Performance analysis and optimization of a cold standby system subject to δ-shocks and imperfect repairs

https://doi.org/10.1016/j.ress.2020.107330Get rights and content

Highlights

  • The effect of frequency of shocks is considered in this paper.

  • GP models are utilized to characterize the lifetime and repair time.

  • Some reliability measures of the system are obtained.

  • The optimal replacement policy is formulated based on the cost rate.

Abstract

This paper analyzes and optimizes the economic performance of a cold standby system subject to δ-shocks and imperfect repairs. Specially, the system consists of two components with different reliability characteristics. It is assumed that shocks arrive according to a Poisson process. When a component is active during operation, it will fail whenever the interarrival time between successive shocks is less than a threshold influenced by the number of repairs performed on this component. For reliability analysis, geometric process models are utilized to characterize the lifetime and repair time of the active component subject to δ-shocks and imperfect repairs. Using the supplementary variable method, some reliability measures of the system are obtained. Moreover, an explicit expression for the long-run cost per unit time is derived to quantify the system's economic performance. Finally, the optimal repair policy is determined based on this economic performance measure, and a sensitivity analysis is conducted to provide managerial insights for the efficient operation of such a system.

Introduction

Repairable systems subject to random shocks are quite common in practice. Such systems are widely seen in many fields, such as computer network, electric grid, manufacturing, and communication systems. For example, network computers often operate in an environment subject to invasion of viruses and malicious attacks. Such events arriving randomly can be treated as stochastic shocks, and frequent occurrences of such events will cause the network to crush if timely actions are not executed. In an electric grid, power surges are an unavoidable occurrence and may be severe enough to damage components, especially when surges occur too frequently. Due to the practical values, performance analysis and optimization of such systems have been studied by many researchers [1], [2], [3], [4].

Many studies on repairable systems assume that a system is good as new after repair. However, this assumption is not always true in practice. Due to aging and wear, the performance of the system is often getting worse. Lam [5,6] first introduced a geometric process (GP) to describe deteriorating systems. Specially, a stochastic process {ξn,n=1,2,} is called a GP, if there exists a real number a>0 such that {an1ξn,n=1,2,} forms a renewal process. The number a is called the ratio of the GP. Lam adopted a maintenance strategy N, and further discussed the uniqueness and monotonicity of the optimal maintenance strategy. Later, Zhang [7] studied the optimal binary replacement strategy (T*,N*) from the theoretical and numerical points of view and proved that under some conditions, the policy is better than a policy focusing only on T*orN*. As an extension, the GP has been applied to multi-component systems. By introducing two supplementary variables, Lam and Zhang [8] determined some reliability indices of GP model for a series system. Using a GP model, Zhang [9] considered the optimal replacement policy for a cold standby system. Zhang and Wu [10] derived reliability indices and the optimal replacement policy for a k/n(F) repairable system. With a bivariate mixed replacement policy(R,N), Wang and Zhang [11] studied the optimal policy for a simple repairable system subject to preventive repair and failure repair. Wang and Zhang [12] considered an optimal repair-replacement problem for a repairable system whose failures can be detected only by periodic inspections.

Stanley [13] obtained explicit expressions for the long run average cost per unit time by assuming that the survival times, the repair times and the magnitude of shocks after each failure constitute GPs. Lam and Zhang [14] studied a GP maintenance model with shocks, where shocks can reduce the system's operating time but cannot cause a failure. Castro and Pérez-Ocón [15] considered a reward optimization problem for a system subject to repairable and non-repairable failures. Wang and Zhang [16] analyzed a repairable system with two types of failures. Chen and Li [17] discussed a new extreme shock model for a deteriorating system where the system's deterioration was caused by both the external shocks and the internal load. Wu and Wu [18] and Wu [19] considered the reliability indices and optimal maintenance policy for a cold standby system under Poisson shocks.

