Article citation information:
Breznická, A., Mikuš,
P., Krbaťa, M., Kohutiar, M., Timárová,
Ľ., Fekiač, J.J., Kakošová,
L., Jeluš, A. Analytical and simulation approach to
estimating the failure probability of technical components using an
interference model. Scientific Journal of
Silesian University of Technology. Series Transport. 2026, 132, 57-71. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2026.132.3
Alena BREZNICKÁ[1],
Pavol MIKUŠ[2],
Michal KRBAŤA[3],
Marcel KOHUTIAR[4],
Ľudmila TIMÁROVÁ[5],
Jozef Jaroslav FEKIAČ[6],
Lucia KAKOŠOVÁ[7],
Alex JELUŠ[8]
ANALYTICAL AND SIMULATION
APPROACH TO ESTIMATING THE FAILURE PROBABILITY OF TECHNICAL COMPONENTS USING AN
INTERFERENCE MODEL
Summary. This paper presents an analytical
and simulation-based approach to estimating the probability of failure of
technical components using a stress – strength interference model. The proposed
methodology integrates the statistical processing of experimental or
operational data with probabilistic modeling and simulation
implemented within the MATLAB environment. The input variables – operational
stress and structural resistance – are described using appropriate probability
distributions, enabling the evaluation of their mutual interaction and its
impact on system reliability. The developed simulation framework allows for
both graphical and numerical assessment of the interference region, which
represents critical states in which failure may occur. The influence of
variability in input parameters, particularly the standard deviation, is analyzed in order to evaluate the sensitivity and
robustness of the model. The results demonstrate that increased dispersion of
stress and resistance significantly affects the probability of failure and the
system safety margin. The application of the proposed method to a selected
technical component revealed a failure probability exceeding the admissible
limit, confirming the need for optimization measures. Adjustments to the
resistance characteristics led to a partial improvement in reliability;
however, additional design or operational modifications were required to meet
the prescribed requirements. The main contribution of this work lies in the
development of a flexible simulation tool that supports reliability assessment,
sensitivity analysis, and decision-making processes in engineering practice.
This approach is particularly beneficial under conditions of limited
availability of empirical data and can be effectively applied in industries
such as automotive engineering, where stochastic loading and material variability
play a crucial role in system reliability.
Keywords: reliability, resistance
dependability, stress, interference method, failure probability,
stress–strength model, risk analysis
1. INTRODUCTION
The interference theory of
reliability represents a statistical framework for evaluating system
reliability by examining the extent to which the distribution of system demands
overlaps with the distribution of its actual performance or durability [1]. This
overlap serves as a principal indicator for determining the probability of
failure. In reliability assessment, core analytical tools include the
examination of failure-free operation and the evaluation of service life, both
of which provide essential insights into system performance over time. The
theoretical basis of this approach is rooted in the deterministic safe-life
concept, which relies on predefined reliability parameters and strict adherence
to specified limit values.
Within this deterministic
perspective, a failure event is considered to occur when the imposed load
exceeds the resistance threshold of the component or system – denoted as R – which
represents its capacity to sustain operational stresses. In contrast, the
probabilistic (stochastic) formulation of reliability assumes that both the
applied stress and resistance to failure are inherently random variables
governed by specific probability distribution functions. These quantities may
span a wide numerical domain, resulting in scenarios where their distributions
either do not intersect at all or exhibit a degree of mutual interference that
directly affects the associated reliability estimate [2].
Distribution of random variables:
The first random variable represents
the system’s operational conditions and the corresponding operating stress, S
(system stress). This stress arises from the combined effects of external loads
together with the conditions associated with the specific modes in which the
system is used.
The second random variable describes
the system’s resistance, R. Resistance to the acting stress S
denotes the capability of the system or its components to withstand physical,
chemical, and biological influences which, through their cumulative or
instantaneous effects, may induce degradation processes leading to component
failures [3].
Both parameters incorporated into
the model represent random variables, each defined by stochastic quantities or
processes. Their empirical behavior may be
illustrated using histograms or – following appropriate statistical evaluation
– described through probability distribution functions [4]. Numerous analytical
approaches have been proposed in the literature to quantify reliability based
on interference, particularly for systems in which the probability density
functions of stress, fₛ(S), and resistance, fᵣ(R), follow exponential, normal,
Weibull, gamma, or lognormal distributions.
