A mechanical or electrical system consists of a set of units or subsystems. To determine system reliability, it is necessary to study the reliability of each component of the system as well as the structure of the system. If R(t) is system reliability at time t, R(t₀) represented by R_sys will be system reliability at time t₀. It is assumed that the system consists of n units in which reliability is represented by R1, R2, ..., Rn. As mentioned, R_sys depends on R₁ and R₂, ..., R_n and system structure. The basic hypothesis in this section is the independence of the constituent units of a system [8]. Systems are generally divided into two categories: series and parallel systems, which are described in the following sections.

Suppose a system consists of two units of c₁ and c₂. The system will be sequential if the system fails with the failure of one of its components. In other words, the system is active if both units are active. Fig. (3) shows a sequential system.

The reliability of a sequential system is defined as Eq. (1):

(1)

Assuming that a system consists of two units of c₁ and c₂, this system is a parallel system in which performance requires the operation of at least one of the units c₁ and c₂. In other words, the system fails when both units fail. Fig. (4) shows a parallel system.

The reliability of a parallel system is defined as Eq. (2):

(2)

In this method, reliability assessment is generally performed for repairable systems using MTBF and for non-repairable systems using MTTF [10-11]. The methodology of the reliability modeling of the failure data is presented in detail in Fig. (5). This figure shows the step-by-step flowchart for reliability analysis by using the statistical analysis method.

3.1.3. Evaluation of Independence and Identical Distribution

After collecting the data and before fitting the distribution to the data, the basis for the assumption of independence and identical distribution of the failure data should be examined. In statistics and probability, sequences of random variables are called Independent and Identical Distributions (IID), if they are all equally distributed and mutually independent. In order to analyze this assumption, two common methods of trend test and serial correlation test are used [3, 19, 20].

Table 1. Terms in reliability issues [4].

Definition	Term
Mean time to failure (mean working time)
Mean repair time (mean stop times)
Mean time between failure

Fig. (1). Bathtub curve [5].

Fig. (2). Relationship between MTTR, MTTF, MTBF [4].

Fig. (3). A sequential system or series [9].

Fig. (4). A parallel system.

Fig. (5). Algorithm of statistical analysis method [11].

3.1.6. Statistical Distributions and Fitting Method

Statistical techniques enable us to obtain a proper insight into the diversity of subsystems, system components, and methods of maintenance by fitting the appropriate statistical distribution to the failure data. There are broad statistical distributions to describe the equipment life cycle, which are generally divided into two categories: stable and unstable. Exponential, normal, lognormal, Weibull, gamma, and Power-law model functions are the most commonly used functions in the reliability field [15, 20]. Goodness-of-fit tests are used to validate the fitted functions and select the best distribution among them. Table 3 shows the valid tests used in this regard.

Many studies have been performed using the statistical analysis method in the reliability analysis of tunneling and civil engineering equipment. The reliability, availability, and maintainability analysis of EPB were discussed [21, 22]. In order to model the EPB reliability, the device is divided into five separate subsystems including mechanical, electrical, hydraulic, pneumatic, and hydro subsystems in a series configuration. According to the trend test and series correlation, renewal processes have been applied to analyze all subsystems. After calculating the reliability and maintenance functions for all subsystems, it was found that the mechanical subsystem with the highest failure frequency had the least reliability and maintenance.

Reliability analysis of a drum shearer machine was investigated using the failure data obtained from Tabas Coal Mine [1]. Among the renewal process, homogeneous Poisson process, and non-homogeneous Poisson process, the best simulation approach was selected for each subsystem and the reliability of subsystems was evaluated. The results of this study indicated that the most critical subsystem of the drum shearer machine is water spray in which its reliability reaches zero before other subsystems. The plots of the drum Shearer machine are shown in Fig. (6).

The reliability-based maintenance planning of the hydraulic system of rotary drilling machines was performed in a study [23]. In this research, data analysis shows that the time between failures of some machines follows the two-parameter and three-parameter Weibull distribution. Furthermore, the time between failures of other machines follows the normal log distribution. According to the hydraulic systems reliability plan reliability-based preventive maintenance intervals for machine reliability levels of 80% were 10 hours. In another study, reliability and maintenance analysis of rotary drilling machines pneumatic system was performed [23] After modeling the pneumatic system reliability, maintenance and repair schedules were presented based on different reliability levels. The pneumatic system of the machine under study was divided into four sub-sections A, B, C, and D. The results showed that the reliability of the pneumatic system and subsystems A and B reached 80% after about 7 hours, subsystem C after about 103 hours, and subsystem D after 44 hours of drilling.

Fig. (6). Reliability plots of each subsystem of drum shearer [1].

Table 2. Trend test methods.

