Monte-Carlo Simulation of Reliability of System with Complex Interconnections

Abstract

Modern automotive systems must satisfy strict reliability requirements. Most real vehicle systems and safety-critical networks have complex interconnections. The sensitivities and probabilistic uncertainties of the reliability of systems with complex interconnections (SwCIs) can be investigated by Monte-Carlo Simulation (MCS). This paper focuses on the sensitivities and parametrical uncertainties of SwCIs’ reliability. The proposed method can be implemented in the investigation of the uncertainties of SwCI reliability, i.e., in the determination of critical system elements and the estimation of the required number of spare parts (RNSP) of the system, which depends on the probability of allowable spare equipment shortage.

1. Introduction

Nowadays, transportation and vehicle engineering play an increasingly important role in everyday life and are subject to continuous development. The reliability of vehicle systems is an important factor in traffic and general human safety. Vehicles use a variety of sensors and sensor networks to gather information. Szűcs and Hézer provided an overview of the current development trends and challenges and the necessary improvements in vehicle technologies [1].

Cognitive mobility studies are, in fact, the interconnection of the following research areas: transportation, vehicle engineering, artificial intelligence, information technology and social sciences [2,3]. Vehicle-to-Everything (V2X) technology, which implements communication between the infrastructure and sensor networks of vehicles, is one of the key areas of cognitive mobility research.

Vehicular Ad hoc NETworks (VANET) were used for cognitive mobility. They are a subcategory of Mobile Ad hoc NETworks (MANET) and use wireless communication between vehicles as well as Road Side Units (RSUs) [4,5].

Maintenance management and vehicle engineers must meet strong technical reliability and economy requirements.

On the one hand, the maintenance cost can be the second largest component of company budget [6]. From a financial viewpoint, therefore, it is paramount to determine the optimal number of spare parts.

On the other hand, the developers must determine which components can be expediently technically modified to increase the reliability of the entire system. Analyzing the correlations between the reliabilities of the system and its elements is helpful in selecting the elements that have the greatest impact on system reliability.

There is considerable research on maintenance and reliability theory, with a great number of publications. For example, referring to the reliability of a technical object in a broad sense, the authors of [7] defined it as the ability of equipment to be trouble-free in operation, during its life cycle and in the execution of the set task. The subject of [8] was an analysis of the maintenance of railway vehicles used in rail passenger transport. The analysis used data on the failure rates of vehicles and it was conducted using reliability parameter indicators.

Reliability is the probability that equipment will meet the intended standards of performance and deliver the desired results within a specified period of time under specified (environmental) conditions [9].

The failures of vehicle systems are stochastic processes which are influenced by many environmental effects; so, they have parametric uncertainties. These processes can be characterized by the probability distributions of the time between failures.

The objective of Payette and Abdul-Nour’s work was to review the concepts of reliability engineering and to highlight the importance of an integrated approach in the analyzing of complex systems [10].

From a reliability point of view, there are two types of systems. Simple systems are systems with simple interconnections (SwSIs) that can be divided into a sequence of parallel and/or series subsystems [11]. The systems that cannot be divided into identical sequences are named systems with complex interconnection (SwCIs). The reliability of these systems cannot be determined by fault tree analysis (FTA) and Reliability Block Diagram (RBD) methods. Myers investigated the reliability of digital fly-by-wire aircraft control as an SwCI [12]. According to Iordache, complex systems are composed of subsystems that have nonlinear interactions, resulting in multiple levels of organization [13]. The applicable approach to correctly computing the availability of an SwCI is the Bayesian True Table Method (BTTM). The Bayesian True Table is the sum of the probabilities of all possible states of the investigated system [14].

Spare part provision is a complex problem. The reliability-based spare part provision can be used to quantify the effect of the operational environment [15]. In this method, the operational environment effects on the reliability characteristic and, consequently, on the required number of spare parts can be analyzed.