When stress and resistance conform
to different probability distributions, the computation of the resulting
two-dimensional joint integral becomes mathematically complex, making an
analytical determination of reliability difficult. Under such circumstances, it
is advantageous to employ mathematical or simulation-based modeling
techniques as an alternative to direct analytical evaluation [5]. For this
reason, the present paper focuses specifically on the computational methodology
associated with the interference theory of reliability.
2. DEFINITION OF THE MODEL
P(reliability) = P (R > S)
This is the probability that the
system's resistance is greater than the stress.
R∼N(μR,σ2R) a S∼N(μS,σ2S)
(1)
Where:
µR, σ - mean value and standard deviation of
resistance,
µS, σS - mean value and standard deviation of
stress,
Φ - distribution function of standard normal
distribution.
The operational stress S and
the structural resistance R are represented using their respective
probability distribution functions and probability density functions [6]. Let
the probability density function associated with the stochastic stress variable
S be denoted by fₛ(S), and the probability density function
corresponding to the random resistance variable R by fᵣ(R). Similarly, let Fₛ(S)
describe the cumulative distribution function of the stress variable S,
while Fᵣ(R)
denotes the cumulative distribution function for the resistance variable R.
Since both S and R are random
quantities, they follow specific probability distribution laws, typically
continuous or discrete in nature [7].
These variables may influence one
another – an interaction referred to as interference – and this property can be
effectively utilized when assessing the reliability of systems or structural
elements. The boundary points at which the distributions intersect define the
region of mutual interaction between the two variables.
The mathematical formulation of the
model can be further described by its fundamental analytical structure,
specifically through the following theoretical basis. The interference approach
makes it possible to account for the fact that both load and strength are
random variables whose mutual interaction directly affects the probability of
failure. Analyzing the overlap of their probability
distributions provides a more realistic representation of reliability than
traditional deterministic methods.
This approach also helps to identify
critical regions in which the probability of failure is highest and enables
more accurate modeling of the effects of varying
environmental or operational conditions. In practice, this means that a system
can be optimized even before its physical implementation by adjusting design
parameters or operational strategies to minimize the risk of failure.
Generally, also for other
distributions [8]:
Where:
fS(S) - stress probability density,
fR(R - resistance probability density.
For general probability
distributions of stress and resistance, this probability is expressed by a
double integral of the probability density functions fₛ(S)
and fᵣ(R).
Mathematically, this represents the integration of the stress probability
density over its entire domain, where, for each stress value, the probability
that the system resistance exceeds this value is determined.
Failure rate (failure probability):
Failure intensity:
The dynamics of the failure process
over time are characterized by the failure intensity, λ(t), which represents the
conditional probability of failure per unit time, assuming failure-free
operation up to time t. The failure intensity is defined as the ratio of
the probability density function of the time to failure, f(t), to the
reliability function, R(t):
Where:
f(t) - probability density of time to failure,
R(t) - probability of failure-free operation.
These relationships form the basis
for the analytical assessment of system reliability using SSI, where the
variance and mean values of component stress and resistance are compared. This
approach will be utilized in the reliability analysis presented in this paper.
The assessed system or object, Mₖ, which is exposed to loading
during the monitored time period, is considered reliable if the given operating
stress, L, does not exceed the strength with a specified probability.
Mk = Pr (S > L)
(5)
Where:
Mk - System or object,
S - Stress [%/MPa],
L - Load Stress [%/MPa].
The presented mathematical framework
enables a quantitative assessment of the safe service life of the system and
forms the basis for the probabilistic analysis of failures and the lifetime of
technical objects.
Figure 1 illustrates the conceptual
principle underlying the area that is proportional to the probability of
failure. The shaded region represents the domain in which both random variables
exert mutual influence [9]. This area corresponds to the probability of failure
and reflects the fact that, due to the stochastic nature of the quantities
involved – particularly the dispersion of their values – there exists a
non-zero likelihood that the applied stress will exceed the available
resistance, resulting in a failure event [10].