Method	Statistics or Method
Graphical [14]	Draw a cumulative graph of failure times (TBFs) or repair times (TTRs) by the cumulative number of failures
	The straight line indicates no trend in data and the convex or concave curves, respectively, indicating a decrease or an increase in the failure rate. The increasing failure rate indicates the occurrence of premature failure.
MIL-HDBK-189 [16]
	If U<xc2, the assumption H0 (data have no trend) is rejected.
Laplace [17]
	, TBFs on the rise (have trend) TBFs are on the decline (have trend)

Table 3. Goodness-of-fit Test [4].

Model	Test	Statistics	Acceptance Condition H0
stable	[20] Chi-square test
	Kolmogorov-Smirnov test(K-S)
	Anderson-Darling test(A-D)
	Mann-Scheuer-Fertig test
unstable	Cramer-Von-Mises test
	Drawing method	Logarithmic graph of the number of failures ln (N (tj)) compared to the cumulative time ln (tj) plotted (j failure number), if the straight line is observed the PLP model may be appropriate.

For a dragline machine, the analysis of reliability and evaluation of failure rate for critical subsystems was done [24]. The results showed that the dragline’s bucket is the most critical subsystem of this machine. In addition to the above-mentioned research, several studies have also been carried out using the statistical analysis method to conduct equipment reliability analysis in tunneling and civil engineering [25-39].

The Failure Mode and Effects Analysis method (FMEA) is an analytical technique based on the pre-occurrence prevention law that is used to identify potential causes of failure. This technique focuses on enhancing the security factor by preventing failures. FMEA is a low-risk tool used to predict problems and deficiencies in the process of designing or developing processes and services in an organization [5, 40]. One of the major differences between FMEA and other qualitative techniques is that FMEA is an action, not a reaction. In many cases, facing the problem may be defined and implemented to eliminate those problems. These actions are a reaction to what has happened. In the implementation of FMEA, measures to eliminate or reduce potential problems occurrence are defined and implemented by predicting them and calculating their risk. This preventative approach is an action against what may happen in the future and will certainly require much less cost and time to take corrective measures in the early stages of product or process design. The purpose of FMEA is to search for all that can cause a product or process to fail before that product has reached production or the process is ready for production.

The Military Standard (MIL STD1629A), the Non-Automobile Standard (SAE ARP5580) and the Design, Manufacture, and Assembly of Machinery standards (SAE J1739) describe the FMEA implementation method in failure mode and effects analysis. In many studies, the Risk Priority Number (RPN) is calculated and then the safety status is examined. Potential failure states for each system are identified and the cause of each failure is predicted. A score is assigned for the severity, occurrence, and detection of each failure and all corrective actions are identified and at the last stage, the necessary recommendations are made [41].

This method uses the risk matrix to prioritize risks. The risk matrix is developed using the severity, occurrence, and detection parameters. The RPN is determined using these parameters and the Mean Time Between Failures (MTBF) is used to rank the probability. The severity parameter is rated according to the recommendations of the Quality Management System in the American Automotive Industry (QS9000) and SAE J1739. In this way, the risks are classified using RPN. The RPN is calculated using the following descriptive (Equation. 3):

(3)

In the FMEA method, failure characteristics checking including critical conditions, control probabilities, safety or severity, and acceptable RPN are considered as a decision criterion for correction or prevention of failure and failure states with higher RPN are classified as critical [42].

By means of the FMEA method, the reliability of a tunnel drilling machine was analyzed [43]. For this purpose, 48 failure modes were assumed for the main machine system and all subsystems, and then the effects of each failure were determined. Finally, necessary corrective actions were taken to prevent or reduce the failure. This method was also used to quantify the contribution of maintenance activities to offshore oil structures by [44, 45].

Risk assessment of a road tunnel construction was conducted using FMEA [46]. This study resulted that the instability of the working face is the most probable risk, whereas, other possible risks include underground water inflow, existing of karst bulbs, mixed tunnel face, and face instability and squeezing.

For each given system, a Markov model contains a list of possible states of that system, possible transition paths between those states, and the parameter rates of those transitions. In reliability analysis, transition usually involves failures and repairs. When a graphical Markov model is expressed, each state is usually represented as a bubble, with arrows indicating the transition path between the states. Fig. (7) shows a single-component Markov state that contains only two healthy and corrupted states [47, 48].

In Fig. (7), λ represents the transition rate parameter from zero to one. In addition, Pj (T) means the probability of the system being in state j at time t. If the health of the device is specific at some early time in T = 0, the initial probabilities include two P₀ (0) = 1 and P₁ (0) = 0 modes [49]. Subsequently, the probability of a zero state decreases at a constant rate λ, which means that if the system is zero at any given time, the probability of switching to one during the next increase in time dt is equal to λ x dt as Eq. (4).

(4)

It is assumed that X(t) represents a set of random variables that represent the Markov process. Then the probability of P_ij, transition from state i at time t = 0 to state j is defined as Eq. (5):

(5)

Given the set of possible states for j, the total probability of a transition to any j from i, plus the probability of remaining at i, must be added to 1. According to Fig. (8), in the third state of K, X can have transition t_i it after j.