The paper by Hamoud and Yiu describes a simple and practical reliability model for estimating the number of spare parts required in order to meet a pre-determined level of the group’s availability [16]. Their model is based on a stationary Markov process.

One of the applicable methods is Monte-Carlo Simulation.

The Monte-Carlo method is an effective mathematical tool for solving deterministic problems with a series of random events. It was Metropolis and Ulam who first named this as Monte-Carlo Simulation (MCS) [17]. Pokorádi applied MCS to the investigation of uncertainties of maintenance processes [18] and to the determination of the accessibilities of temporal systems [19].

Cui et al., based on the reliability analysis of the historical maintenance data of aviation materials, used the least square method to calculate the optimal distribution rule of the life of parts, after which the equation for the support rate, life reliability and inventory demand could be calculated. Finally, the Monte-Carlo method is used to calculate the spare part requirements of full appliances [20].

This paper proposes a methodology of MCS-based investigation of the uncertainty of SwCI reliability. The results of the simulation can be used to perform the following tasks:

• To determine the required number of spare parts (RNSP) of equipment with complex interconnections such as vehicle sensors depending on the uncertainties of system blocks (see Section 3.2 and Section 4.1);
• To choose the most critical elements of SwCIs (such as V2X, VANETs and vehicle sensors and sensor networks) from a system reliability point of view (see Section 3.3 and Section 4.2).

It is important to mention that non-realistic reliability data are used to describe the method so that the results can be clearly illustrated.

This paper is organized as follows: Section 2 presents the core reliability model of MCS. Section 3 lays out the methodology of the structural analysis. Section 4 discusses the functional analysis. Section 5 offers the conclusions deduced from the results of the simulations. Finally, the author summarizes the main findings of this study and outlines some future research directions.

2. Core Model—Reliability of System with Complex Interconnections

For MCS, the first step is to create the so-called core model.

The system and all its elements can have two states. In the case of the operating state (designated by “+” in

Table 1. Truth Table of investigated system.

j	Component					System State	${\mathit{Q}}_{\mathit{j}}$
	A	B	C	D	E
1	−	−	−	−	−	−	$p_A{p}_B{p}_C{p}_D{p}_E$
2	+	−	−	−	−	−	$r_A{p}_B{p}_C{p}_D{p}_E$
3	−	+	−	−	−	−	$p_A{r}_B{p}_C{p}_D{p}_E$
4	+	+	−	−	−	−	$r_A{r}_B{p}_C{p}_D{p}_E$
5	−	−	+	−	−	−	$p_A{p}_B{r}_C{p}_D{p}_E$
6	+	−	+	−	−	−	$r_A{p}_B{r}_C{p}_D{p}_E$
7	−	+	+	−	−	−	$p_A{r}_B{r}_C{p}_D{p}_E$
8	+	+	+	−	−	−	$r_A{r}_B{r}_C{p}_D{p}_E$
9	−	−	−	+	−	−	$p_A{p}_B{p}_C{r}_D{p}_E$
10	+	−	−	+	−	+	$r_A{p}_B{p}_C{r}_D{p}_E$
11	−	+	−	+	−	+	$p_A{r}_B{p}_C{r}_D{p}_E$
12	+	+	−	+	−	+	$r_A{r}_B{p}_C{r}_D{p}_E$
13	−	−	+	+	−	−	$p_A{p}_B{r}_C{r}_D{p}_E$
14	+	−	+	+	−	+	$r_A{p}_B{r}_C{r}_D{p}_E$
15	−	+	+	+	−	+	$p_A{r}_B{r}_C{r}_D{p}_E$
16	+	+	+	+	−	+	$r_A{r}_B{r}_C{r}_D{p}_E$
17	−	−	−	−	+	−	$p_A{p}_B{p}_C{p}_D{r}_E$
18	+	−	−	−	+	−	$r_A{p}_B{p}_C{p}_D{r}_e$
19	−	+	−	−	+	+	$p_A{r}_B{p}_C{p}_D{r}_E$
20	+	+	−	−	+	+	$r_A{r}_B{p}_C{p}_D{r}_E$
21	−	−	+	−	+	+	$p_A{p}_B{r}_C{p}_D{r}_E$
22	+	−	+	−	+	+	$r_A{p}_B{r}_C{p}_D{r}_E$
23	−	+	+	−	+	+	$p_A{r}_B{r}_C{p}_D{r}_E$
24	+	+	+	−	+	+	$r_A{r}_B{r}_C{p}_D{r}_E$
25	−	−	−	+	+	−	$p_A{p}_B{p}_C{r}_D{r}_E$
26	+	−	−	+	+	+	$r_A{p}_B{p}_C{r}_D{r}_E$
27	−	+	−	+	+	+	$p_A{r}_B{p}_C{r}_D{r}_E$
28	+	+	−	+	+	+	$r_A{r}_B{p}_C{r}_D{r}_E$
29	−	−	+	+	+	+	$p_A{p}_B{r}_C{r}_D{r}_E$
30	+	−	+	+	+	+	$r_A{p}_B{r}_C{r}_D{r}_E$
31	−	+	+	+	+	+	$p_A{r}_B{r}_C{r}_D{r}_E$
32	+	+	+	+	+	+	$r_A{r}_B{r}_C{r}_D{r}_E$