Fig. 1. Illustration of the range of
interference
When determining the probability of
failure-free operation, Rₜ, and assuming that the random variables
S and R are statistically independent,
the calculation can rely on the well-established principle that the probability
of the simultaneous occurrence of two independent events is equal to the
product of their individual probabilities. These methodological foundations
form the basis for constructing analytical and mathematical models of both
failure-prone and failure-free elements using the SSI interference method [11].
2. SSI SIMULATION MODEL
The input quantities – stress S
and resistance R – are random variables derived from experimental
measurements. These measurements produce a non-parametric data distribution
that can be further statistically processed, for example, in the form of a
histogram, or converted into a suitable parametric distribution [12]. Both
approaches allow for the generation of input data that are subsequently used to
assess the probability of occurrence of critical events.
On this basis, the probability of
failure or failure-free performance of technical components can be evaluated
through the interference-based reliability approach. A crucial prerequisite for
its correct application is the rigorous statistical processing of input data,
which are typically obtained from experimental measurements or operational
monitoring. In real engineering systems, both stress S and resistance R
are affected by numerous stochastic influences, including variations in
material properties, measurement uncertainties, fluctuations in operational
conditions, and degradation mechanisms occurring throughout the service life of
the component. Consequently, representing these parameters by single
deterministic values is generally inadequate and may lead to an underestimated
assessment of failure risk.
The expression of random variables S
and R by means of relative frequency histograms and corresponding
probability density distributions is illustrated in Figure 2.
During the analysis of measured
datasets, two methodological strategies may be adopted:
non-parametric methods, which rely on empirical probability distributions
derived directly from observed data, and parametric methods, in which
experimental observations are approximated by selecting an appropriate
theoretical distribution model – for example, normal, lognormal, or Weibull
distributions. The selection of a suitable distribution depends on several
factors, including the physical characteristics of the variable being examined,
the size and quality of the available dataset, and the underlying mechanisms
governing the behavior of the studied phenomenon.
Fig. 2. Expression of random
variables S and R by relative frequency histogram and probability density
distribution
The procedure for the simulation
model operation:
For the range of experimental or
generated values, the corresponding values of the distribution functions, Fᵣ(R) and Fₛ(S), are determined for
different values of resistance R [13]. The resulting failure-free
performance is influenced by the values of stress S and resistance R
across their entire range.
If these two quantities are
represented graphically in terms of their mutual dependence, the point of
intersection corresponds to a combination of two independent phenomena. The
area under the curve in the graph corresponds to the probability of failure-free
operation, expressed as a percentage. Conversely, the area above this curve
represents the probability of failure.
Fig. 3. Probability of failure
The modeling
procedure was compiled as follows:
Initially, it is necessary to obtain
input data in the form of histograms or probability distributions of the
operational stress (S) and resistance (R) of the analyzed element. Subsequently, random values of stress and
resistance are generated, thereby creating a specific simulated case [14].
In the present study, a specific
analysis is considered, in which resistance represents values obtained from
operation, namely the values of selected reliability characteristics within the
monitored period: component life (time resistance) and the number of failures
over time. The stress associated with the complaint process is represented by
the stress parameter.
In the next step, it is determined
to which region – failure or failure-free – the given point belongs, and the
point is simultaneously recorded graphically. The boundary between these
regions is defined by the condition R ≥ S and is illustrated in
the graph by a red line (see Figure 3). If this condition is not satisfied, the
state is classified as a failure.
During the simulation, both the
total number of iterations (N) and the number of failure cases (n)
are monitored. The generated data are then statistically processed, yielding
probability density functions and distribution functions, which are
subsequently presented graphically to provide a clearer understanding of the
interference between S and R.
The MATLAB software environment is
employed for mathematical modeling, providing not
only high-quality graphical outputs but also the capability to develop
interactive applications. Within this environment, the user can modify the
distribution parameters and evaluate which combinations of stress and
resistance are appropriate for a given design via a user interface. The program
enables the selection of distribution types for stress and resistance, the
adjustment of their parameters, and the definition of the number of
simulations.
If the results indicate that the
required reliability criteria are not satisfied, the user can repeat the
simulations with modified input parameters until the desired level of
failure-free operation is achieved.