Possible transitions between different states of a Markov process can be easily described by a transition state diagram as in Fig. (8). A probability matrix P(t) can be constructed by adjusting its elements to the corresponding probability transition. For example, the probability of transition from i to j is equal to P_ij in row i and column j of P(t). Given the m+1 possible state of X, the probability matrix P(t) is constructed as follows:

The Markov method was used to estimate the reliability of auxiliary ventilation systems in the construction of long tunnels by [50]. In this study, active and standby jet fans were modeled as a random process. Therefore, the probability of replacement of any disabled jet fan with a standby jet was estimated by using the Markov chain theory. Also, Markov chain reliability analysis based on random process principles supported by mathematical rules was presented. A Markov Chain is a special state of the Markov process that is used to study the behavior of a particular random short-term and long-term system behavior. The Markov method was also used in the reliability analysis of drilling operations in open mines [51]. In this study, the failure rate and repair rate of all machines were calculated using available data. Then, 16 possible operating modes were defined and the probability of drilling fields in each case was calculated using Markov theory. The results of the study showed that about 77 percent of all drilling machines were in operational condition. This means that given 360 working days a year, the drilling operation in a reliable condition was 278 days.

The reliability analysis for an Earth Pressure Balance-Tunnel Boring Machine (EPB-TBM) was conducted using Markov modelling [52] and the failure and repair rates of the different subsystems were determined. Fig. (9) shows the transition diagram for the EPB-TBM and its subsystems (b1, b2, b3, and b4) based on a reliability block diagram. In this figure, I1/4 and λ are repair rate and failure rate of subsystems, respectively. The results showed that the availability of this machine was 61% which increased to 70% by proper maintenance and planning.

Fig. (7). Markov states.

Fig. (8). Markov Process [49].

Fault tree analysis is one of the common methods used to analyze the reliability of engineering systems. This method was developed at Bell Telephone Laboratories in the early 1960s to analyze the standby control system due to reliability and safety [5]. The fault tree analysis method is a top-down logical and graphical diagram describing the failure and its causes [53]. The fault tree analysis diagram shows all the failures of the system, subsystem, and collection that use a set of signs and symbols to represent the relationships between the failures and their causes [8]. One of the benefits of this method is that while identifying all the intermediate and final events, it is possible to calculate their probability of occurrence. It can also be used to reconfigure a system to reduce its sensitivity and vulnerability [54]. Table 4 shows the steps for developing a fault tree.

Events in a fault tree are associated with statistical probabilities. For example, a component failure may occur at some constant rates λ (a constant risk function). In this simple case, the probability of failure depends on the rate λ and the time t as Eq. (6):

(6)

A fault tree is often normalized to a given interval and the probability of an event depends on the relationship between the event risk function and this interval. A series of gates is used to estimate the reliability of this method that these gates in fault tree are the output probability of a set of Boolean logic operations. The gate output event probability depends on the input event probability. An AND gate represents a combination of independent events. Mathematically, this gate is equivalent to a subset of input events, and the probability of output of the AND gate is as Eq. (7):

(7)

On the other hand, an OR gate belongs to the assembly set and its output probability is as Eq. (8):

(8)

Fig. (9). Transition diagram of EPB-TBM and subsystem [52].

Vertical drilling machine reliability analysis was performed by using the fault tree method [55]. In this research, the components and subsystems of vertical drilling machine were firstly classified as a tree and investigated using Boolean law of faults. Finally, the reliability of the tunnel drilling machine using the fault tree analysis method was 0.53. The reliability of the hydraulic excavator system using the fault tree was also discussed [56]. In this research, a reliability block diagram of the excavator system fault tree was developed. Furthermore, an algorithm was presented to obtain the minimum set of cuts as well as the minimum set of paths from the fault tree and the reliability of machines and its subsystems over time. Using Boolean and fuzzy laws, it was found that power generation has the highest reliability among subsystems. The reliability analysis of a conveyor system using compound data was also performed [57], in which the proposed method estimates the probability of major event failure using statistical analysis of recorded field failures. Under these circumstances that past failure records do not exist, the method follows a fuzzy theoretical set evaluation based on the expert judgment of the failure intervals. Analyzing the results of the proposed method illustrates the practical role of the experience of experts in providing reliable information.

The potential risk analysis of undesirable events for shield driven tunnels was conducted using fault tree analysis and Analytic Hierarchy Process (AHP). The possible risks of tunneling were considered into four groups: Machine blockage or hold-up, mucking problems, cutter-related malfunction and finally, segment defects [58]. The risk analysis indicated that the related risk to the cutters could reduce the tunnel advance rate. In addition to the mentioned studies, much research has been conducted using the fault tree method to conduct the reliability and risk analysis [59-69].