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Security of cyber-physical systems (CPS) has been a prime concern due to the ever-increasing integration of communication networks with physical devices. In this paper, we propose a zero − sum stochastic game model to capture the system dynamics resulting from the attacker − defender interactions in an attacked CPS. In addition to predicting the attacker behavior in terms of optimal strategies, the best strategies for counteracting the malicious activities are also determined. The lifetime of the attacked CPS is characterized based on a continuous time Markov chain (CTMC) and consequently, mean time to failure of the system is derived. Further, we address dependability issues of the system by formulating a semi-Markov model and obtaining different dependability measures such as reliability, availability and confidentiality. The studies identify critical states of the system with respect to its dependability. Theoretical results are also validated via simulation. The results of the studies are expected to be helpful in developing efficient and robust defense mechanisms.

The reliability–redundancy allocation problem (RRAP) involves the selection of components reliability and redundancy levels, with the aim of maximizing system reliability. Generally, RRAPs are formulated with the assumption of active redundancy strategy. In this paper, the problem is formulated by considering a cold-standby strategy. In most practical cases, complicated decision-making problems are encountered due to the presence of several conflicting objectives such as maximizing system reliability and simultaneously minimizing the total cost, weight or volume of the systems. Therefore, it is essential to consider these conflicting criteria as the main objectives of the optimization problem. Hence, in this paper, a bi-objective RRAP is developed. The proposed mathematical model is developed for three benchmark problems and solved by using a multi-objective evolutionary algorithm (NSGA-II). In order to calculate the exact reliability values for the problems, a recently introduced Markov-based approach is used. The results indicate that, in comparison with recent studies, the proposed NSGA-II shows superior performance and leads to better results in all benchmark problems. In other words, for each problem, a variety of structures are presented which outperform the best active structure and at least one structure is obtained that outperforms the best standby one in the literature.

The main focus of this paper is on the development of reliability measures for a repairable multi-state system which operates under dynamic regimes under the discrete-time hypothesis. The switching process of regimes is governed by a Markov chain, and the functioning process of the system follows another Markov chain with different transition probability matrices under different regimes. In terms of two chains as above, a new Markov chain is constructed to depict the evolution process of the dynamic system. For the regime consideration, some novel reliability indices are essential and firstly introduced in this paper. By means of hierarchical partitions for the new state space, Ion-Channel modeling theory and discrete-time Markov chain, the traditional and novel reliability and availability functions for the system under random regimes are easily obtained with the closed form solutions, such as two types of system reliabilities, two types of system point availabilities, two types of system multiple-point availabilities and the associated system multi-interval availabilities and so on. In addition, some probability distributions of sojourn times we are interested in are discussed and computed here. Finally, a numerical example is given to illustrate the results obtained in the paper.

This research paper presents practical stochastic models for designing and analyzing the time-dependent reliability of nonrepairable systems. The models are formulated for nonrepairable systems with heterogeneous components having phase-type time-to-failure distributions by a structured continuous time Markov chain (CTMC). The versatility of the phase-type distributions enhances the flexibility and practicality of the systems. By virtue of these benefits, studies in reliability engineering can be more advanced than the previous studies. This study attempts to solve a redundancy allocation problem (RAP) by using these new models. The implications of mixing components, redundancy levels, and redundancy strategies are simultaneously considered to maximize the reliability of a system. An imperfect switching case in a standby redundant system is also considered. Furthermore, the experimental results for a well-known RAP benchmark problem are presented to demonstrate the approximating error of the previous reliability function for a standby redundant system and the usefulness of the current research.

The emergence of the smart grid has brought great innovation in the new distribution power system, facilitated a green and sustainable energy-based society, and mitigated the growing energy crisis. The smart grid is regarded as the next generation electrical power grid. It not only improves the power distribution systems with the techniques of distributed generation, but also makes the systems more complicated than the traditional ones. Thus it is important to evaluate the availability of smarts grids and to indicate the vulnerable parts of the systems. In this paper, a hierarchical model consisting of a layered fault tree (FT) and continuous-time Markov chains (CTMCs) is presented to model a smart grid system with two different power supply modes. We also analyze the component importance of the system, aiming to find the weak parts of the system thereby improving the system design. The importance analysis of components is based on parametric sensitivities and binary decision diagram (BDD) representation for the FTs. In a numerical illustration, we quantify the availability of the system with two power supply modes and also evaluate the importance of all components in the system.