), the system or an item can perform its required function. If the system or an item is in the non-operating state (designated by “−” in BTT), it cannot perform its required function.

Table 1 shows the BTT of the investigated SwCI (see Figure 1), where Q_j denotes the probabilities of the jth system state.

From the “end-user” point of view, the availability of equipment can be characterized by the Time to Failure (TTF) that accumulates during operating time from the first use, or from restoration, until failure. Generally, maintenance experts can easily record the performance indicators (e.g., working hours) between two consecutive failures. The mean value of the TTF is the Mean Time to Failure (MTTF). The failure rate λ_i is numerically equal to the reciprocal of the MTTF, i.e.,

$${\lambda }_i={\displaystyle \frac{1}{MTT{F}_i}}.$$

(1)

Since the failure rate λ_i of the ith element is known, its reliability (the probability that the element can accomplish its task) can be determined by the following equation:

$$r_i(t)=e^{-{\lambda }_{i}t}.$$

(2)

where t is the performance indicator interval of the element. (During simulations, the performance indicator interval will be 8760 working hours, which is 1 year). The connection between reliability and the probability of failure is given below:

$$r_i+p_i=1, R_{sys}+P_{sys}=1.$$

(3)

For the investigation of the sensitivity and uncertainty of SwCI reliability, the MTBF of elements will be used.

In Table 1, there are 19 operable (gray rows) and 13 non-operable (white rows) system states. Pokorádi con-firmed that out of the two situations described above, the one which has the lower number of possible system states must be chosen to investigate system reliability [14]. This statement will be important during MCS of system reliabilities. Applying Table 1 and Equation (3) as well as the experience of Pokorádi [14], the system’s reliability can be determined by the following equation:

$$R_{sys}=1-(Q_1+Q_2+Q_3+Q_4+Q_5+Q_6+Q_7+Q_8+Q_9+Q_{13}+Q_{17}+Q_{18}+Q_{25}).$$

(4)

Figure 2 shows that the system’s reliability R_sys depends on different MTTFs of the components and the performance indicator interval.

3. Structural Analysis

The “structural sensitivity” of a technical system’s reliability refers to the effect of the reliability of a given system element from a structural point of view [17]. The value of its coefficient is determined only by localization in the system structure—not by the reliability value—of the given component.

During MCS, the concepts of “momentary TTF” and “momentary failure rate” must be introduced. Following the direction laid out in Equations (1) and (2), the momentary failure rate and reliability of an element can be calculated.

