A multi-state shock model with mutative failure patterns

Highlights

• A multi-state shock model with three types of shocks is proposed.

• The failure pattern mutates when the system state drops to a lower one.

• The finite Markov chain imbedding approach is used to obtain reliability indices.

• The phase-type distribution is used to depict the interarrival time of two shocks.

• Three different replacement policies are proposed for the new model.

Abstract

In this paper, to accommodate the known fact that a multi-state system is more likely to fail when its state gets worse, a multi-state shock model is proposed, in which the system failure patterns are mutative for different states. Shocks are classified into type I, type II and type III by magnitude. A type III shock has a destructive effect on the system and causes a complete failure whenever it occurs. A type II shock triggers the system down to a lower state. The system completely fails when the additive number of type II shocks exceeds a threshold. A type I shock has different effects for different system states. The system completely fails if the cumulative number of type I shocks exceeds a threshold which decreases as the system state gets lower. Distributions of the lasting time until the end of each system state, the lifetime and the residual lifetime of the system are derived when the interarrival times between successive shocks follow a common continuous phase-type distribution. Three different replacement policies are proposed to fit the characteristic of the new model, and corresponding optimization models are constructed to gain the optimal quantities.

Introduction

Shock models have been widely used in engineering fields. They are mainly used to study the systems which are subject to a sequence of shocks over time. Shocks can be considered as the impacts of sudden changes in the system operation or external effects of the working environment, and they can cause the system failure or reduce its residual lifetime. Various shock models have been defined and studied in the previous literature, which can be classified into five groups [1]: extreme shock model [2], cumulative shock model [3], run shock model [4], δ-shock model [5], [6], [7] and mixed shock model [8]. In the extreme shock model, a system breaks down because of an individual shock with the magnitude that exceeds a critical level. In recent years, the extreme shock model has been extended by Cha and Finkelstein [9], Cirillo and Hüsler [10] and Eryilmaz [1]. According to the cumulative shock model, a system fails when the cumulative effect of shocks exceeds a threshold. Several extended cumulative shock models have also been proposed by Bai et al. [11], Montoro-Cazorla and Pérez-Ocón [12] as well as Eryilmaz [13]. A mixed shock model is obtained by combining two different shock models. For example, a system fails if either the cumulative effect or an extreme shock reaches its critical level, whichever comes first [8]. Such mixed shock model has been extended into a more realistic case by Gut and Hüјsler [14]. Other extended mixed shock models and related studies have also been done in recent years [15], [16], [17]. Additionally, the dependency of degradation and shock is also studied by many researchers [18], [19], [20].

However, the abovementioned extreme, cumulative and mixed shock models mostly consider a binary-state system, where both the system and components only have two states, i.e., perfect functioning state and complete failure state. However, many engineering systems belong to a class called Multi-state system (MSS), in which both the system and its components may exhibit more than two states, ranging from perfect functioning state to complete failure state [21], [22], [23]. MSSs have wide applications in many areas such as engineering reliability, population dynamics, game theoretical models, and medicine [22], [24], [25], [26]. Multi-state k-out-of-n system and multi-state consecutive-k-out-of-n system were proposed by Tian et al. [27] and Huang et al. [28], respectively. Multi-state systems with common bus performance sharing were presented by Levitin [29]. A heuristic approach was used to solve the redundancy optimization problem in multi-state series-parallel systems by Agarwal and Gupta [30]. Additionally, numerous studies that are relevant to MSS have also been done in recent years [31], [32], [33].

A multi-state system under an extreme shock model is firstly proposed by Eryilmaz [1]. In such a shock model, the system fails when a shock with large magnitude occurs. If the magnitude of a shock is between two critical levels, it has partial damage and leads the system to a lower state. However, there are two main limitations in this model. One is that the number of states in such a multi-state system is random. Specifically, although the working state gets lower and lower, the system never breaks down, no matter how many shocks with partial damage occur before a large shock. Another limitation is that the failure pattern is invariable no matter which state the system is in. In a more realistic case, the working performance might be worse when the system state gets lower so that the system is more fragile and increasingly possible to fail. Thus, the failure pattern is likely to mutate during its operation. In the previous studies, this situation is barely considered, and the failure pattern is invariable. In order to improve the applicability of the previous model, we extend it into a more realistic situation. In this paper, we are the first to propose a multi-state shock model with mutative failure patterns by combining extreme and cumulative shock models, and employing mutative failure patterns according to different system states.