The polygonization of high-speed train wheels is commonly considered as a serious threat to reliability of high-speed trains all over the world. Traditional studies in polygonization of wheels focus on modeling its mechanism via physics-based models. However, to the knowledge of the authors, the polygonization process has not been fully explained by existing studies due to the complex degradation mechanism and varying operation environment which is difficult to monitor and measure. In this article, the data-driven multi-state system (MSS) models are developed for the reliability assessment of high-speed train wheels based on the analysis of actual operation data. The results can be used to support practical maintenance decision making. Another contribution of this paper is to prove that the environmental factors exert significant impacts on the reliability of Chinese high-speed train wheels. This finding has been included in the development of the MSS models. The results by these models are used to suggest a more reasonable maintenance schedule for Chinese high-speed train wheels. The obtained maintenance intervals are expected to assist the railway companies to reduce the operational risk as well as improve the maintenance efficiency.

This paper revisits the redundancy allocation problem (RAP) from the viewpoint of the mixed redundancy strategy. This powerful strategy has some major weaknesses originated from its complicated formulation. Previously, for estimating the reliability of this strategy a lower bound formulation was introduced. In an attempt to improve the complicated, time-consuming, and imprecise lower bound formulation previously proposed, in this paper, an exact Markov based approach is developed. Being a powerful and robust tool, it is especially advantageous for its short computation time. This is demonstrated by its implementation to solve a well-known benchmark test problem. Based on the results, the new approach finds better solutions with higher reliability values than before, with a significant reduction in the computation time.

This paper innovatively proposes a new model of multi-type production system which can process different types of production. The multi-type production system consists of several independent sequentially ordered subsystems which undertaking different tasks. Different types of production require different manufacturing lines: required subsystems will be powered on, while others powered off. The system can switch into different production line based on detection results. A binary matrix is suggested to describe the composition of each production line of the multi-type production system. Reliability-redundancy allocation on each subsystem is considered for achieving higher reliabilities of different production lines. The reliability-redundancy allocation problem for the proposed multi-type production system is essentially multi-objective subject to resource constraints. This paper formulates the multi-objective optimization problem and considers the optimal redundancy strategy, either active or cold standby. The exact reliabilities of cold standby redundant subsystems with imperfect detector/switch are determined by introducing an approach based on continuous time Markov chain. Then, a multi-objective evolutionary algorithm is suggested to solve the proposed reliability-redundancy allocation problem. Finally, numerical examples are presented to illustrate the proposed methods.

Despite the fact that the fields of pull type production control policies and condition-based preventive maintenance have much in common contextually, they have evolved independently up to now. In this investigation, an attempt is made to bridge the gap between these two branches of knowledge by introducing the single-stage Kanban system with deterioration failures and condition-based preventive maintenance. The formalism of continuous time Markov chains is used to model the system and expressions for eight performance metrics are derived. Two important, from a managerial perspective, constrained optimization problems for the proposed model are defined where the objective is the simultaneous optimization of the Kanban policy, the preventive maintenance policy and the inspection schedule under conflicting performance criteria. Multiple instances of each optimization problem are solved by means of the augmented Lagrangian genetic algorithm. The results from the optimization trials coupled by the results from extensive numerical examples facilitate the thorough investigation of the system’s behaviour.

This paper presents a multi-state Markov model for a coal power generating unit. The paper proposes a technique for the estimation of transition intensities (rates) between the various generating capacity levels of the unit based on field observation. The technique can be applied to such units where output generating capacity is uniformly distributed. In order to estimate the transition intensities a special Markov chain embedded in the observed capacity process was defined. By using this technique, all transition intensities can be estimated from the observed realization of the unit generating capacity stochastic process. The proposed multi-state Markov model was used to calculate important reliability indices such as the Forced Outage Rate (FOR), the Expected Energy Not Supplied (EENS) to consumers, etc. These indices were found for short-time periods (about 100 h). It was shown that these indices are sensibly different from those calculated for a long-term range. Such Markov models could be very useful for power system security analysis and short-term operating decisions.