According to different effects caused by shocks, shock models can be classified into different kinds, such as the cumulative shock model and extreme shock model. Readers are referred to [20], [21], [22] for more details. Sometimes, a system failure is caused by the short time interval (i.e., high frequency) between successive shocks. For example, flooding occurs in an area because two rainstorms are too close and water cannot be drained in time. Similar phenomena are also crucial to the resilience of critical infrastructure subject to disruptions and the operation of a healthcare system during a pandemic. To describe the effect of shock frequency on a system, Li first [23] introduced a δ-shock model. Lam and Zhang [24] studied a δ-shock model for the maintenance of a one-component deteriorating system. By assuming that shocks arrive according to a Poisson process and the threshold of a deadly shock is increasing with the number of repairs, they determined an optimal replacement policy. The model studied by Lam and Zhang [24] has been extended by Tang and Lam [25,26], where the interarrival times of shocks follow a Weibull, Gamma, or log-normal distribution.

It is worth pointing out that the δ-shock models studied in [24], [25], [26] considered only one component. So far, no work has been done on the use of GP models for a two-component system subject to δ-shock. In this paper, we generalize the model given in [24] by considering a two-component system. In particular, we consider a two-component cold standby system subject to δ-shocks. The shocks arrive according to a Poisson process. The threshold of a deadly shock increases with the number of repairs performed. The successive repair times on each component can be described by an increasing GP. The two components will be replaced by new ones when the number of repairs on Component 1 reaches a specific value N.Using supplementary variables, some reliability indices are determined. Moreover, the explicit expression for the long-run cost per unit time of the system is also derived. An optimal policy N* is proposed to minimize the long-run cost per unit time. Sensitivity analysis is also provided.

The remainder of this paper is organized as follows. Section 2 provides a detailed description of the two-component cold standby system subject to δ-shocks and imperfect repairs. In Section 3, different reliability measures of the system are derived based on a collection of differential equations related to the system's state probabilities. Section 4 derives the explicit expression for the long-run expected reward per unit time under the replacement policy N based on the renewal reward theory. A numerical example is presented in Section 5 to illustrate the use of the proposed model in practice. Finally, Section 6 concludes this paper.

To introduce clearly, first we give some notations of symbols.

Notation

Section snippets

Model Description

In this paper, we consider a δ-shock model for a two-component cold standby system. The assumptions of the model are as follows.

  • (1)

    The system consists of two different components and is repaired by a repairman. At the beginning, both components are new, component 1 is active, and component 2 is under cold standby.

  • (2)

    If the active component fails, it will be replaced by the standby component when available, and the failed component will be repaired. If the standby component is not available at

Reliability Measures

In this section, we first construct the differential equations of state probabilities of the system using the supplementary variable method, and solve the differential equations to calculate the state probabilities of the system.

Let S(t) be the state of the two-component cold standby system with the following possibilities:S(t)={01component1isworking,component2iscoldstandby.02component2isworking,component1iscoldstandby.11component1isworking,component2isrepaired.12component2isworking,component1is

Cost Analysis

In this section, we first derive the explicit expression for the long-run expected reward per unit time under the replacement policy N using the renewal reward theory.

Let T1 be the first replacement time of the system and Tn(n2) be the time between the(n1)th replacement and the nth replacement of the system. {T1,T2,} forms a renewal process, and the interarrival time between two consecutive replacements is called a renewal cycle. Then, the renewal process and the reward in each cycle

Numerical Example

In this subsection, we first provide a numerical example to illustrate how to determine the optimal replacement policy N* that minimizes the long-run cost per unit time. Next, we perform a sensitivity analysis on the optimal replacement policy N* and the optimal long-run cost per unit time of the system C(N*) along with the effects of changing different pairs of parameters: (b1,b2) and (θ,δ).

To describe a deteriorating system, we assume α1=1.10, α2=1.05,b1=0.95, and b2=0.90. When αi>1, i=1,2,

Conclusion

In this paper, we study a δ-shock model for a two-unit cold standby system subject to imperfect repairs. First, the state probabilities of the system are obtained using the supplementary variable method, and the reliability indices of the system are derived. By using the renewal reward theory, the expected cost per unit time of the system is derived and used in a replacement policy N. The procedure of finding the optimal maintenance strategy is illustrated in a numerical example. The

CRediT authorship contribution statement

Bing Zhao: Methodology, Writing - original draft. Dequan Yue: Conceptualization. Haitao Liao: Supervision, Writing - review & editing. Yuanhui Liu: Resources. Xiaohong Zhang: Software.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 11301458) and the National Natural Science Foundation of China (Grant No. 71701140).

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