Reliability Evaluation and Optimization of a System with Mixed Run Shock

Considering maintenance resources assignments of the large-scale complex equipment, the policy that single repairman has multiple vacations is utilized to analyze a warm standby repairable system. The former models that the known conditions are usually assumed to follow exponential distribution or other typical distributions have many constraint conditions. It is assumed that the online component lifetime, standby components lifetime, repair time and repairman vacation time are subject to different continuous Phase-type (PH) distribution, respectively. The reliability model with high versatile is acquired. The reliability features, such as system reliability function, system stationary availability, stationary failure frequency and mean time between failures (MTBF), are provided by the model. Finally, the applicability of the model is implemented by numerical application, and the influences of standby components counts and vacations or non-vacation of a repairman on the system reliability laws are demonstrated.

In He et al.[8], continuous phase-type (PH) distributions are constructed to approximate finite discrete probability distributions and discrete PH-distributions. The approximations are based on Erlangization with a fixed number of phases. In this article, we first introduce continuous PH approximations with Erlang distributions of different orders. Then we develop an algorithm to find the continuous PH approximation with the minimum variance, among all such PH approximations with the same total number of phases. Thus, the proposed continuous PH approximations lead to a smaller gap between the variances of the Erlangization-based approximations and the original discrete random variables, which is achieved without adding more phases. The new approximations are useful to mitigate the burden in computation caused by the large number of phases needed in Erlangization approximation. Stochastic dominance is shown between the original (discrete) distributions and the approximations, which leads to bounds on the quantities of original distributions and/or stochastic models (e.g., reliability models). The approximation method is applied to analyze reliability models and a COVID-19 isolation program.

This paper proposes a bivariate (T, L2) replacement policy with double repair-cost limits for a system subject to increasing failure rate model. External shocks are divided into two types: type-I shock and type-II shock. Each type-II shock makes the system suffer a minor failure, while each type-I shock increases the failure rate of the system by a certain amount and it will induce the system into critical failure. When a minor failure occurs, the repair cost is evaluated and minimal repair is executed if the repair cost is less than a single repair-cost limit L1 and the accumulated repair cost is less than a cumulative repair-cost limit L2, otherwise, the system is replaced by a new one. In addition, the system is replaced at scheduled time T or at critical failure. By formulating the optimisation problem, the optimal T* and L2* which minimise the long-run expected cost per unit time are found, respectively. We develop the corresponding computational algorithm to obtain the optimal replacement policy and present a numerical example to illustrate the effectiveness of the proposed model.

Phase-type (PH) distributions are being used to model a wide range of phenomena in performance and dependability evaluation. The resulting models may be employed in analytical as well as in simulation-driven approaches. Simulations require the efficient generation of random variates from PH distributions. PH distributions have different representations and different associated computational costs for pseudo random-variate generation (PRVG). In this paper we study the problem of efficient representation and efficient generation of PH distributed variates. We introduce various PH representations of different sizes and optimize them according to different cost functions associated with PRVG.

Reliability of a multi-state system subject to shocks using phase-type distributions

Effective and efficient emergency medical services (EMS) are a critical part of a national healthcare system. This paper describes research to model EMS by better capturing ambulance service times using Coxian phase-type (PH) distributions. Distributions are fitted to both the overall cycle time for different classes of patient priorities, as well as to sub-cycles. Sub-cycles are the distinct identifiable parts of the ambulance cycle time, such as travel times, time on scene and turnaround time at the hospital. The Coxian PH fits have then been used within a priority simulation model to provide guidance on the number of ambulances required to meet response time targets. Results from using various numbers of phases from the fitted Coxian distributions are compared. The proposed benefit of using sub-cycle fits is that it more readily permits scenario modelling within the simulation, such as evaluating the impact of reducing the turnaround time on the overall response times in the ambulance system.

In this paper, the sensitivity of the handoff dwell time (HDT) model on wireless mobile network performance is studied. We present an approach on the basis of the universal approximation property of phase-type (PH) distribution in order to address the intractable numerical issue caused by HDT. We first obtain the steady state probability under the general HDT. The performance metrics from the perspective of base station (BS) as well as from the point view of call are developed. The PH-type HDT is then assumed to derive the numerically tractable expressions for the performance metrics. Finally, the truncated Gaussian HDT is approximately fitted by the PH distribution via expectation maximization algorithm to reuse the closed form analytical expression. The comparison shows that, in terms of the BS performance, the handoff blocking probability exhibits a significant discrepancy whenever one uses the exponential or truncated Gaussian HDT distribution, while the new call blocking probability is less sensitive to different HDT. In addition, HDT has little impact on the call performance with respect to the call completion probability and total waiting time. Both the analytical model and the approximation approach are verified by simulation.

This paper studies a series repairable system with a single repair facility in which the component lifetime and repair time are subject to general distribution that can be represented as Phase-type (PH) distribution. Based on the basic assumption, the proposed model is more suitable to characterize the real situation. The process of solving system steady-state probability is provided, and the analytical expressions for system stationary availability, mean time between failures (MTBF), mean down time (MDT), reliability and failure rate functions are expressed. Finally, the validity and applicability of the model are illustrated with a numerical example.

Movements in palm oil price give important signals to various stakeholders of the palm oil industry in Malaysia. Thus, understanding external and internal factors that may affect the palm oil price is vital to the industry players for sustainability of their activities. This study investigates relative importance of external and internal shocks on the movement of palm oil price in Malaysia. Employing a structural vector autoregressive (SVAR) model on quarterly data from 1990 to 2019, the findings reveal that external shocks are more dominant in affecting the palm oil price. Shocks to the crude oil price, the prices of substitution goods (soybeans oil, rapeseed oil, and sunflower oil), the world palm oil price, and foreign income significantly affect the palm oil price in the short and medium run. The results also indicate that a shock to soybean oil price has a more profound effect on the palm oil price than a shock to rapeseed oil or sunflower oil prices, respectively. Likewise, shocks to incomes from India as well as from Netherlands create greater impacts on the palm oil price than a shock to income from the other trading partners, respectively. The study has shown the importance of external factors in affecting the palm oil industry.