3. SIMULATION MODEL STEPS
It is necessary to determine an
appropriate number of simulations and to obtain input data describing the
parameters of the probability density distributions for the operational stress
and the resistance of the analyzed element. A
visualization of the selection of the number of simulations is presented in
Figure 4. Determining an appropriate number of simulations for the interference
method depends on the desired accuracy of the results, the variability of the
input data, and the available computational resources. The objective is to
achieve stable and reliable estimates (e.g., of the probability of failure)
without resorting to an unnecessarily large number of simulations.
Fig. 4. Determining the number of simulations
in a mathematical model
First, the operating stress and
resistance values are generated and subsequently statistically processed into
probability density functions and distribution functions. The representation of
the determination of operating stress is shown in Figure 5. In the next step,
the system reliability function is calculated, taking into account the time
dependence of the applied stress and the potential reduction of resistance due
to degradation processes. Based on the reliability function, the failure
intensity can also be determined, which characterizes the instantaneous
susceptibility of the system to failure during operation.
Fig. 5. Determination of operating stress and
resistance of a component
The minimum value of the resistance
function and the maximum value of the operational stress function are
determined. At the same time, the interdependence between the resistance (R)
and stress (S) values is illustrated. Finally, the area under the curve
is calculated, representing the degree of probability overlap, and this area is
shown in Figure 6a and Figure 6b.
Fig. 6a. Probability density functions and
distribution functions
Fig. 6b. Probability density functions and
distribution functions
The next step in the experiment is to analyze the impact of variations in standard deviation
values. This analysis focuses on how changes in data variability (dispersion)
influence the results of statistical evaluations and decision-making processes.
The primary objectives of this step are to assess the robustness of the models
– for example, how changes in variability affect regression results; to
evaluate the reliability of the results – whether conclusions remain valid
under higher or lower levels of uncertainty; and to support risk analysis,
particularly in fields such as finance, manufacturing, and medicine, where data
fluctuations may have significant consequences.
The analysis of the impact of variations in standard
deviation within manufacturing systems focuses on assessing how variability in
input or output parameters affects the stability and performance of the entire
production process. Increased dispersion of technological quantities, such as
product life, failure rates, or the number of non-conforming units in
production, may lead to a higher probability of defective products, reduced
component reliability, or inefficient utilization of production capacity.
For this reason, quantitative analysis of variance
represents a key tool in process optimization, quality management, and the
reduction of production-related risks.
Fig. 7. Principle of impact of changes in the
resistance parameter
A statistical probability model can
be developed, and the relationship between stress and resistance can be
visualized using histograms of randomly generated values through the scatterhist function. This function is employed
within the MATLAB environment to perform simulations and to provide a combined
graphical representation of the data. It integrates a scatter plot with
histograms into a single figure (Figure 8).
The central part of the
visualization consists of a two-dimensional scatter plot, which illustrates the
relationship between two variables (e.g. stress versus resistance). Along the
upper edge, a histogram of one variable – stress – is displayed, while the
right-hand side presents a histogram of the second variable, namely resistance.
This visualization effectively
represents the relationship between randomly generated stress and resistance
values. The main plot captures the correlation between the variables, whereas
the marginal histograms provide insight into their individual probability
distributions. Such a representation enables an efficient assessment of the
interrelationship and overlap of the distributions of quantities influencing
system reliability.
Based on the resulting graphical
output, it is possible to identify regions with a high concentration of stress
and resistance values, as well as potentially critical areas of their mutual
overlap. These regions correspond to states in which a significant reduction in
the system safety margin occurs, leading to an increased probability of
failure.
Furthermore, analysis of the scatter
plot allows for the assessment of the degree of dependence between stress and
resistance. If a correlation exists between the variables, it can be identified
from the inclination and shape of the point cloud.
This information is particularly
important when refining the probabilistic reliability model, as the assumption
of independence between variables may not always be satisfied under real
operating conditions. For an indicative assessment, a reduced number of
simulation runs is selected. Although increasing the number of iterations would
enhance the statistical robustness and precision of the simulation outputs, it
would simultaneously reduce the clarity and visual interpretability of the
resulting graphical representations. The simulation results, together with
their graphical illustrations, clearly demonstrate the significant impact of
variability in the individual resistance and stress parameters. The underlying
principle of the analytical procedure employed is illustrated in Figure 7,
which presents a graphical evaluation of how variations in the resistance
parameter influence the resulting reliability characteristics.