Consider a system which is subject to shocks with random magnitudes at random times. Shocks are divided into three types according to their magnitude. A type III shock, whose magnitude is the largest, is considered as an extreme shock with a catastrophic effect on the system and causes a complete failure whenever it occurs. A type II shock, with a medium magnitude, leads the system to a lower working state, and the system completely fails when a certain number of type II shocks occur. A type I shock has the smallest magnitude and cumulative effect, implying that a certain amount of type I shocks will also lead to system failure. The critical number of type I shocks to break down the system decreases when the system state gets lower, because the system is more fragile and more likely to fail in a lower state.

Such a model is motivated by an actual engineering problem of a sliding spool according to a similar situation suggested by Fan et al. [34]. A sliding spool consists of a spool and a sleeve, and is an important component to control the hydraulic oil flows in a hydraulic system [35], [36]. A sliding spool is subject to different failure mechanisms [37]. An important failure cause is clamping stagnation, in which case the spool is stuck in the sleeve. In practice, the sudden appearance of contaminant in the hydraulic oil is a major cause of the clamping stagnation, and this failure process can be modeled by a random shock model [38].

According to many experimental results, the most severe effect of shocks is generated by the wear debris [38], [39], [40]. The wear debris can be divided into three types according to its size: small-sized, medium-sized and large-sized debris, which can be regarded as the type I, type II and type III shocks in the proposed model. Wear debris with different sizes can cause different failure patterns: immediate stagnation and cumulative stagnation [39]. A large-sized particle (a type III shock), whose size is supposed to be close to the clearance of the sliding spool, may cause an immediate stagnation whenever it enters the clearance. Small-sized particles (type I shocks) cannot cause failure immediately, but a certain amount of them can form a filter cake cumulatively. When the filter cake becomes large enough, cumulative stagnation occurs [34]. Besides, a medium-sized particle (a type II shock) can decrease the system operational performance and lead to a lower working state, so the system can be regarded as a multi-state system, which is not considered by Fan et al. [34]. A medium-sized particle also has an effect to accelerate the formation of a filter cake, because small-sized particles are easier to adhere to this type of particles. When a medium-sized particle enters the clearance, it leads the sliding spool to a lower state, in which the required number of small-sized particles to break down the system is lower than that in a higher state.

The studies of maintenance policies are also extremely important for many engineering systems, which suffer from inevitable failures because of complex deterioration processes and adverse environmental conditions [41], [42], [43]. The unexpected failures may cause enormous losses, including massive product losses, high corrective replacement costs and safety hazards to the environment and personnel. Thus, the maintenance is of great significance to avoid failures, thereby saving servicing costs. Many studies involve the maintenance policies of engineering systems, and the maintenance of shock model is introduced in detail by Nakagawa [44]. In recent years, many relevant researches have been done [45], [46], [47]. In this paper, the classical age replacement policy is employed as the first replacement policy for the new model. Moreover, two more flexible replacement policies are proposed according to the characteristics of the presented multi-state shock model with mutative failure patterns.

The remaining parts of the paper are organized as follows. In Section 2, we formulate the model and derive expressions for the lasting time until the end of each state (T_l) and the lifetime (T) of the system. In Section 3, a Markov chain is constructed to describe the shock process. The distributions of T_l and T, and the mean residual lifetime are derived based on the assumption that the interarrival times between successive shocks follow a common continuous phase-type distribution. Section 4 is devoted to propose three different replacement policies and construct three unconstrained optimization models to achieve the optimal replacement times. In Sections 5 and 6, numerical illustrations and conclusions are presented respectively.

Notation

N_l

random variable representing the number of shocks that have occurred until the end of the state l

N

random variable representing the total number of shocks that have occurred until the system fails

T_l

lasting time until the end of state l

T

lifetime of the system

X_i

interarrival time between ($i-1$)th and ith shocks, $i=1,2,3,...$

Y_i

magnitude of the ith shock, $i=1,2,3,...$

c₁, c₂

two critical levels to define the types of shocks

k_1l

a predetermined number of type I shocks that cause the system failure at state l

k₂

a predetermined number of type II shocks that cause the system failure

p₁

probability of the occurrence of a type I shock

p₂

probability of the occurrence of a type II shock

p₃

probability of the occurrence of a type III shock, $p_3=1-p_1-p_2$

Λ

one-step transition probability matrix

c_f

corrective replacement cost

c_p

preventive replacement cost

T_P1

replacement age for policy 1

T_P2

replacement time for policy 2

T_P3

replacement time for policy 3

PH_d

discrete phase-type distribution

PH_c

continuous phase-type distribution