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

Efficient and accurate methods are indispensible in the evaluation of complex systems. Methods of various abstraction levels may be applied, ranging from measurements in test-beds over simulation to analytical approaches. These methods vary in their efficiency and accuracy. In order to obtain realistic results, methods from all abstraction levels should be applied. Furthermore, real-world phenomena must be represented in a suitable way at all abstraction levels. This approach requires the use of stochastic models for the faults that may affect a system. This thesis explores several important issues of system evaluation using stochastic fault models. The first part of the thesis provides a survey of stochastic fault-models for system evaluation. We survey a broad range of existing fault-models and provide a new classification scheme for fault models in service-oriented systems. In the second part of the thesis we address the use of phase-type (PH) distributions as fault models in system evaluation. We study algorithms for random-variate generation and discuss their computational costs. An optimisation method for reducing the cost of random-variate generation by transformation of the representation is presented and an optimality result for the sub-class of acyclic phase-type distributions is derived. We then develop a new algorithm for efficient and user-friendly fitting of PH distributions to data sets using clustering. The algorithm is implemented as part of the HyperStar tool. We additionally illustrate application of HyperStar as a graphical user interface for prototype algorithms using a new algorithm for fitting general PH distributions. The third part illustrates application of the results from the first and second parts. We first study the application of stochastic fault models for IP packet loss in fault-injection and investigate the interaction between fault models and traffic arrival streams and its impact on evaluation results. We then describe a library for random-variate generation from phase-type distributions and show its use in two case-studies.

Phase-type (PH) fitting is a technique to approximate any general distribution as a PH distribution, which is a probability distribution representing an absorbing time of a Markov chain. Since the PH distribution is described as a discrete- or continuous-time Markov chain (CTMC), the PH fitting can provide approximate Markov models to any non-exponential stochastic models. Thus, the PH fitting is helpful for model-based performance evaluation. On the other hand, from the statistical point of view, the PH fitting is categorized as parameter estimation from data. Some efficient PH fitting techniques are based on the maximum-likelihood principle. Therefore, it is crucial to evaluate statistical errors, i.e., the variance and covariance of estimators. In maximum-likelihood estimation, the Fisher information matrix is a well-known method to compute the variance and covariance of estimators, and is obtained as the second derivative of the log-likelihood function (LLF). In this article, we propose an algorithm for efficiently computing the Fisher information matrix in PH fitting. By applying the uniformization technique to a CTMC, we design the algorithm for computing the second derivatives of LLF in PH fitting.

The paper investigates the dynamic relationship between external and internal shocks and the current account in selected small islands developing states. External shocks, defined as terms of trade fluctuations, explain a significant proportion of the variation in the current account balances. The external shocks have a temporary negative impact on the current account balances with a subsequent improvement, generating a J-curve type reaction. In contrast, real output shocks have a positive and significant effect on the current account.

A single component nonrepairable system suffering from both an internal stochastic degradation process and external random shocks is investigated in this paper. More specifically, the Wiener process with a positive drift coefficient is introduced to describe the gradual deterioration and the arrival number of external shocks is counted with a nonhomogeneous Poisson process (NHPP). Meanwhile, fault tolerant design is incorporated into the stochastically deterioration system so as to protect it from shock failures to some extent and is consummately addressed via a generalized m − δ shock model. From the actual engineering point of view, external shocks are typically classified into two distinct categories in this current research, that is, a minor shock (Type I shock) increasing the damage load on current degradation level and a traumatic shock (Type II shock) resulting in system catastrophic failure immediately. The closed‐form expression of system survival function is derived analytically and is viewed as the generalization of existing reliability function for systems subject to dependent and competing failure processes. Based on which, two time‐based maintenance (TBM) policies including an age replacement model and a block replacement model are scheduled, where the expected long‐run cost rate (ELRCR) in each model is, respectively, optimized to seek the optimal replacement interval. In the illustrative example part, a subsea blowout preventer (BOP) control system is arranged to validate the theoretical results numerically. To compare which policy is more profitable under different conditions, the relative gain on optimal maintenance cost rate of the two TBM policies is presented.

We consider a shock and wear model in which the inter-shock arrival process is a phase-type (PH) renewal process, and the system’s lifetime is generally distributed. The system has two competing failure modes. One failure mode is due to random shocks, which cause failure by overloading the system. The other failure mode is owning to wear-out failures, which usually happen after the system has run for many cycles. System failure is not self-announcing and remains undiscovered unless an inspection is performed. The intervals between successive inspections are identical and equal to T time units. If a system failure is detected, the corrective repair or replacement is conducted immediately. If the system is found working at inspection, preventive maintenance will be carried out to prolong its useful life. Furthermore, to model the occurrence of events with an underlying monotonic trend, the extended geometric process (EGP) is employed to account for the impact of different types of failures on the system’s degradation. Moreover, for establishing the cost rate function in our model, the counting process generated from a PH renewal shock process is studied in detail using the roots method and formula for calculating residues. Based on these results, the survival function and other characteristics of the system are further investigated. Finally, numerical examples that determine the optimal inspection period T[Formula: see text] and the optimal replacement policy N[Formula: see text], which minimizes the long-run average cost rate, are presented.