Fig. 8. Relationship between stress and
resistance using a scatter–histogram representation
The next step is the analysis of the probability of
failure based on a simulation experiment with parameters defined as follows:
Input:
Number of simulations: 1000
Stress parameters: exponential distribution,
mean value 50
Resilience parameters: Weibull distribution,
standard deviation 20
Output:
Probability of failure-free operation: 0.919
Probability of failure: 0.081
Fig. 9. Graphical representation of 2D faulted
and faultless set of realizations at
100,000 simulations
Figure 9 shows the results of a 2D
simulation model of the interference of load (stress) and resistance of a
technical element. This is a classic stress–strength interference model that
allows quantifying the probability of system failure based on the overlap of
two random variables – the applied load and the available load capacity of the
element.
Interpretation of the graph
The X-axis (Resistance) represents
the random variable of the load capacity of the structure or component.
This variable is usually determined
by material properties, manufacturing tolerances, and degradation during the
life cycle.
The Y-axis (Stress) represents the
magnitude of the applied load, which can be deterministic or stochastic, while
the model assumes its variability over time or due to operating conditions.
The green points (Reliability
region) indicate the states for which:
"Load
(Stress)"<"Strength (Resistance)"
In this region, the system is
reliable, since the load capacity exceeds the load.
The red points (Failure region)
represent situations where:
"Load
(Stress)"≥"Strength (Resistance)"
Failure occurs in this region
because the applied load exceeds the load-bearing capacity of the element. To
apply this model in practice, an extensive simulation of random values for both
quantities is performed. The resulting scatter plot illustrates the
two-dimensional distribution of load and load capacity across a range of
operating states. Each simulated state therefore represents one potential
configuration of the real system.
The region in which the simulated
load values are lower than the corresponding load capacity values represents
failure-free operation. These points correspond to cases in which the structure
satisfies the required safety and reliability criteria.
Conversely, cases in which the
simulated load exceeds the load capacity fall within the failure region. This
condition arises as a result of combinations of extreme loads, material
degradation, or unfavorable parameter values that
lead to the load-bearing capacity of the system being exceeded.
The interference region – the zone
in which stress and resistance values overlap – thus represents a critical
factor in reliability assessment. It is the density of points within this
region that determines the number of simulated cases resulting in failure.
4. CONCLUSION
The simulation results, together
with their graphical interpretation, indicate an unacceptably high probability
of failure of 0.156, which substantially exceeds the established admissible
limit of 0.06. A reduction in this failure probability can be achieved
primarily by optimizing manufacturing processes and implementing higher-grade
materials, thereby increasing the structural resistance of the examined
component. This effect is illustrated in Figure 8, which demonstrates the
sensitivity of the resulting failure probability to variations in the
resistance parameter. Although increasing the number of simulation iterations
generally enhances numerical accuracy, it may also reduce the visual clarity of
graphical outputs, which is relevant when interpreting results for design or
educational purposes.
The improved failure probability of
0.081 obtained after modifying the resistance distribution reflects partial
progress; however, it remains above the required reliability threshold. This
indicates that the system, in its current configuration, does not meet the
prescribed reliability standards. The presented simulation model proves to be
an effective tool for overcoming the limitations inherent in classical
analytical approaches, particularly under conditions of limited empirical data
availability. The model enables a detailed examination of the interaction
between the stochastic characteristics of stress and resistance distributions,
thereby supporting a more realistic assessment of reliability in complex
technical systems.
Figure 9 provides a combined
visualization of safe and unsafe operational regions, where the red region
represents the domain of failure and the green region corresponds to
failure-free operation. This visual distinction is particularly valuable for
both technical diagnostics and pedagogical purposes, as it offers an intuitive
understanding of the mathematical relationships underlying the SSI interference
method. Moreover, the graphical outputs facilitate comprehension of how
different probability distributions of random variables influence system
reliability, which is especially beneficial in the teaching of stochastic
reliability concepts.
