space formalism are also discussed.