For the MCS-based determination of the structural sensitivity of an SwCI—assuming a normal probability distribution reliability—the core model was used with the following nominal input (mean value and standard deviation of components’ TTF) data:

$$m_{TTF}=4\cdot {10}^4\left[\mathrm{w}·\mathrm{hour}\right](R_{sys}=0.943116), \mathrm{and} s_{TTF}=4\cdot {10}^3\left[\mathrm{w}·\mathrm{hour}\right].$$

(5)

The tasks are as follows:

• To determine the RNSP for 1 year (8760 working hours) of 50,000 pieces of equipment if the allowable probability of shortage of spare parts is 0.5%;
• To identify which element should be modified technically to increase system reliability.

In accordance with common engineering practice, normal distributions of failure rates were assumed.

3.1. Simulation

The first task of MCS is to determine the optimal number of excitations. In the case of a small number of excitations, the elapsed time of the simulation is short, but the obtained results cannot have sufficient accuracy. In the case of a higher than optimal number of excitations, the obtained simulation results are accurate, but the elapsed time can be unreasonably long. In this study, the number of excitations (random sample size) was determined as 1,000,000.

Table 2. Statistical parameters of structural simulation.

Element	mTTFi [w·hour]	δmTTFi	sTTFi [w·hour]	δsTTFi
A	4.000078 × 104	1.949730 × 10−5	3.997412 × 103	6.468521 × 10−4
B	3.999908 × 104	2.299380 × 10−5	4.001015 × 103	2.538572 × 10−4
C	4.000215 × 104	5.362357 × 10−5	3.997899 × 103	5.251031 × 10−4
D	3.999789 × 104	5.271307 × 10−5	4.003549 × 103	8.874094 × 10−4
E	3.999878 × 104	3.034227 × 10−5	3.999747 × 103	6.307987 × 10−5

summarizes the mean values and standard deviations of the excitations, as well as their relative differences from the nominal ones, as seen below:

$$\delta m_{TTFi}=\left|{\displaystyle \frac{m_{TTFi}-m_{TTF}}{m_{TTF}}}\right|, \mathrm{and} \delta s_{TTFi}=\left|{\displaystyle \frac{s_{TTFi}-s_{TTF}}{s_{TTF}}}\right|.$$

(6)

The normally distributed probabilistic uncertainty of system reliability can be determined by statistical analysis of the MCS result:

$$m_{Rsys}=0.943116 \mathrm{and} s_{Rsys}= 5.968841\mathrm{}\cdot 10^{-3}.$$

(7)

3.2. The Determination of the Requested Number of Spare Parts

One of the key tasks of maintenance management is to determine the required number of spare parts (RNSP). If one has a low number of spare parts, it may not be possible to replace or repair faulty equipment because there are not enough spare parts. However, greater than the optimal number of spare parts can lead to unnecessary maintenance costs and storage requirements. System reliability can be determined with uncertainty, depicted by the probability function determined above. To determine the RNSP, the allowed probability of shortage must be known—depending on technical, economic and safety aspects.

For technological reasons, the end users of the investigated SwCI can only replace the system in one go. However, during their repair, it is possible to identify which elements have failed and caused the equipment to stop working. So, to determine the RNSP, the uncertainties of the full system’s reliability must be investigated.

Knowing the density function f(R_sys) of system reliability, the RNSP can be determined. First, the so-called probability of the RNSP R_RNSP must be determined using the mathematical relation

$$P_{AS}={\displaystyle \underset{-\infty }{\overset{R_{RNSP}}{\int }}f\left(R_{sys}\right)}d{R}_{sys},$$

(8)

where P_AS is the allowable probability of shortage. Knowing the distribution function f(R_sys), R_RNSP can be determined depending on P_AS. (For example, in the case of Gauss distribution, a table of standard normal distribution can be used.)

Using the methodology of [18], RNSP is rounded to the upper integer depending on R_RNSP:

$$RNSP=⌈\left({\displaystyle \frac{1}{R_{RNSP}}}-1\right)N⌉$$

(9)

where N is the number of pieces of equipment in the system (in the present case, N = 50,000).