In an earlier paper, a closed form expression was obtained for the joint interval reliability of a Markov system with a partitioned state space S = U ∪ D , i.e. for the probability that the system will reside in the set of up states U throughout the union of some specific disjoint time intervals I l = [ θ l , θ l + ζ l ] , l = 1 , … , k . The deterministic time intervals I l formed a demand pattern specifying the desired active periods. In the present paper, we admit stochastic demand patterns by assuming that the lengths of the active periods, ζ l , as well as the lengths of the neutral periods, θ l - ( θ l - 1 + ζ l - 1 ) , are random. We explore two mechanisms for modelling random demand: (1) by alternating renewal processes; (2) by sojourn times of some continuous time Markov chain with a partitioned state space. The first construction results in an expression in terms of a revised version of the moment generating functions of the sojourns of the alternating renewal process. The second construction involves the probability that a Markov chain follows certain patterns of visits to some groups of states and yields an expression using Kronecker matrix operations. The model of a small computer system is analysed to exemplify the ideas.

Quantitative safety assessment of a safety system plays an important role in comparing design alternatives at design stage and deciding appropriate design options to apply for safety systems. There are a number of such indices given in the literature. Most of the safety indices consider only system parameters (hazard rate, repair rate, diagnosis, coverage, etc.) along with proof-tests (or inspection). This paper extends the underlying model to incorporate demand rate and imperfect proof-tests. It also introduces a new safety index, average probability of failure on actual demand (PFaD), and an availability index, manifested availability (mAv). This paper uses Markov regenerative process-based analysis for state probabilities. Based on state-probability values of various states of the underlying Markov chain, solutions are derived for safety index PFaD and availability mAv.

Software rejuvenation is an important way to counteract the phenomenon of software aging and system failures. It is a preventive and proactive technique, which consists of periodically restarting an application at a clean internal state. Starting an application generally means that an amount of memory is captured and closing an application engenders the release of an amount of memory. In general, when an application is initiated an amount of memory is captured and when terminated an amount of memory is released. In this paper a model describing the amount of free memory on a system is presented. The modelling is formulated under a continuous time Markov chain framework. Additionally the cost of performing rejuvenation is also taken into consideration, a cost function for the model is produced and a rejuvenation policy is proposed. The contribution of this paper consists of using a cyclic non-homogeneous Markov chain in order to study the overall behaviour of the system capturing time dependence of the rejuvenation rates and deriving an optimal rejuvenation policy. Finally, a case study is presented in order to illustrate the results of the cost analysis.

A non-homogeneous semi-Markov process is considered as an approach to model reliability characteristics of components or small systems with complex test resp. maintenance strategies. This approach generalizes previous results achieved for ordinary inhomogeneous Markov processes. This paper focuses on the following topics to make the application of semi-Markovian models feasible: rather than transition probabilities Qij(t), which are used in normal mathematical text books to define semi-Markov processes, transition rates λ ij ( ) are used, as is usual for ordinary Markov processes. These transition rates may depend on two types of time in general: on process time and on sojourn time in state i. Such transition rates can be followed from failure and repair rates of the underlying technical components, in much the same way, as this is known for ordinary Markov processes. Rather than immediately starting to solve the Kolmogorov equations, which would result in N2 integral equations, a system of N integral equations for frequency densities of reaching states is considered. Once this system is solved, the initial value problem for state probabilities can be solved by straightforward integration. An example involving 14 states has been solved as an illustration using the approach.

A new approach for computing the sensitivity of fault-tolerant systems modeled by a Markov process, as the derivative of the transient solution with respect to a system parameter, is proposed. We compare this approach with respect to the Standard Uniformization technique from a computation time point of view. This new method is faster than the Standard Uniformization method if the mission time is long and the state space is not too large.

In this paper, the limiting distribution of the residual lifetime of a Markov repairable system is proved to be an exponential distribution. Further, we give the parameters of the exponential distributions for several special Markov repairable systems.