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Finite state Markov processes and their aggregated Markov processes have been extensively studied, especially in ion channel modeling and reliability modeling. In reliability field, the asymptotic behaviors of repairable systems modeled by both processes have been paid much attention to. For a Markov process, it is well-known that limiting measures such as availability and transition probability do not depend on the initial state of the process. However, for an aggregated Markov process, it is difficult to directly know whether this conclusion holds true or not from the limiting measure formulas expressed by the Laplace transforms. In this paper, four limiting measures expressed by Laplace transforms are proved to be independent of the initial state through Tauber's theorem. The proof is presented under the assumption that the rank of transition rate matrix is one less than the dimension of state space for the Markov process, which includes the case that all states communicate with each other. Some numerical examples and discussions based on these are presented to illustrate the results directly and to show future related research topics. Finally, the conclusion of the paper is given. [ABSTRACT FROM AUTHOR]

Dependency plays a key role in the design stage of multi-component manufacturing systems. To capture stochastic dependencies between subsystems, this paper considers a load-sharing k -out-of- n : G system which contains non-identical multi-state subsystems. The state evolution of each subsystem follows a continuous-time homogeneous Markov chain. Most existing research on k -out-of- n : G systems focuses on independent subsystems. However, a subsystem failure usually leads to a higher failure rate for each surviving subsystem within the system. This paper aims to study the interaction among subsystems and analyse the system reliability performance. The transition rate matrix for a given subsystem is assumed to depend on the total number of the remaining surviving subsystems, which can be represented as the sum of a baseline transition rate matrix and a stochastic dependency matrix. This paper proves that the transition rate matrix of the system is the generalised Kronecker sum of transition rate matrices of subsystems, and develops an explicit method to calculate the generalised Kronecker sum. The theory of aggregated stochastic processes is employed to obtain closed-form formulas for reliability indexes. A case study of multi-engine aircraft systems is provided where numerical examples are given to illustrate the developed model and obtained results. [ABSTRACT FROM AUTHOR]

In this paper, state space of a time-homogeneous Markov process is split into several ordered subspaces. Then, there are three kinds of transitions between states—transitions from a higher-order subspace to a lower-order subspace, transitions within the same subspace, and transitions from a lower-order subspace to a higher-order subspace. Considering time interval omission problem and first passage time considered, we define some related counting processes for the Markov process and discuss their associated probabilities, expectations and generating functions by using Laplace transform. The main results are presented in matrix forms. The relationships among counting processes and their special cases are also discussed briefly. Some simple numerical examples are presented to illustrate the established results. These results will be useful in different problems arising in reliability, economics and social science fields. [ABSTRACT FROM AUTHOR]

Reliability evaluation under shock models has attracted considerable attention in the literature. Existing research on shock models has focused primarily on damage effects of shocks treated as constants after the arrivals of shocks. Due to the presence of self‐healing, deterioration, and variation over time, the damage effects caused by the shocks may have some evolutions. In this work, we focus on arrivals of shocks that follow a continuous‐time renewal process, and a Markov process that describes the evolution of shock damage. In this regard, we develop here two kinds of shock models and then derive system reliabilities under these two kinds of shock models. The asymptotic behavior of the damage evolution process of each shock is then discussed, and the probabilities of each shock eventually disappearing and destroying the system are derived through the use of aggregated stochastic processes. Finally, the results and numerical examples are presented for three special cases, namely: (1) when the inter‐arrival times of the shocks follow exponential distribution, (2) when the inter‐arrival times of the shocks follow Erlang distribution, and (3) when the inter‐arrival times of the shocks follow uniform distribution. [ABSTRACT FROM AUTHOR]

Maintenance planning and optimizations are very important issues in both theory and practice. In this paper, Hawkes processes, a tool for description of self-excited failure effects, are used to model the failure processes of single-unit systems under the assumption of neglected repair times. For such degraded single-unit systems, some maintenance planning estimations such as the distribution, mean and variance of failures or repairs by time t , reliability, mean cost of maintenance, mean and variance of system lifetime are presented under the maximal entropy assumption on repair effects. Meanwhile, two optimization problems are developed by considering the lifetime mean of the single-unit system and total maintenance cost as objective functions and constraints alternatively. We have proved that both optimization problems can reduce to two simplified optimal cases, respectively, and their optimal policies are increasing stepwise sequences of increasing rates of failure rates, which greatly reduce the scope of optimal solutions. The relationships between the estimations and the optimizations are discussed. Furthermore, the detailed optimal maintenance policies for a special case are given as well including an algorithm. [ABSTRACT FROM AUTHOR]

Shock models have been widely studied in reliability area, because many real phenomena in reliability engineering can be depicted by the shock models. In general, the damage effects of shocks are treated as constants after the shocks arrive. In fact, the damage effects resulted from shocks may have some evolutions due to some self-healings, deteriorations and variations over time. In the present paper, three types of shock models with damage effect evolutions following Markov processes are developed in discrete time situations, then the system reliability formulas and the means and variances of their lifetimes are given. The relationship among the introduced shock models is discussed, and some special cases for three types of shock models are also studied. The asymptotic behaviors of damage evolution processes are presented for a special case. Some numerical examples are presented to illustrate the results obtained in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate systems subject to random shocks that are classified into critical and noncritical categories, and develop two novel critical shock models. Classical extreme shock models and run shock models are special cases of our developed models. The system fails when the total number of critical shocks reaches a predetermined threshold, or when the system stays in an environment that induces critical shocks for a preset threshold time, corresponding to failure mechanisms of the developed two critical shock models respectively. Markov renewal processes are employed to capture the magnitude and interarrival time dependency of environment‐induced shocks. Explicit formulas for systems under the two critical shock models are derived, including the reliability function, the mean time to failure and so on. Furthermore, the two critical shock models are extended to the random threshold case and the integrated case where formulas of the reliability indexes of the systems are provided. Finally, a case study of a lithium‐ion battery system is conducted to illustrate the proposed models and the obtained results. [ABSTRACT FROM AUTHOR]