The RNSPs of different assessed uncertainties are given in

Table 3. Required number of spare parts depending on estimated uncertainty.

Allowable Probability of Shortage	Number of Required Spare Parts
10%	3446
7.5%	3501
5%	3569
2%	3710
1%	3807
0.5%	3890
0.2%	3997
0.1%	4074
0.05%	4133

and Figure 3.

3.3. Correlation Analysis

The sensitivity of a system’s reliability is characterized by its correlation with the reliability of a given element. The strength of linear stochastic interdependencies of two random variables can be characterized by their correlation coefficient. If these two parameters have a strong positive correlation coefficient, their values should most probably change in the same direction; conversely, in the case of a strong negative correlation, the coefficient will change in the opposite direction. However, zero coefficient means that these two parameters are independent of each other.

The correlation coefficient r_ημ can be estimated by the equation

$$r_{\mathsf{\eta }\mathsf{\mu }}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\left(x_i-{\displaystyle \sum _{j=1}^n{x}_j}\right)}\left(y_i-{\displaystyle \sum _{j=1}^n{y}_j}\right)}{\sqrt{{{\displaystyle \sum _{i=1}^n\left(x_i-{\displaystyle \sum _{j=1}^n{x}_j}\right)}}^2{\left(y_i-{\displaystyle \sum _{j=1}^n{y}_j}\right)}^2}}},$$

(10)

where x₁; x₂; … x_n and y₁; y₂; … y_n samples belong to the variables η and μ [21]. MCS can be used to generate the sample sets mentioned above.

To determine the required number of spare parts, the reliability of the system has to be examined. Therefore, during this analysis, the correlation coefficients of the reliabilities of the elements and the system are determined and analyzed, but not the correlations of their TTFs.

The author worked out the adaptation of the linear mathematical diagnostic modeling methodology for setting up the Linear Sensitivity Model of System Reliability (LSMoSR) [14].

Table 4. Correlation and sensitivity coefficients.

A	B	C	D	E
MCS
0.170191	0.2665901	0.172167	0.652113	0.651235
LSMoSR [14]
0.017100	0.026100	0.017100	0.107100	0.107100

and Figure 4 display the correlation coefficients of MCS and—for comparison—the linearized sensitivity coefficient of the LSMoSR.

4. Functional Analysis

The functional sensitivity coefficient of a component—in contradistinction to the structural sensitivity coefficient—is influenced not only by its location in the system, but mainly by its reliability resulting from its functional role and physical location in the system.

Elements A, B and C, and D and E have different failure parameters due to their different functions and physical localization (different ambient physical impacts). Therefore, the sensitivity of the investigated SwCI becomes asymmetric.

Table 5. Statistical parameters of functional simulation.

Element	mTTFi [w·hour]		δmTTFi	sTTFi [w·hour]		δsTTFi
	stat.	MCS		stat.	MCS
A	39,487.66	39,487.33	8.36 × 10−6	4184.33	4180.36	9.49 × 10−4
B	39,901.59	39,902.39	2.00 × 10−5	3479.14	3477.19	5.60 × 10−4
C	41,061.69	41,061.80	2.68 × 10−6	4174.01	4172.57	3.45 × 10−4
D	35,403.23	35,401.38	5.23 × 10−5	3948.09	3948.74	1.65 × 10−4
E	39,476.73	39,477.90	2.96 × 10−5	3731.43	3735.14	9.94 × 10−4

presents the main statistical data of the TTFs determined by data received from the repairing teams and data of the functional simulation. Based on the above-stated information in Section 3.1, the excitation number is defined as one million. For simulation purposes, the core model is used.

4.1. The Determination of the Requested Number of Spare Parts

The average value and standard deviation of the reliability of the system for a period of 1 year (8760 h) is as follows:

m_Rsys = 0.937806 and s_Rsys = 6.723347 × 10⁻³.