Based on the conducted simulations,
it is evident that the current safety margin of the analyzed
system is insufficient, highlighting the need for adjustments in either design
or operational parameters. Increasing system resistance through the use of
higher-quality materials is one possible approach; alternatively, reducing
applied stress through optimized operating conditions, load redistribution, or
the implementation of additional protective mechanisms may also contribute to
achieving the desired reliability level.
A significant advantage of the
proposed simulation framework lies in its flexibility and its capability to
perform sensitivity analyzes. By modifying the mean
values or variances associated with stress and resistance distributions, it
becomes possible to identify input parameters that exert the greatest influence
on the resulting probability of failure. Such information is highly valuable in
engineering decision-making, as it enables designers to determine which
measures will yield the most substantial improvements in reliability.
Furthermore, the model supports comparative evaluations of alternative design
solutions, allowing engineers to assess different configuration scenarios using
a unified reliability metric.
When complemented by graphical
outputs, the simulation methodology becomes a practical decision-support tool.
The results are not only quantitatively measurable but also visually
interpretable, thereby significantly enhancing the transparency and clarity of
reliability assessments for engineering teams and stakeholders.
From a long-term perspective, the
interference-based simulation model offers the capability to predict the
time-dependent evolution of system reliability, taking into account gradual
material degradation, wear mechanisms, and variations in operational conditions
throughout the service life of a component. This predictive potential can
improve maintenance planning, optimize inspection intervals, and reduce the
likelihood of unexpected failures, thereby contributing to enhanced operational
safety and cost-effective asset management.
The findings obtained from the
simulation analysis have direct relevance for the automotive industry, where
reliability assessment plays a critical role in ensuring vehicle safety,
operational robustness, and long-term durability of mechanical, electronic, and
structural subsystems. The interference-based simulation approach provides a
valuable framework for quantifying the probability that operational loading
will exceed the resistance capacity of components, which is essential for
identifying failure-critical regions in automotive systems.
In automotive applications, the
stochastic nature of stress and resistance parameters is particularly evident
due to:
These factors contribute to an
expansion of the interference region and consequently increase the probability
of failure, as demonstrated by the simulation results.
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Received 15.04.2026; accepted in revised form 05.07.2026
Scientific Journal of Silesian University of
Technology. Series Transport is licensed under a Creative Commons Attribution
4.0 International License
[1] Faculty of Special Technology, Alexander Dubček University of Trenčín, Ku kyselke 469, 911 06 Trenčín, Slovakia. Email: alena.breznicka@tnuni.sk. ORCID: https://orcid.org/0000-0003-3138-2040
[2] Faculty of Special Technology, Alexander Dubček University of Trenčín, Ku kyselke 469, 911 06 Trenčín, Slovakia. Email: pavol.mikus@tnuni.sk. ORCID: https://orcid.org/0000-0002-7314-9258
[3] Faculty of Special Technology, Alexander Dubček University of Trenčín, Ku kyselke 469, 911 06 Trenčín, Slovakia. Email: michal.krbata@tnuni.sk. ORCID: https://orcid.org/0000-0002-0520-8180
[4] Faculty of Special Technology, Alexander Dubček University of Trenčín, Ku kyselke 469, 911 06 Trenčín, Slovakia. Email: marcel.kohutiar@tnuni.sk. ORCID: https://orcid.org/0000-0002-4710-5913
[5] Faculty of Special Technology, Alexander Dubček University of Trenčín, Ku kyselke 469, 911 06 Trenčín, Slovakia. Email: ludmila.timarova@tnuni.sk. ORCID: https://orcid.org/0000-0003-2192-3253
[6] Faculty of Special Technology, Alexander Dubček University of Trenčín, Ku kyselke 469, 911 06 Trenčín, Slovakia. Email: jozef.fekiac@tnuni.sk. ORCID: https://orcid.org/0009-0001-6392-8389
[7] Faculty of Special Technology, Alexander Dubček University of Trenčín, Ku kyselke 469, 911 06 Trenčín, Slovakia. Email: lucia.kakosova@tnuni.sk. ORCID: https://orcid.org/0009-0002-3041-4290
[8] Faculty of Special Technology, Alexander Dubček University of Trenčín, Ku kyselke 469, 911 06 Trenčín, Slovakia. Email: alex.jelus@tnuni.sk. ORCID: https://orcid.org/0009-0007-2717-1474