(11)

The simulation results indicate that in order to guarantee less than a 0.005 (0.5%) probability of shortage, a minimum of 4596 spare parts should be purchased for a 1-year time interval. For economic, maintenance and safety decisions (which are not the subject of this model study).

Table 6. Required number of spare parts depending on allowable probability of shortage.

Estimating Uncertainty	Number of Spare Parts
10%	3810
7.5%	3873
5%	3951
2%	4112
1%	4222
0.5%	4317
0.2%	4440
0.1%	4528
0.05%	4596

and Figure 5 outline the RNSP for different allowable probabilities of shortage.

4.2. Correlation Analysis

To determine which element must be modified technically for the increase in system reliability, the correlation analysis (see Section 3.2) is used. The results of the analysis are seen in

Table 7. Comparison of functional and structural correlation coefficients.

A	B	C	D	E
Functional
0.158650	0.208326	0.161111	0.717366	0.609552
Structural
0.170191	0.2665901	0.172167	0.652113	0.651235

and Figure 6.

5. Discussion

By analyzing the results of the simulations and analyses from an engineering point of view, the author has drawn the following conclusions:

5.1. Conclusions About Core Model

{1.1}

If the MTTFs of elements increase, the system’s reliability will increase too.

{1.2}

The system’s reliability decreases if the performance indicator interval increases.

{1.3}

The diagrams of Figure 2 demonstrate that the investigated system’s reliability is nonlinear and depends on the MTTFs of its elements and the performance indicator intervals.

• A system is linear f its operator has the following property: W(Cu) = CW(u).

{1.4}

The BTT method can be used as a core model of MCSs.

{1.5}

The disadvantage of this method is that the number of system states grows by an exponential function depending on the number of elements.

• This may pose a problem during the following MCSs.

5.2. Conclusions About Monte-Carlo Simulations

{2.1}

In the case of identical MTTFs, in the “first line”, system reliability has greater sensitivity to the reliability of element B.

{2.2}

In the case of identical MTTFs, system reliability has greater sensitivity to the “second-line” D and E elements than the "first-line" elements.

{2.3}

Comparing the results of the correlation analysis and LMoSR [14], this revealed that the obtained sensitivity results of the two used methods have similar properties (see Figure 3).

{2.4}

The fundamental source of the magnitude difference between the two results mentioned above is the nonlinearity of the system’s reliability model (see Figure 2).

• The LSMoSR method linearizes the core model, so it may result in numerical uncertainties. MCS can manage this problem and provides a more satisfying result. However, MCS requires much more computing capacity than the LSMoSR. If the aim of the analysis is “only” the selection of the most critical elements from a system reliability point of view, both methods yield adequate results.

{2.5}

The comparison of the structural and functional correlation analysis results (see Figure 6) highlights the effects of the different functional roles and physical locations of elements on system reliability.

• There is a significant difference between the impacts of the reliability of elements D and E on system reliability. This is the consequence of the difference between their functions in the system in the case of the analyzed work of the system.

{2.6}

The results of MSC can be used to determine the RNSP and investigate the sensitivity of system reliability – but it requires considerable calculation capacities.

The main conclusion is that the proposed MCS-based analysis method is suitable for determining the NRNSP depending on the allowable estimation inaccuracy. Furthermore, the sensitivities of the reliability of an SwCI can be examined based on the results of MCS.

6. Summary and Future Work

There are numerous SwCIs (such as vehicle sensor networks, V2Xs and VANETs) in modern transportation. This paper proposed an MCS-based method that can be applied to determine the RNSP of the equipment of vehicle sensors and the critical elements of SwCIs (such as V2X, VNETs and vehicle sensors and sensor networks) from a system reliability point of view.

Based on the engineering evaluation of the results of the proposed simulation, important conclusions can be drawn regarding the reliability of the examined system, which can be read in detail in Section 5.

The author’s future scientific research related to this field of applied mathematics and technical reliability will include creating new methodologies to increase the reliability of vehicle and transport systems and networks.